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sal : let 's see if we can calculate the definite integral from zero to one of x squared times two to the x to the third power d x . like always i encourage you to pause this video and see if you can figure this out on your own . i 'm assuming you 've had a go at it . there 's a couple of interesting things here . the first thing , at least that my brain does , it says , `` i 'm used to taking derivatives and anti-derivatives of e to the x , not some other base to the x . '' we know that the derivative with respect to x of e to the x is e to the x , or we could say that the anti-derivative of e to the x is equal to e to the x plus c. since i 'm dealing with something raised to , this particular situation , something raised to a function of x , it seems like i might want to put , i might want to change the base here , but how do i do that ? the way i would do that is re-express two in terms of e. what would be two in terms of e ? two is equal to e , is equal to e raised to the power that you need to raise e to to get to two . what 's the power that you have to raise two to to get to two ? well that 's the natural log of two . once again the natural log of two is the exponent that you have to raise e to to get to two . if you actually raise e to it you 're going to get two . this is what two is . now what is two to the x to the third ? well if we raise both sides of this to the x to the third power , we raise both sides to the x to the third power , two to the x to the third is equal to , if i raise something to an exponent and then raise that to an exponent , it 's going to be equal to e to the x to the third , x to the third , times the natural log of two , times the natural log of two . that already seems pretty interesting . let 's rewrite this , and actually what i 'm going to do , let 's just focus on the indefinite integral first , see if we can figure that out . then we can apply , then we can take , we can evaluate the definite ones . let 's just think about this , let 's think about the indefinite integral of x squared times two to the x to the third power d x. i really want to find the anti-derivative of this . well this is going to be the exact same thing as the integral of , i 'll write my x squared still , but instead of two to the x to the third i 'm going to write all of this business . let me just copy and paste that . we already established that this is the same thing as two to the x to the third power . copy and paste , just like that . then let me close it with a d x. i was able to get it in terms of e as a base . that makes me a little bit more comfortable but it still seems pretty complicated . you might be saying , `` okay , look . `` maybe u substitution could be at play here . '' because i have this crazy expression , x to the third times the natural log of two , but what 's the derivative of that ? well that 's going to be three x squared times the natural log of two , or three times the natural log of two times x squared . that 's just a constant times x squared . we already have a x squared here so maybe we can engineer this a little bit to have the constant there as well . let 's think about that . if we made this , if we defined this as u , if we said u is equal to x to the third times the natural log of two , what is du going to be ? du is going to be , it 's going to be , well natural log of two is just a constant so it 's going to be three x squared times the natural log of two . we could actually just change the order we 're multiplying a little bit . we could say that this is the same thing as x squared times three natural log of two , which is the same thing just using logarithm properties , as x squared times the natural log of two to the third power . three natural log of two is the same thing as the natural log of two to the third power . this is equal to x squared times the natural log of eight . let 's see , if this is u , where is du ? oh , and of course we ca n't forget the dx . this is a dx right over here , dx , dx , dx . where is the du ? well we have a dx . let me circle things . you have a dx here , you have a dx there . you have an x squared here , you have an x squared here . so really all we need is , all we need here is the natural log of eight . ideally we would have the natural log of eight right over here , and we could put it there as long as we also , we could multiply by the natural log of eight as long as we also divide by a natural log of eight . we can do it like right over here , we could divide by natural log of eight . but we know that the anti-derivative of some constant times a function is the same thing as a constant times the anti-derivative of that function . we could just take that on the outside . it 's one over the natural log of eight . let 's write this in terms of u and du . this simplifies to one over the natural log of eight times the anti-derivative of e to the u , e to the u , that 's the u , du . this times this times that is du , du . and this is straightforward , we know what this is going to be . this is going to be equal to , let me just write the one over natural log of eight out here , one over natural log of eight times e to the u , and of course if we 're thinking in terms of just anti-derivative there would be some constant out there . then we would just reverse the substitution . we already know what u is . this is going to be equal to , the anti-derivative of this expression is one over the natural log of eight times e to the , instead of u , we know that u is x to the third times the natural log of two . and of course we could put a plus c there . now , going back to the original problem . we just need to evaluate the anti-derivative of this at each of these points . let 's rewrite that . given what we just figured out , let me copy and paste that . this is just going to be equal to , it 's going to be equal to the anti-derivative evaluated at one minus the anti-derivative evaluated at zero . we do n't have to worry about the constants because those will cancel out . so we are going to get , we are going to get one -- let me evaluate it first at one . you 're going to get one over the natural log of eight times e to the one to the third power , which is just one , times the natural log of two , natural log of two , that 's evaluated at one . then we 're going to have minus it evaluated it at zero . it 's going to be one over the natural log of eight times e to the , well when x is zero this whole thing is going to be zero . well e to the zero is just one , and e to the natural log of two , well that 's just going to be two , we already established that early on , this is just going to be equal to two . we are left with two over the natural log of eight minus one over the natural log of eight , which is just going to be equal to one over the natural log of eight . and we are , and we are done .
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you might be saying , `` okay , look . `` maybe u substitution could be at play here . '' because i have this crazy expression , x to the third times the natural log of two , but what 's the derivative of that ?
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is there no simpler substitution that could be made ?
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sal : let 's see if we can calculate the definite integral from zero to one of x squared times two to the x to the third power d x . like always i encourage you to pause this video and see if you can figure this out on your own . i 'm assuming you 've had a go at it . there 's a couple of interesting things here . the first thing , at least that my brain does , it says , `` i 'm used to taking derivatives and anti-derivatives of e to the x , not some other base to the x . '' we know that the derivative with respect to x of e to the x is e to the x , or we could say that the anti-derivative of e to the x is equal to e to the x plus c. since i 'm dealing with something raised to , this particular situation , something raised to a function of x , it seems like i might want to put , i might want to change the base here , but how do i do that ? the way i would do that is re-express two in terms of e. what would be two in terms of e ? two is equal to e , is equal to e raised to the power that you need to raise e to to get to two . what 's the power that you have to raise two to to get to two ? well that 's the natural log of two . once again the natural log of two is the exponent that you have to raise e to to get to two . if you actually raise e to it you 're going to get two . this is what two is . now what is two to the x to the third ? well if we raise both sides of this to the x to the third power , we raise both sides to the x to the third power , two to the x to the third is equal to , if i raise something to an exponent and then raise that to an exponent , it 's going to be equal to e to the x to the third , x to the third , times the natural log of two , times the natural log of two . that already seems pretty interesting . let 's rewrite this , and actually what i 'm going to do , let 's just focus on the indefinite integral first , see if we can figure that out . then we can apply , then we can take , we can evaluate the definite ones . let 's just think about this , let 's think about the indefinite integral of x squared times two to the x to the third power d x. i really want to find the anti-derivative of this . well this is going to be the exact same thing as the integral of , i 'll write my x squared still , but instead of two to the x to the third i 'm going to write all of this business . let me just copy and paste that . we already established that this is the same thing as two to the x to the third power . copy and paste , just like that . then let me close it with a d x. i was able to get it in terms of e as a base . that makes me a little bit more comfortable but it still seems pretty complicated . you might be saying , `` okay , look . `` maybe u substitution could be at play here . '' because i have this crazy expression , x to the third times the natural log of two , but what 's the derivative of that ? well that 's going to be three x squared times the natural log of two , or three times the natural log of two times x squared . that 's just a constant times x squared . we already have a x squared here so maybe we can engineer this a little bit to have the constant there as well . let 's think about that . if we made this , if we defined this as u , if we said u is equal to x to the third times the natural log of two , what is du going to be ? du is going to be , it 's going to be , well natural log of two is just a constant so it 's going to be three x squared times the natural log of two . we could actually just change the order we 're multiplying a little bit . we could say that this is the same thing as x squared times three natural log of two , which is the same thing just using logarithm properties , as x squared times the natural log of two to the third power . three natural log of two is the same thing as the natural log of two to the third power . this is equal to x squared times the natural log of eight . let 's see , if this is u , where is du ? oh , and of course we ca n't forget the dx . this is a dx right over here , dx , dx , dx . where is the du ? well we have a dx . let me circle things . you have a dx here , you have a dx there . you have an x squared here , you have an x squared here . so really all we need is , all we need here is the natural log of eight . ideally we would have the natural log of eight right over here , and we could put it there as long as we also , we could multiply by the natural log of eight as long as we also divide by a natural log of eight . we can do it like right over here , we could divide by natural log of eight . but we know that the anti-derivative of some constant times a function is the same thing as a constant times the anti-derivative of that function . we could just take that on the outside . it 's one over the natural log of eight . let 's write this in terms of u and du . this simplifies to one over the natural log of eight times the anti-derivative of e to the u , e to the u , that 's the u , du . this times this times that is du , du . and this is straightforward , we know what this is going to be . this is going to be equal to , let me just write the one over natural log of eight out here , one over natural log of eight times e to the u , and of course if we 're thinking in terms of just anti-derivative there would be some constant out there . then we would just reverse the substitution . we already know what u is . this is going to be equal to , the anti-derivative of this expression is one over the natural log of eight times e to the , instead of u , we know that u is x to the third times the natural log of two . and of course we could put a plus c there . now , going back to the original problem . we just need to evaluate the anti-derivative of this at each of these points . let 's rewrite that . given what we just figured out , let me copy and paste that . this is just going to be equal to , it 's going to be equal to the anti-derivative evaluated at one minus the anti-derivative evaluated at zero . we do n't have to worry about the constants because those will cancel out . so we are going to get , we are going to get one -- let me evaluate it first at one . you 're going to get one over the natural log of eight times e to the one to the third power , which is just one , times the natural log of two , natural log of two , that 's evaluated at one . then we 're going to have minus it evaluated it at zero . it 's going to be one over the natural log of eight times e to the , well when x is zero this whole thing is going to be zero . well e to the zero is just one , and e to the natural log of two , well that 's just going to be two , we already established that early on , this is just going to be equal to two . we are left with two over the natural log of eight minus one over the natural log of eight , which is just going to be equal to one over the natural log of eight . and we are , and we are done .
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you might be saying , `` okay , look . `` maybe u substitution could be at play here . '' because i have this crazy expression , x to the third times the natural log of two , but what 's the derivative of that ?
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is there no simpler substitution that could be made ?
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sal : let 's see if we can calculate the definite integral from zero to one of x squared times two to the x to the third power d x . like always i encourage you to pause this video and see if you can figure this out on your own . i 'm assuming you 've had a go at it . there 's a couple of interesting things here . the first thing , at least that my brain does , it says , `` i 'm used to taking derivatives and anti-derivatives of e to the x , not some other base to the x . '' we know that the derivative with respect to x of e to the x is e to the x , or we could say that the anti-derivative of e to the x is equal to e to the x plus c. since i 'm dealing with something raised to , this particular situation , something raised to a function of x , it seems like i might want to put , i might want to change the base here , but how do i do that ? the way i would do that is re-express two in terms of e. what would be two in terms of e ? two is equal to e , is equal to e raised to the power that you need to raise e to to get to two . what 's the power that you have to raise two to to get to two ? well that 's the natural log of two . once again the natural log of two is the exponent that you have to raise e to to get to two . if you actually raise e to it you 're going to get two . this is what two is . now what is two to the x to the third ? well if we raise both sides of this to the x to the third power , we raise both sides to the x to the third power , two to the x to the third is equal to , if i raise something to an exponent and then raise that to an exponent , it 's going to be equal to e to the x to the third , x to the third , times the natural log of two , times the natural log of two . that already seems pretty interesting . let 's rewrite this , and actually what i 'm going to do , let 's just focus on the indefinite integral first , see if we can figure that out . then we can apply , then we can take , we can evaluate the definite ones . let 's just think about this , let 's think about the indefinite integral of x squared times two to the x to the third power d x. i really want to find the anti-derivative of this . well this is going to be the exact same thing as the integral of , i 'll write my x squared still , but instead of two to the x to the third i 'm going to write all of this business . let me just copy and paste that . we already established that this is the same thing as two to the x to the third power . copy and paste , just like that . then let me close it with a d x. i was able to get it in terms of e as a base . that makes me a little bit more comfortable but it still seems pretty complicated . you might be saying , `` okay , look . `` maybe u substitution could be at play here . '' because i have this crazy expression , x to the third times the natural log of two , but what 's the derivative of that ? well that 's going to be three x squared times the natural log of two , or three times the natural log of two times x squared . that 's just a constant times x squared . we already have a x squared here so maybe we can engineer this a little bit to have the constant there as well . let 's think about that . if we made this , if we defined this as u , if we said u is equal to x to the third times the natural log of two , what is du going to be ? du is going to be , it 's going to be , well natural log of two is just a constant so it 's going to be three x squared times the natural log of two . we could actually just change the order we 're multiplying a little bit . we could say that this is the same thing as x squared times three natural log of two , which is the same thing just using logarithm properties , as x squared times the natural log of two to the third power . three natural log of two is the same thing as the natural log of two to the third power . this is equal to x squared times the natural log of eight . let 's see , if this is u , where is du ? oh , and of course we ca n't forget the dx . this is a dx right over here , dx , dx , dx . where is the du ? well we have a dx . let me circle things . you have a dx here , you have a dx there . you have an x squared here , you have an x squared here . so really all we need is , all we need here is the natural log of eight . ideally we would have the natural log of eight right over here , and we could put it there as long as we also , we could multiply by the natural log of eight as long as we also divide by a natural log of eight . we can do it like right over here , we could divide by natural log of eight . but we know that the anti-derivative of some constant times a function is the same thing as a constant times the anti-derivative of that function . we could just take that on the outside . it 's one over the natural log of eight . let 's write this in terms of u and du . this simplifies to one over the natural log of eight times the anti-derivative of e to the u , e to the u , that 's the u , du . this times this times that is du , du . and this is straightforward , we know what this is going to be . this is going to be equal to , let me just write the one over natural log of eight out here , one over natural log of eight times e to the u , and of course if we 're thinking in terms of just anti-derivative there would be some constant out there . then we would just reverse the substitution . we already know what u is . this is going to be equal to , the anti-derivative of this expression is one over the natural log of eight times e to the , instead of u , we know that u is x to the third times the natural log of two . and of course we could put a plus c there . now , going back to the original problem . we just need to evaluate the anti-derivative of this at each of these points . let 's rewrite that . given what we just figured out , let me copy and paste that . this is just going to be equal to , it 's going to be equal to the anti-derivative evaluated at one minus the anti-derivative evaluated at zero . we do n't have to worry about the constants because those will cancel out . so we are going to get , we are going to get one -- let me evaluate it first at one . you 're going to get one over the natural log of eight times e to the one to the third power , which is just one , times the natural log of two , natural log of two , that 's evaluated at one . then we 're going to have minus it evaluated it at zero . it 's going to be one over the natural log of eight times e to the , well when x is zero this whole thing is going to be zero . well e to the zero is just one , and e to the natural log of two , well that 's just going to be two , we already established that early on , this is just going to be equal to two . we are left with two over the natural log of eight minus one over the natural log of eight , which is just going to be equal to one over the natural log of eight . and we are , and we are done .
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you might be saying , `` okay , look . `` maybe u substitution could be at play here . '' because i have this crazy expression , x to the third times the natural log of two , but what 's the derivative of that ?
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could sal also have made u=x^3 and then do 2^u du ?
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sal : let 's see if we can calculate the definite integral from zero to one of x squared times two to the x to the third power d x . like always i encourage you to pause this video and see if you can figure this out on your own . i 'm assuming you 've had a go at it . there 's a couple of interesting things here . the first thing , at least that my brain does , it says , `` i 'm used to taking derivatives and anti-derivatives of e to the x , not some other base to the x . '' we know that the derivative with respect to x of e to the x is e to the x , or we could say that the anti-derivative of e to the x is equal to e to the x plus c. since i 'm dealing with something raised to , this particular situation , something raised to a function of x , it seems like i might want to put , i might want to change the base here , but how do i do that ? the way i would do that is re-express two in terms of e. what would be two in terms of e ? two is equal to e , is equal to e raised to the power that you need to raise e to to get to two . what 's the power that you have to raise two to to get to two ? well that 's the natural log of two . once again the natural log of two is the exponent that you have to raise e to to get to two . if you actually raise e to it you 're going to get two . this is what two is . now what is two to the x to the third ? well if we raise both sides of this to the x to the third power , we raise both sides to the x to the third power , two to the x to the third is equal to , if i raise something to an exponent and then raise that to an exponent , it 's going to be equal to e to the x to the third , x to the third , times the natural log of two , times the natural log of two . that already seems pretty interesting . let 's rewrite this , and actually what i 'm going to do , let 's just focus on the indefinite integral first , see if we can figure that out . then we can apply , then we can take , we can evaluate the definite ones . let 's just think about this , let 's think about the indefinite integral of x squared times two to the x to the third power d x. i really want to find the anti-derivative of this . well this is going to be the exact same thing as the integral of , i 'll write my x squared still , but instead of two to the x to the third i 'm going to write all of this business . let me just copy and paste that . we already established that this is the same thing as two to the x to the third power . copy and paste , just like that . then let me close it with a d x. i was able to get it in terms of e as a base . that makes me a little bit more comfortable but it still seems pretty complicated . you might be saying , `` okay , look . `` maybe u substitution could be at play here . '' because i have this crazy expression , x to the third times the natural log of two , but what 's the derivative of that ? well that 's going to be three x squared times the natural log of two , or three times the natural log of two times x squared . that 's just a constant times x squared . we already have a x squared here so maybe we can engineer this a little bit to have the constant there as well . let 's think about that . if we made this , if we defined this as u , if we said u is equal to x to the third times the natural log of two , what is du going to be ? du is going to be , it 's going to be , well natural log of two is just a constant so it 's going to be three x squared times the natural log of two . we could actually just change the order we 're multiplying a little bit . we could say that this is the same thing as x squared times three natural log of two , which is the same thing just using logarithm properties , as x squared times the natural log of two to the third power . three natural log of two is the same thing as the natural log of two to the third power . this is equal to x squared times the natural log of eight . let 's see , if this is u , where is du ? oh , and of course we ca n't forget the dx . this is a dx right over here , dx , dx , dx . where is the du ? well we have a dx . let me circle things . you have a dx here , you have a dx there . you have an x squared here , you have an x squared here . so really all we need is , all we need here is the natural log of eight . ideally we would have the natural log of eight right over here , and we could put it there as long as we also , we could multiply by the natural log of eight as long as we also divide by a natural log of eight . we can do it like right over here , we could divide by natural log of eight . but we know that the anti-derivative of some constant times a function is the same thing as a constant times the anti-derivative of that function . we could just take that on the outside . it 's one over the natural log of eight . let 's write this in terms of u and du . this simplifies to one over the natural log of eight times the anti-derivative of e to the u , e to the u , that 's the u , du . this times this times that is du , du . and this is straightforward , we know what this is going to be . this is going to be equal to , let me just write the one over natural log of eight out here , one over natural log of eight times e to the u , and of course if we 're thinking in terms of just anti-derivative there would be some constant out there . then we would just reverse the substitution . we already know what u is . this is going to be equal to , the anti-derivative of this expression is one over the natural log of eight times e to the , instead of u , we know that u is x to the third times the natural log of two . and of course we could put a plus c there . now , going back to the original problem . we just need to evaluate the anti-derivative of this at each of these points . let 's rewrite that . given what we just figured out , let me copy and paste that . this is just going to be equal to , it 's going to be equal to the anti-derivative evaluated at one minus the anti-derivative evaluated at zero . we do n't have to worry about the constants because those will cancel out . so we are going to get , we are going to get one -- let me evaluate it first at one . you 're going to get one over the natural log of eight times e to the one to the third power , which is just one , times the natural log of two , natural log of two , that 's evaluated at one . then we 're going to have minus it evaluated it at zero . it 's going to be one over the natural log of eight times e to the , well when x is zero this whole thing is going to be zero . well e to the zero is just one , and e to the natural log of two , well that 's just going to be two , we already established that early on , this is just going to be equal to two . we are left with two over the natural log of eight minus one over the natural log of eight , which is just going to be equal to one over the natural log of eight . and we are , and we are done .
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well that 's going to be three x squared times the natural log of two , or three times the natural log of two times x squared . that 's just a constant times x squared . we already have a x squared here so maybe we can engineer this a little bit to have the constant there as well .
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at the very very end why do n't we have to add a possible constant ( + c ) ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize .
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what is the probability that you picked the right door out of 100 doors ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation .
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- what are your chances of winning if you never switch ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out .
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- what are your chances of winning if you always switch ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out .
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are we required to say `` i will always switch '' or `` not switch '' even before choosing the first time ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch .
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would the probability still concentrate ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one .
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how does the theory work if there is only one goat ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains .
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how did monty hall become so great and what if your choice is the right prize and you switch ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ?
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did sal mean sticking to your gut instead of guns ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation .
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what you had was the first item chosen has a 1/3 chance of winning , but the second item has a 1/2 chance , so 1/2 is bigger than 1/3 , so switching is better , right ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say .
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what does p ( w ) mean ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there .
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1- what are my odds of picking the right door ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch .
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what would the probabilities be in the following similar scenario ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch .
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how can this square with the idea that switching curtains improves the odds of winning ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ?
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are you saying the act of revealing your choice is what gives switching better odds ?
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let 's now tackle a classic thought experiment in probability , called the monte hall problem . and it 's called the monty hall problem because monty hall was the game show host in let 's make a deal , where they would set up a situation very similar to the monte hall problem that we 're about to say . so let 's say that on the show , you 're presented with three curtains . so you 're the contestant , this little chef-looking character right over there . you 're presented with three curtains -- curtain number one , curtain number two , and curtain number three . and you 're told that behind one of these three curtains , there 's a fabulous prize , something that you really want , a car , or a vacation , or some type of large amount of cash . and then behind the other two , and we do n't know which they are , there is something that you do not want . a new pet goat or an ostrich or something like that , or a beach ball , something that is not as good as the cash prize . and so your goal is to try to find the cash prize . and they say guess which one , or which one would you like to select ? and so let 's say that you select door number one , or curtain number one . then the monte hall and let 's make a deal crew will make it a little bit more interesting . they want to show you whether or not you won . they 'll then show you one of the other two doors , or one of the other two curtains . and they 'll show you one of the other two curtains that does not have the prize . and no matter which one you pick , there 'll always be at least one other curtain that does not have the prize . there might be two , if you picked right . but there will always be at least one other curtain that does not have the prize . and then they will show it to you . and so let 's say that they show you curtain number three . and so curtain number three has the goat . and then they 'll ask you , do you want to switch to curtain number two ? and the question here is , does it make a difference ? are you better off holding fast , sticking to your guns , staying with the original guess ? are you better off switching to whatever curtain is left ? or does it not matter ? it 's just random probability , and it 's not going to make a difference whether you switch or not . so that is the brain teaser . pause the video now . i encourage you to think about it . in the next video , we will start to analyze the solution a little bit deeper , whether it makes any difference at all . so now i 've assumed that you 've unpaused it . you 've thought deeply about it . perhaps you have an opinion on it . now let 's work it through a little bit . and at any point , i encourage you to pause it and kind of extrapolate beyond what i 've already talked about . so let 's think about the game show from the show 's point of view . so the show knows where there 's the goat and where there is n't the goat . so let 's door number one , door number two , and door number three . so let 's say that our prize is right over here . so our prize is the car . our prize is the car , and that we have a goat over here , and over here we also have -- maybe we have two goats in this situation . so what are we going to do as the game show ? remember , the contestants do n't know this . we know this . so the contestant picks door number one right over here . then we ca n't lift door number two because there 's a car back there . we 're going to lift door number three , and we 're going to expose this goat . in which case , it probably would be good for the person to switch . if the person picks door number two , then we as the game show can show either door number one or door number three , and then it actually does not make sense for the person to switch . if they picked door number three , then we have to show door number one because we ca n't pick door number two . and in that case , it actually makes a lot of sense for the person to switch . now , with that out of the way , let 's think about the probabilities given the two strategies . so if you do n't switch , or another way to think about this strategy is you always stick to your guns . you always stick to your first guess . well , in that situation , what is your probability of winning ? well , there 's three doors . the prize is equally likely to be behind any one of them . so there 's three possibilities . one has the outcome that you desire . the probability of winning will be 1/3 if you do n't switch . likewise , your probability of losing , well , there 's two ways that you can lose out of three possibilities . it 's going to be 2/3 . and these are the only possibilities , and these add up to one right over here . so do n't switch , 1/3 chance of winning . now let 's think about the switching situation . so let 's say always -- when you always switch . let 's think about how this might play out . what is your probability of winning ? and before we even think about that , think about how you would win if you always switch . so if you pick wrong the first time , they 're going to show you this . and so you should always switch . so if you pick door number one , they 're going to show you door number three . you should switch . if you picked wrong door number three , they 're going to show you door number one . you should switch . so if you picked wrong and switch , you will always win . let me write this down . and this insight actually came from one of the middle school students in the summer camp that khan academy was running . it 's actually a fabulous way to think about this . so if you pick wrong , so step one , so initial pick is wrong , so you pick one of the two wrong doors , and then step two , you always switch , you will land on the car . because if you picked one of the wrong doors , they 're going to have to show the other wrong door . and so if you switch , you 're going to end up on the right answer . so what is the probability of winning if you always switch ? well , it 's going to be the probability that you initially picked wrong . well , what 's the probability that you initially picked wrong ? well , there 's two out of the three ways to initially pick wrong . so you actually have a 2/3 chance of winning . there 's a 2/3 chance you 're going to pick wrong and then switch into the right one . likewise , what 's your probability of losing , given that you 're always going to switch ? well , the way that you would lose is you pick right , you picked correctly . in step two , they 're going to show one of the two empty or non-prize doors . and then step three , you 're going to switch into the other empty door . but either way , you 're definitely going to switch . so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 . so you see here , it 's sometimes counter-intuitive , but hopefully this makes sense as to why it is n't . you have a 1/3 chance of winning if you stick to your guns , and a 2/3 chance of winning if you always switch . another way to think about it is , when you first make your initial pick , there 's a 1/3 chance that it 's there , and there 's a 2/3 chance that it 's in one of the other two doors . and they 're going to empty out one of them . so when you switch , you essentially are capturing that 2/3 probability . and we see that right there . so hopefully you enjoyed that .
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so the only way to lose , if you 're always going to switch , is to pick the right the first time . well , what 's the probability of you picking right the first time ? well , that is only 1/3 .
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what is the probability that the marks of the first four persons taken from left are in descending order and the first two persons taken from right are in descending order ?
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better . and this one and this one right over here . now what fraction of the whole have we shaded in ? well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ? well we have one , two , three , four shaded in . so now we have a total of 4/9 shaded it . 4 of the 9 equal sections are shaded in . so 4/9 of the whole is shaded in . now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections . let me write this down . i have five equal sections . and let me shade in five of them . so one , two , three , four , five . we already know that each of these sections , each of these situated in sections represent 1/5 . so 1/5 , another way of saying that is a fifth , is 1/5 . but now how much do i have shaded in ? well i have five out of the five equal sections shaded , or i have 5/5 shaded in . and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole . now what i want you to do is pause this video and write down on a piece of paper or at least think in your head , what fraction of each of these wholes is shaded in ? so let 's go to this first one . we have one , two , three , four , five , six equal sections . and we see that one , two , three , four are shaded in . so 4/6 of this figure is shaded in . let 's go over here . we have one , two , three , four , five equal sections . and one , two , three , four are shaded in . so here , 4/5 of this circle is shaded in . now in this figure , i have two equal sections and both of them are shaded in . and this we would say two halves of this figure are shaded in . and once again , if two halves are shaded in , that means everything is shaded in , that this represents a whole . now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger . it actually looks like it 's bigger than the other three combined . so you do not have four equal sections here . so at least based on how it 's drawn , you ca n't say that 3/4 is actually filled in .
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well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth .
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1/1 is equal to 1 right ?
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better . and this one and this one right over here . now what fraction of the whole have we shaded in ? well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ? well we have one , two , three , four shaded in . so now we have a total of 4/9 shaded it . 4 of the 9 equal sections are shaded in . so 4/9 of the whole is shaded in . now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections . let me write this down . i have five equal sections . and let me shade in five of them . so one , two , three , four , five . we already know that each of these sections , each of these situated in sections represent 1/5 . so 1/5 , another way of saying that is a fifth , is 1/5 . but now how much do i have shaded in ? well i have five out of the five equal sections shaded , or i have 5/5 shaded in . and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole . now what i want you to do is pause this video and write down on a piece of paper or at least think in your head , what fraction of each of these wholes is shaded in ? so let 's go to this first one . we have one , two , three , four , five , six equal sections . and we see that one , two , three , four are shaded in . so 4/6 of this figure is shaded in . let 's go over here . we have one , two , three , four , five equal sections . and one , two , three , four are shaded in . so here , 4/5 of this circle is shaded in . now in this figure , i have two equal sections and both of them are shaded in . and this we would say two halves of this figure are shaded in . and once again , if two halves are shaded in , that means everything is shaded in , that this represents a whole . now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger . it actually looks like it 's bigger than the other three combined . so you do not have four equal sections here . so at least based on how it 's drawn , you ca n't say that 3/4 is actually filled in .
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well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth .
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how do you do for exp : find 1/100 of 15 ?
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better . and this one and this one right over here . now what fraction of the whole have we shaded in ? well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ? well we have one , two , three , four shaded in . so now we have a total of 4/9 shaded it . 4 of the 9 equal sections are shaded in . so 4/9 of the whole is shaded in . now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections . let me write this down . i have five equal sections . and let me shade in five of them . so one , two , three , four , five . we already know that each of these sections , each of these situated in sections represent 1/5 . so 1/5 , another way of saying that is a fifth , is 1/5 . but now how much do i have shaded in ? well i have five out of the five equal sections shaded , or i have 5/5 shaded in . and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole . now what i want you to do is pause this video and write down on a piece of paper or at least think in your head , what fraction of each of these wholes is shaded in ? so let 's go to this first one . we have one , two , three , four , five , six equal sections . and we see that one , two , three , four are shaded in . so 4/6 of this figure is shaded in . let 's go over here . we have one , two , three , four , five equal sections . and one , two , three , four are shaded in . so here , 4/5 of this circle is shaded in . now in this figure , i have two equal sections and both of them are shaded in . and this we would say two halves of this figure are shaded in . and once again , if two halves are shaded in , that means everything is shaded in , that this represents a whole . now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger . it actually looks like it 's bigger than the other three combined . so you do not have four equal sections here . so at least based on how it 's drawn , you ca n't say that 3/4 is actually filled in .
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well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better .
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if you have 15 squares and you shaded in 13.007 would you have 13 7/100 shaded in , right ?
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better . and this one and this one right over here . now what fraction of the whole have we shaded in ? well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ? well we have one , two , three , four shaded in . so now we have a total of 4/9 shaded it . 4 of the 9 equal sections are shaded in . so 4/9 of the whole is shaded in . now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections . let me write this down . i have five equal sections . and let me shade in five of them . so one , two , three , four , five . we already know that each of these sections , each of these situated in sections represent 1/5 . so 1/5 , another way of saying that is a fifth , is 1/5 . but now how much do i have shaded in ? well i have five out of the five equal sections shaded , or i have 5/5 shaded in . and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole . now what i want you to do is pause this video and write down on a piece of paper or at least think in your head , what fraction of each of these wholes is shaded in ? so let 's go to this first one . we have one , two , three , four , five , six equal sections . and we see that one , two , three , four are shaded in . so 4/6 of this figure is shaded in . let 's go over here . we have one , two , three , four , five equal sections . and one , two , three , four are shaded in . so here , 4/5 of this circle is shaded in . now in this figure , i have two equal sections and both of them are shaded in . and this we would say two halves of this figure are shaded in . and once again , if two halves are shaded in , that means everything is shaded in , that this represents a whole . now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger . it actually looks like it 's bigger than the other three combined . so you do not have four equal sections here . so at least based on how it 's drawn , you ca n't say that 3/4 is actually filled in .
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections .
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how did the pictures suddenly change and why ?
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better . and this one and this one right over here . now what fraction of the whole have we shaded in ? well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ? well we have one , two , three , four shaded in . so now we have a total of 4/9 shaded it . 4 of the 9 equal sections are shaded in . so 4/9 of the whole is shaded in . now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections . let me write this down . i have five equal sections . and let me shade in five of them . so one , two , three , four , five . we already know that each of these sections , each of these situated in sections represent 1/5 . so 1/5 , another way of saying that is a fifth , is 1/5 . but now how much do i have shaded in ? well i have five out of the five equal sections shaded , or i have 5/5 shaded in . and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole . now what i want you to do is pause this video and write down on a piece of paper or at least think in your head , what fraction of each of these wholes is shaded in ? so let 's go to this first one . we have one , two , three , four , five , six equal sections . and we see that one , two , three , four are shaded in . so 4/6 of this figure is shaded in . let 's go over here . we have one , two , three , four , five equal sections . and one , two , three , four are shaded in . so here , 4/5 of this circle is shaded in . now in this figure , i have two equal sections and both of them are shaded in . and this we would say two halves of this figure are shaded in . and once again , if two halves are shaded in , that means everything is shaded in , that this represents a whole . now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger . it actually looks like it 's bigger than the other three combined . so you do not have four equal sections here . so at least based on how it 's drawn , you ca n't say that 3/4 is actually filled in .
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections .
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what is a mixed number ?
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better . and this one and this one right over here . now what fraction of the whole have we shaded in ? well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ? well we have one , two , three , four shaded in . so now we have a total of 4/9 shaded it . 4 of the 9 equal sections are shaded in . so 4/9 of the whole is shaded in . now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections . let me write this down . i have five equal sections . and let me shade in five of them . so one , two , three , four , five . we already know that each of these sections , each of these situated in sections represent 1/5 . so 1/5 , another way of saying that is a fifth , is 1/5 . but now how much do i have shaded in ? well i have five out of the five equal sections shaded , or i have 5/5 shaded in . and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole . now what i want you to do is pause this video and write down on a piece of paper or at least think in your head , what fraction of each of these wholes is shaded in ? so let 's go to this first one . we have one , two , three , four , five , six equal sections . and we see that one , two , three , four are shaded in . so 4/6 of this figure is shaded in . let 's go over here . we have one , two , three , four , five equal sections . and one , two , three , four are shaded in . so here , 4/5 of this circle is shaded in . now in this figure , i have two equal sections and both of them are shaded in . and this we would say two halves of this figure are shaded in . and once again , if two halves are shaded in , that means everything is shaded in , that this represents a whole . now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger . it actually looks like it 's bigger than the other three combined . so you do not have four equal sections here . so at least based on how it 's drawn , you ca n't say that 3/4 is actually filled in .
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and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole .
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would 6/8 be an equivalent fraction to 12/16 ?
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better . and this one and this one right over here . now what fraction of the whole have we shaded in ? well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ? well we have one , two , three , four shaded in . so now we have a total of 4/9 shaded it . 4 of the 9 equal sections are shaded in . so 4/9 of the whole is shaded in . now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections . let me write this down . i have five equal sections . and let me shade in five of them . so one , two , three , four , five . we already know that each of these sections , each of these situated in sections represent 1/5 . so 1/5 , another way of saying that is a fifth , is 1/5 . but now how much do i have shaded in ? well i have five out of the five equal sections shaded , or i have 5/5 shaded in . and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole . now what i want you to do is pause this video and write down on a piece of paper or at least think in your head , what fraction of each of these wholes is shaded in ? so let 's go to this first one . we have one , two , three , four , five , six equal sections . and we see that one , two , three , four are shaded in . so 4/6 of this figure is shaded in . let 's go over here . we have one , two , three , four , five equal sections . and one , two , three , four are shaded in . so here , 4/5 of this circle is shaded in . now in this figure , i have two equal sections and both of them are shaded in . and this we would say two halves of this figure are shaded in . and once again , if two halves are shaded in , that means everything is shaded in , that this represents a whole . now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger . it actually looks like it 's bigger than the other three combined . so you do not have four equal sections here . so at least based on how it 's drawn , you ca n't say that 3/4 is actually filled in .
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now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections .
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does the part of the whole you shade first have to be in the middle ?
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better . and this one and this one right over here . now what fraction of the whole have we shaded in ? well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ? well we have one , two , three , four shaded in . so now we have a total of 4/9 shaded it . 4 of the 9 equal sections are shaded in . so 4/9 of the whole is shaded in . now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections . let me write this down . i have five equal sections . and let me shade in five of them . so one , two , three , four , five . we already know that each of these sections , each of these situated in sections represent 1/5 . so 1/5 , another way of saying that is a fifth , is 1/5 . but now how much do i have shaded in ? well i have five out of the five equal sections shaded , or i have 5/5 shaded in . and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole . now what i want you to do is pause this video and write down on a piece of paper or at least think in your head , what fraction of each of these wholes is shaded in ? so let 's go to this first one . we have one , two , three , four , five , six equal sections . and we see that one , two , three , four are shaded in . so 4/6 of this figure is shaded in . let 's go over here . we have one , two , three , four , five equal sections . and one , two , three , four are shaded in . so here , 4/5 of this circle is shaded in . now in this figure , i have two equal sections and both of them are shaded in . and this we would say two halves of this figure are shaded in . and once again , if two halves are shaded in , that means everything is shaded in , that this represents a whole . now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger . it actually looks like it 's bigger than the other three combined . so you do not have four equal sections here . so at least based on how it 's drawn , you ca n't say that 3/4 is actually filled in .
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well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ?
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how did sal call 1/9 , ninth ?
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better . and this one and this one right over here . now what fraction of the whole have we shaded in ? well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ? well we have one , two , three , four shaded in . so now we have a total of 4/9 shaded it . 4 of the 9 equal sections are shaded in . so 4/9 of the whole is shaded in . now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections . let me write this down . i have five equal sections . and let me shade in five of them . so one , two , three , four , five . we already know that each of these sections , each of these situated in sections represent 1/5 . so 1/5 , another way of saying that is a fifth , is 1/5 . but now how much do i have shaded in ? well i have five out of the five equal sections shaded , or i have 5/5 shaded in . and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole . now what i want you to do is pause this video and write down on a piece of paper or at least think in your head , what fraction of each of these wholes is shaded in ? so let 's go to this first one . we have one , two , three , four , five , six equal sections . and we see that one , two , three , four are shaded in . so 4/6 of this figure is shaded in . let 's go over here . we have one , two , three , four , five equal sections . and one , two , three , four are shaded in . so here , 4/5 of this circle is shaded in . now in this figure , i have two equal sections and both of them are shaded in . and this we would say two halves of this figure are shaded in . and once again , if two halves are shaded in , that means everything is shaded in , that this represents a whole . now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger . it actually looks like it 's bigger than the other three combined . so you do not have four equal sections here . so at least based on how it 's drawn , you ca n't say that 3/4 is actually filled in .
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and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole .
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what is the square root of `` pi '' ?
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better . and this one and this one right over here . now what fraction of the whole have we shaded in ? well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ? well we have one , two , three , four shaded in . so now we have a total of 4/9 shaded it . 4 of the 9 equal sections are shaded in . so 4/9 of the whole is shaded in . now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections . let me write this down . i have five equal sections . and let me shade in five of them . so one , two , three , four , five . we already know that each of these sections , each of these situated in sections represent 1/5 . so 1/5 , another way of saying that is a fifth , is 1/5 . but now how much do i have shaded in ? well i have five out of the five equal sections shaded , or i have 5/5 shaded in . and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole . now what i want you to do is pause this video and write down on a piece of paper or at least think in your head , what fraction of each of these wholes is shaded in ? so let 's go to this first one . we have one , two , three , four , five , six equal sections . and we see that one , two , three , four are shaded in . so 4/6 of this figure is shaded in . let 's go over here . we have one , two , three , four , five equal sections . and one , two , three , four are shaded in . so here , 4/5 of this circle is shaded in . now in this figure , i have two equal sections and both of them are shaded in . and this we would say two halves of this figure are shaded in . and once again , if two halves are shaded in , that means everything is shaded in , that this represents a whole . now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger . it actually looks like it 's bigger than the other three combined . so you do not have four equal sections here . so at least based on how it 's drawn , you ca n't say that 3/4 is actually filled in .
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well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth .
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what is the decimal number of 1/3 ?
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i have a square here divided into one , two , three , four , five , six , seven , eight , nine equal sections . and we 've already seen that if we were to shade in one of these sections , if we were to select one of these sections , let 's say the middle one right over here , this is one out of the nine equal sections . so if someone said , what fraction of the whole does this purple square represent ? well , you would say , well , that represents 1/9 of the whole . this thing right over here represents 1/9 . now what would happen if we shaded in more than that ? so let 's say we shaded in this one and this one , let me shade it in a little bit better . and this one and this one right over here . now what fraction of the whole have we shaded in ? well , each of these , we 've already seen , each of these represent 1/9 . so that 's 1/9 , that 's 1/9 . when i say 1/9 , i could also say a ninth . so this is 1/9 or a ninth , so each of these represents a ninth . but how many of these ninths do we have shaded in ? well we have one , two , three , four shaded in . so now we have a total of 4/9 shaded it . 4 of the 9 equal sections are shaded in . so 4/9 of the whole is shaded in . now let 's make things a little bit more interesting . let 's shade in . so here i have five equal sections . let me write this down . i have five equal sections . and let me shade in five of them . so one , two , three , four , five . we already know that each of these sections , each of these situated in sections represent 1/5 . so 1/5 , another way of saying that is a fifth , is 1/5 . but now how much do i have shaded in ? well i have five out of the five equal sections shaded , or i have 5/5 shaded in . and you might be saying , wait , wait , if i gave five out of the five equal sections shaded in , if i have 5/5 shaded in , i 've got the whole thing shaded in . and you would be absolutely right . 5/5 is equal to the whole . now what i want you to do is pause this video and write down on a piece of paper or at least think in your head , what fraction of each of these wholes is shaded in ? so let 's go to this first one . we have one , two , three , four , five , six equal sections . and we see that one , two , three , four are shaded in . so 4/6 of this figure is shaded in . let 's go over here . we have one , two , three , four , five equal sections . and one , two , three , four are shaded in . so here , 4/5 of this circle is shaded in . now in this figure , i have two equal sections and both of them are shaded in . and this we would say two halves of this figure are shaded in . and once again , if two halves are shaded in , that means everything is shaded in , that this represents a whole . now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger . it actually looks like it 's bigger than the other three combined . so you do not have four equal sections here . so at least based on how it 's drawn , you ca n't say that 3/4 is actually filled in .
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now this one right over here it might be tempting to say , i have one , two , three , four sections and one , two , three have been shaded in so maybe the red represents 3/4 of the figure . but remember , the sections have to be equal sections . and this red section is way bigger .
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is there a way to make a fraction out of uneven sections ?
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in order to solve problems and make decisions , you often have to access information already stored in your brain . but how is that information stored ? there are a lot of possibilities , but the one we 're going to focus on in this video is the semantic network approach , which argues that concepts are organized in your mind in terms of connected ideas . you can kind of think of this as parallel to how information might be stored in a computer . you have different nodes , which represent your concepts , and those nodes are connected by links . depending on how connected the nodes are , the links might be shorter , for closely related ideas , or longer , for less related ideas . let me show you an example to make this a little more concrete . the first semantic network model was hierarchical , meaning that they thought concepts were organized from higher order categories down to lower order categories and their exemplars . so let 's start with a general category : animal . so that 's this node here . animal might be linked to other nodes , such as bird or fish . and bird might be further linked to canary , bluebird , or more distantly , ostrich . ostrich is probably not the first thing you think of when you think of a bird , so that 's why its link is longer . but simple labels are n't the only type of knowledge we store . we can store characteristics and properties of concepts at each node . according to the principle of cognitive economy , which just means that our brain is efficient , we store these properties at the highest possible node . for example then , instead of storing can breathe at each animal 's node , we store that property just at the animal node . more specific characteristics such as sings , or long legs , would be stored at lower level nodes . one pretty interesting piece of supporting evidence for this type of hierarchical organization comes from how long it takes people to verify certain statements . in this kind of test , you say a statement , and ask people to tell you if it 's true or not . as you can imagine , people verify a canary is a canary pretty quickly . it takes them a little longer to verify a canary is a bird , and even longer to verify that a canary is an animal . so this is some support that we store things in a hierarchical manner , because the longer it takes us to verify a connection between two nodes , then the longer those links are , or the more nodes we have to go through to make that link . however , this is n't true for all types of animals , or even all types of categories . for example , people tend to verify that a pig is an animal faster than a pig is a mammal . because of this issue and some other problems , collins and loftus proposed a modified version of the semantic network . rather than a hierarchical organization , this model says that every individual semantic network develops based on their experience and knowledge . so , some links might be longer or shorter for different individuals , and there may be direct links from higher order categories to their exemplars . one pretty cool thing about semantic networks is that it means all the ideas in your head are connected together . so when you activate one concept , you 're pulling up related concepts along with it . this general elevation and availability is called spreading activation . for example , if i say `` fire engine , '' not only do you think of a fire engine , but related concepts such as trucks , fire , even the color red become activated , making it easier for you to retrieve or identify those items .
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for example , people tend to verify that a pig is an animal faster than a pig is a mammal . because of this issue and some other problems , collins and loftus proposed a modified version of the semantic network . rather than a hierarchical organization , this model says that every individual semantic network develops based on their experience and knowledge . so , some links might be longer or shorter for different individuals , and there may be direct links from higher order categories to their exemplars .
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is a semantic network still a strong concept in current psychology ?
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in order to solve problems and make decisions , you often have to access information already stored in your brain . but how is that information stored ? there are a lot of possibilities , but the one we 're going to focus on in this video is the semantic network approach , which argues that concepts are organized in your mind in terms of connected ideas . you can kind of think of this as parallel to how information might be stored in a computer . you have different nodes , which represent your concepts , and those nodes are connected by links . depending on how connected the nodes are , the links might be shorter , for closely related ideas , or longer , for less related ideas . let me show you an example to make this a little more concrete . the first semantic network model was hierarchical , meaning that they thought concepts were organized from higher order categories down to lower order categories and their exemplars . so let 's start with a general category : animal . so that 's this node here . animal might be linked to other nodes , such as bird or fish . and bird might be further linked to canary , bluebird , or more distantly , ostrich . ostrich is probably not the first thing you think of when you think of a bird , so that 's why its link is longer . but simple labels are n't the only type of knowledge we store . we can store characteristics and properties of concepts at each node . according to the principle of cognitive economy , which just means that our brain is efficient , we store these properties at the highest possible node . for example then , instead of storing can breathe at each animal 's node , we store that property just at the animal node . more specific characteristics such as sings , or long legs , would be stored at lower level nodes . one pretty interesting piece of supporting evidence for this type of hierarchical organization comes from how long it takes people to verify certain statements . in this kind of test , you say a statement , and ask people to tell you if it 's true or not . as you can imagine , people verify a canary is a canary pretty quickly . it takes them a little longer to verify a canary is a bird , and even longer to verify that a canary is an animal . so this is some support that we store things in a hierarchical manner , because the longer it takes us to verify a connection between two nodes , then the longer those links are , or the more nodes we have to go through to make that link . however , this is n't true for all types of animals , or even all types of categories . for example , people tend to verify that a pig is an animal faster than a pig is a mammal . because of this issue and some other problems , collins and loftus proposed a modified version of the semantic network . rather than a hierarchical organization , this model says that every individual semantic network develops based on their experience and knowledge . so , some links might be longer or shorter for different individuals , and there may be direct links from higher order categories to their exemplars . one pretty cool thing about semantic networks is that it means all the ideas in your head are connected together . so when you activate one concept , you 're pulling up related concepts along with it . this general elevation and availability is called spreading activation . for example , if i say `` fire engine , '' not only do you think of a fire engine , but related concepts such as trucks , fire , even the color red become activated , making it easier for you to retrieve or identify those items .
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so when you activate one concept , you 're pulling up related concepts along with it . this general elevation and availability is called spreading activation . for example , if i say `` fire engine , '' not only do you think of a fire engine , but related concepts such as trucks , fire , even the color red become activated , making it easier for you to retrieve or identify those items .
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so in the modified semantic networks and spreading activation , are these phenomenons affected by our schemas ?
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in order to solve problems and make decisions , you often have to access information already stored in your brain . but how is that information stored ? there are a lot of possibilities , but the one we 're going to focus on in this video is the semantic network approach , which argues that concepts are organized in your mind in terms of connected ideas . you can kind of think of this as parallel to how information might be stored in a computer . you have different nodes , which represent your concepts , and those nodes are connected by links . depending on how connected the nodes are , the links might be shorter , for closely related ideas , or longer , for less related ideas . let me show you an example to make this a little more concrete . the first semantic network model was hierarchical , meaning that they thought concepts were organized from higher order categories down to lower order categories and their exemplars . so let 's start with a general category : animal . so that 's this node here . animal might be linked to other nodes , such as bird or fish . and bird might be further linked to canary , bluebird , or more distantly , ostrich . ostrich is probably not the first thing you think of when you think of a bird , so that 's why its link is longer . but simple labels are n't the only type of knowledge we store . we can store characteristics and properties of concepts at each node . according to the principle of cognitive economy , which just means that our brain is efficient , we store these properties at the highest possible node . for example then , instead of storing can breathe at each animal 's node , we store that property just at the animal node . more specific characteristics such as sings , or long legs , would be stored at lower level nodes . one pretty interesting piece of supporting evidence for this type of hierarchical organization comes from how long it takes people to verify certain statements . in this kind of test , you say a statement , and ask people to tell you if it 's true or not . as you can imagine , people verify a canary is a canary pretty quickly . it takes them a little longer to verify a canary is a bird , and even longer to verify that a canary is an animal . so this is some support that we store things in a hierarchical manner , because the longer it takes us to verify a connection between two nodes , then the longer those links are , or the more nodes we have to go through to make that link . however , this is n't true for all types of animals , or even all types of categories . for example , people tend to verify that a pig is an animal faster than a pig is a mammal . because of this issue and some other problems , collins and loftus proposed a modified version of the semantic network . rather than a hierarchical organization , this model says that every individual semantic network develops based on their experience and knowledge . so , some links might be longer or shorter for different individuals , and there may be direct links from higher order categories to their exemplars . one pretty cool thing about semantic networks is that it means all the ideas in your head are connected together . so when you activate one concept , you 're pulling up related concepts along with it . this general elevation and availability is called spreading activation . for example , if i say `` fire engine , '' not only do you think of a fire engine , but related concepts such as trucks , fire , even the color red become activated , making it easier for you to retrieve or identify those items .
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so when you activate one concept , you 're pulling up related concepts along with it . this general elevation and availability is called spreading activation . for example , if i say `` fire engine , '' not only do you think of a fire engine , but related concepts such as trucks , fire , even the color red become activated , making it easier for you to retrieve or identify those items .
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what is the difference between spreading activation and priming ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations .
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what does `` x '' after a number mean ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence .
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at 0.07 , why did sal write marmoles instead of marbles ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations .
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will the product change if you put the 3 in 3 ( 4+5 ) on the other side , like this : ( 4+5 ) 3 ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for .
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does it matter what side of the parentheses the 3 goes on ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles .
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is n't the '+ ' sign an operation ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations .
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is 16x+10 a reasonable way to represent the number of shells that heather has left to make necklaces with ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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but this is what they 're asking for . they want us to write this expression .
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why does n't sal write x or * to multiply ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that .
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how is the moon large enough to block the sun ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for .
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why is it 3 is triple means 3 ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles .
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do n't you have to put the multiplication sign ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence .
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why does the equation just say 3 ( 4+5 ) without the multiplication sign ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles .
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what is the multiplecation sign on a computer ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for .
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is there another ways to solve 3 ( 37-2 ) -1 ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on .
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what does numerical expression mean ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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but this is what they 're asking for . they want us to write this expression .
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can you write the 3 at the end of the expression ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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but this is what they 're asking for . they want us to write this expression .
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for your expression , how can you write 3 ( tripled ) and not add a multiplication sign after ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on .
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is constructing numerical expressions just adding the numbers and then multiplying the numbers ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling .
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how come sal uses `` * '' and `` + '' but the problem does n't actually say to use the operations ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on .
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what is the difference between a numerical expression and an algebraic expression ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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but this is what they 're asking for . they want us to write this expression .
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how do you write in base ten numeral form ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence .
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why did you put 4 & 5 in brackets ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles .
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what is algebraic all about ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on .
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how do i write a numerical expression representing 1 item at $ 60 , 2 items at $ 70 and 1 item at $ 20 ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that .
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what is a word problem for 5+3x2-6 ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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but this is what they 're asking for . they want us to write this expression .
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hey sal , i wanted you to put '' write an expression of ex 12 doubed from 132 '' please ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles .
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what is the length of an edge of the cube ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles .
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is there an end to the universe ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 .
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should n't you put a multiplication sign after the 9 ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles .
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what is the difference between braces and parentheses ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles .
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what are the energy points for ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations .
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why does n't sal put an x sign after the 3 in 3 ( 4+5 ) ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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but this is what they 're asking for . they want us to write this expression .
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hey what did alan want ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles .
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wait so how do you know when to put the brackets around which numbers ?
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alan found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . write a numerical expression to model the situation without performing any operations . so let 's think about what 's going on . so he already had 5 marbles in his pocket . and then he found 4 more marbles to add to that . so we can add the 4 marbles to the 5 marbles . so 4 marbles plus the 5 marbles . so that 's what happens after the first sentence . he found 4 marbles to add to his 5 marbles currently in his pocket . he then had a competition with his friends and tripled his marbles . so this is how many marbles he had before tripling . and now he 's tripling his marbles . so we want to multiply 3 times the total number of marbles he has now -- times 4 plus 5 . so this right over here is the numerical expression that models the situation without performing any operations . we , of course , could then actually calculate this . he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for . they want us to write this expression .
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he has 9 marbles before tripling . and then you multiply it by 3 and he has 27 . but this is what they 're asking for .
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so should you put a x after the 3 ?
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- in the last two videos , we talked about our theme and how to break your story into beats using the story spine . the next step is to divide your story spine into larger sections , which we call acts . throughout history , storytellers have experimented with everything from one act to eight acts or more , but the most common structure for film is the three act structure . act one consists of the first three steps of our story spine , `` once upon a time . '' this is where we meet our main character , known as the protagonist , and we find out when and where the story takes place . for example , in finding nemo , we 're introduced to marlin and nemo who live in the safety of the reef , and we learn why marlin worries about the dangers lurking in the open ocean . the first act also tells the audience what type of movie they 're about to see . is it a science fiction , a romantic comedy , a historical drama , or something else ? every day . this is where we learn more about how the world works . for example , in finding nemo , we learn about the other creatures who inhabit the reef and what life is like there every day . until one day . this is often called the inciting incident . it 's an event which leads to a key obstacle your protagonist faces and sets the rest of the story in motion . in finding nemo , nemo ignores his father 's instructions , swims out to touch the boat , and is captured by a scuba diver . in order to save him , marlin is forced to face his biggest fear , the open ocean . the first act can also introduce something called the antagonist . you probably know this as a character we sometimes call the villain , but it can take many forms . generally , the antagonist is a force that gets in the way of your character 's wants and needs . marlin 's antagonist is something , something that stands in his way , the ocean and his fear of it . getting this first act figured out is critical , so let 's ask our storytellers for some more information . - in act one , we want to introduce our characters , introduce the story , and get a landscape of where the story is trying to go . - what 's essential in the first act is that you meet the main character in her or his world and you understand their place in the world and you understand their problem in the world . - you learn enough about this character that you like this character and you want to go on this journey with the filmmaker and the character . it 's very important to hook your audience in act one . - for our movies , i think wall-e has one of strongest first acts . the world is set up . i mean , it 's a trash planet and it 's an abandoned dystopia , and yet you have an idealist , you have wall-e , who believes as a robot that love is possible in an environment where he 's meant to clean up the remnants of the opposite world view , and it 's beautiful that he does n't , he has almost no evidence other than a little green leaf that what he believes and feels is real , and then eve comes in and confirms to him his idealistic tendencies that we can rise above our programming and that we can be more than we 're told and we can be more than what 's around us . - sometimes the inciting incident will introduce a conflict that will launch the main character into a journey that will take place throughout the film . - in most cases i can think of , the inciting incident comes toward the end of the first act . you spent the first act setting up who the characters are , what 's important , what the status quo is in the world , and the inciting incident that 's gon na pull the rug out from under that status quo is gon na launch you into act two . - in wall-e , the inciting incident is when eve is taken off of earth into the axiom , and wall-e follows her up to the axiom on the rocket 'cause he now has a goal which is to get eve back . even though he 's a robot and he can not be further from me , i completely empathize with him and i want for him connection and love and the things that he aspires to . to me , it 's an elegant , beautiful , heartbreaking first act . - and i think a successful first act gets you to invest in your character , care about your character , care about what they care about , so when the thing they care about is threatened or the rug is pulled out from under them in some way , you 're rooting for them to launch into the second act and solve that problem . - ultimately , the first act is the setup for the story . it 's where we learn everything we need to know about our main characters and the world , and we find something out which gets us invested in the journey which follows . in the next exercise , you 'll have a chance to identify the first act in your favorite films , as well as start developing the first act for the story you want to create .
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- in the last two videos , we talked about our theme and how to break your story into beats using the story spine . the next step is to divide your story spine into larger sections , which we call acts .
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does your story have to have an antagonist ?
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- in the last two videos , we talked about our theme and how to break your story into beats using the story spine . the next step is to divide your story spine into larger sections , which we call acts . throughout history , storytellers have experimented with everything from one act to eight acts or more , but the most common structure for film is the three act structure . act one consists of the first three steps of our story spine , `` once upon a time . '' this is where we meet our main character , known as the protagonist , and we find out when and where the story takes place . for example , in finding nemo , we 're introduced to marlin and nemo who live in the safety of the reef , and we learn why marlin worries about the dangers lurking in the open ocean . the first act also tells the audience what type of movie they 're about to see . is it a science fiction , a romantic comedy , a historical drama , or something else ? every day . this is where we learn more about how the world works . for example , in finding nemo , we learn about the other creatures who inhabit the reef and what life is like there every day . until one day . this is often called the inciting incident . it 's an event which leads to a key obstacle your protagonist faces and sets the rest of the story in motion . in finding nemo , nemo ignores his father 's instructions , swims out to touch the boat , and is captured by a scuba diver . in order to save him , marlin is forced to face his biggest fear , the open ocean . the first act can also introduce something called the antagonist . you probably know this as a character we sometimes call the villain , but it can take many forms . generally , the antagonist is a force that gets in the way of your character 's wants and needs . marlin 's antagonist is something , something that stands in his way , the ocean and his fear of it . getting this first act figured out is critical , so let 's ask our storytellers for some more information . - in act one , we want to introduce our characters , introduce the story , and get a landscape of where the story is trying to go . - what 's essential in the first act is that you meet the main character in her or his world and you understand their place in the world and you understand their problem in the world . - you learn enough about this character that you like this character and you want to go on this journey with the filmmaker and the character . it 's very important to hook your audience in act one . - for our movies , i think wall-e has one of strongest first acts . the world is set up . i mean , it 's a trash planet and it 's an abandoned dystopia , and yet you have an idealist , you have wall-e , who believes as a robot that love is possible in an environment where he 's meant to clean up the remnants of the opposite world view , and it 's beautiful that he does n't , he has almost no evidence other than a little green leaf that what he believes and feels is real , and then eve comes in and confirms to him his idealistic tendencies that we can rise above our programming and that we can be more than we 're told and we can be more than what 's around us . - sometimes the inciting incident will introduce a conflict that will launch the main character into a journey that will take place throughout the film . - in most cases i can think of , the inciting incident comes toward the end of the first act . you spent the first act setting up who the characters are , what 's important , what the status quo is in the world , and the inciting incident that 's gon na pull the rug out from under that status quo is gon na launch you into act two . - in wall-e , the inciting incident is when eve is taken off of earth into the axiom , and wall-e follows her up to the axiom on the rocket 'cause he now has a goal which is to get eve back . even though he 's a robot and he can not be further from me , i completely empathize with him and i want for him connection and love and the things that he aspires to . to me , it 's an elegant , beautiful , heartbreaking first act . - and i think a successful first act gets you to invest in your character , care about your character , care about what they care about , so when the thing they care about is threatened or the rug is pulled out from under them in some way , you 're rooting for them to launch into the second act and solve that problem . - ultimately , the first act is the setup for the story . it 's where we learn everything we need to know about our main characters and the world , and we find something out which gets us invested in the journey which follows . in the next exercise , you 'll have a chance to identify the first act in your favorite films , as well as start developing the first act for the story you want to create .
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you probably know this as a character we sometimes call the villain , but it can take many forms . generally , the antagonist is a force that gets in the way of your character 's wants and needs . marlin 's antagonist is something , something that stands in his way , the ocean and his fear of it .
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how is the antagonist , which is the force that gets in the way , different from the obstacles that affects the character 's journey ?
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- in the last two videos , we talked about our theme and how to break your story into beats using the story spine . the next step is to divide your story spine into larger sections , which we call acts . throughout history , storytellers have experimented with everything from one act to eight acts or more , but the most common structure for film is the three act structure . act one consists of the first three steps of our story spine , `` once upon a time . '' this is where we meet our main character , known as the protagonist , and we find out when and where the story takes place . for example , in finding nemo , we 're introduced to marlin and nemo who live in the safety of the reef , and we learn why marlin worries about the dangers lurking in the open ocean . the first act also tells the audience what type of movie they 're about to see . is it a science fiction , a romantic comedy , a historical drama , or something else ? every day . this is where we learn more about how the world works . for example , in finding nemo , we learn about the other creatures who inhabit the reef and what life is like there every day . until one day . this is often called the inciting incident . it 's an event which leads to a key obstacle your protagonist faces and sets the rest of the story in motion . in finding nemo , nemo ignores his father 's instructions , swims out to touch the boat , and is captured by a scuba diver . in order to save him , marlin is forced to face his biggest fear , the open ocean . the first act can also introduce something called the antagonist . you probably know this as a character we sometimes call the villain , but it can take many forms . generally , the antagonist is a force that gets in the way of your character 's wants and needs . marlin 's antagonist is something , something that stands in his way , the ocean and his fear of it . getting this first act figured out is critical , so let 's ask our storytellers for some more information . - in act one , we want to introduce our characters , introduce the story , and get a landscape of where the story is trying to go . - what 's essential in the first act is that you meet the main character in her or his world and you understand their place in the world and you understand their problem in the world . - you learn enough about this character that you like this character and you want to go on this journey with the filmmaker and the character . it 's very important to hook your audience in act one . - for our movies , i think wall-e has one of strongest first acts . the world is set up . i mean , it 's a trash planet and it 's an abandoned dystopia , and yet you have an idealist , you have wall-e , who believes as a robot that love is possible in an environment where he 's meant to clean up the remnants of the opposite world view , and it 's beautiful that he does n't , he has almost no evidence other than a little green leaf that what he believes and feels is real , and then eve comes in and confirms to him his idealistic tendencies that we can rise above our programming and that we can be more than we 're told and we can be more than what 's around us . - sometimes the inciting incident will introduce a conflict that will launch the main character into a journey that will take place throughout the film . - in most cases i can think of , the inciting incident comes toward the end of the first act . you spent the first act setting up who the characters are , what 's important , what the status quo is in the world , and the inciting incident that 's gon na pull the rug out from under that status quo is gon na launch you into act two . - in wall-e , the inciting incident is when eve is taken off of earth into the axiom , and wall-e follows her up to the axiom on the rocket 'cause he now has a goal which is to get eve back . even though he 's a robot and he can not be further from me , i completely empathize with him and i want for him connection and love and the things that he aspires to . to me , it 's an elegant , beautiful , heartbreaking first act . - and i think a successful first act gets you to invest in your character , care about your character , care about what they care about , so when the thing they care about is threatened or the rug is pulled out from under them in some way , you 're rooting for them to launch into the second act and solve that problem . - ultimately , the first act is the setup for the story . it 's where we learn everything we need to know about our main characters and the world , and we find something out which gets us invested in the journey which follows . in the next exercise , you 'll have a chance to identify the first act in your favorite films , as well as start developing the first act for the story you want to create .
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in order to save him , marlin is forced to face his biggest fear , the open ocean . the first act can also introduce something called the antagonist . you probably know this as a character we sometimes call the villain , but it can take many forms .
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would dory be an antagonist ?
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by this point , you may be getting kind of sick of these pressure volume loops . but i assure you they 're very , very worthwhile getting to know . in fact , let 's learn one more thing out of these pressure volume loops . let 's try to squeeze all the knowledge we can out of these things . so i 'm going to draw one for you here very quickly . and on this side , we 've got pressure in millimeters of mercury . and on this side we 've got milliliters , right ? we 're just going to use the same units we 've kind of gotten very familiar with . and we 're going to do 50 , and 100 , and 125 or so on this axis . that 's a pretty normal set of terms or numbers . and then on this side , we 've got 0 , 50 , and i 'm going to go up to 120 or so up here . so this would be a pressure volume loop . and i like to kind of start out where the pressures are low and the volumes are really , really high , so somewhere around here . i 'm going to imagine that this is kind of the end of diastole , where my left ventricle is full of blood , right ? it 's going to start rising in pressure slowly . and as it rises , it 's going to get to let 's say about this point . and then the blood is going to start entering the aorta . and as it enters the aorta , i 've got to get to a nice high pressure . i know that that 's the target or thereabouts . and so the pressure kind of rises even higher . and then the volume starts to fall as volume enters the aorta , and so it leaves the left ventricle . and then the left ventricle continues to lose its blood to the aorta . and the blood in the left ventricle that 's left is around 50 . and then you start having relaxation . so you relax all the way down and the pressure falls to about that point . and then it continues falling , but now there 's a little bit of blood kind of entering into the left ventricle . so it 's starting to fall in pressure but continues to now start rising in volume . and it continues to rise until it 's ready to do the whole thing all over again , right ? and now if this is kind of our overall pressure volume loop , what i want to do is kind of focus in on one particular point . let 's focus in on this point right here . and this is the end of systole , right ? we talked about the end of systole being right here . this is where it begins to start relaxing . and when i say `` it , '' i mean the muscle cells . so you 've got a muscle cell over here . and i like to draw them kind of branched just to remind you that they 're muscle cells . and you 've got now this cell completely contracted down , right ? so it looks like this with the actin and myosin completely overlapping , right ? because that 's what you expect to happen at the end of systole . and , of course , i 'm drawing it this way really just to remind you what 's going on inside of the cell , although you know that , of course , a cell has many , many , many sarcomeres , not just one . and of course , this one i 've drawn just having one . but you get the idea that there 's a lot of overlap between the actin and the myosin . in fact , that 's what these little red lines represent , just the major proteins instead of heart cells . so these are my heart cells full of protein , and they 're completely contracted , right ? so sometimes they like to relax , and sometimes they like to contract . and at the end of systole , where i 've drawn that orange arrow , these muscle cell are completely contracted down . i 'm going to write `` contracted '' just to kind of remind you that that 's what 's going on . and what are they waiting for ? i mean , what 's next ? what are they hoping will happen next ? well , they are hoping that they can now get rid of all that calcium , and so they can relax . so you 've got a lot of calcium in this space . and they 're kind of hoping that the calcium will go away and they can relax . and so imagine now that the calcium -- i 'm going to draw calcium as a white circle , right ? and i 'm going to fill it in . this is my calcium , right ? and imagine that i 've got calcium in here . i 'm just going to draw little white circles in here . and instead of allowing my heart cell to relax , i 'm going to kind of do something interesting . and i remember calling it a trick last time . i guess i can call this a trick . and the trick is i 'm going to trick my heart cell into not relaxing . i will basically not allow it to relax , because i load up this cell -- imagine i can somehow do this -- with lots and lots of calcium . and so i just fill it with calcium . it 's chock full of calcium , and it has no way to really get rid of it . so it is going to continue to be contracted , right ? it 's just going to continue to be contracted if i can somehow fill this with calcium and not allow it to go into relaxation . so my heart cell 's completely contracted . that 's the key , right ? i have done this . and so as a result of doing this now , of course , this wo n't happen , this relaxation bit . this wo n't happen and neither will the next bit . so basically i 'm kind of forcing myself to continue to remain contracted , and all this kind of disappears . and so what happens is that now my heart is basically full of blood , right ? i 've got a heart full of blood . i 'm going to draw it over here . i 'm going to just kind of ignore for the time being the left atrium and the aorta . but this is my left ventricle full of blood . so this is chock full of blood . and what i 'm going to do is i 'm going to take a little needle -- watch this . i 'm going to take a little needle , and i 'm going to try to add a little bit of blood or take a little bit of blood away . and i guess i 'm going to start by taking a little bit of blood away just to see what will happen , right ? so i take a little bit of blood away just to see what will happen . and let me choose a different color . let 's choose a green color . and at this point now , my heart is -- again , it 's full of blood , right ? and i 'm going to now take a needle , and i 'm going to take some blood off of the heart . i 'm going to actually just pull it off like that . so now i 've got blood in my syringe . and as a result , what have i really done ? well , i 've lowered the volume , right ? i 've lowered the volume . actually , let me switch to a green color and show you that i 've lowered the volume . and if i lower the volume , basically , what will happen ? well , if i lower the volume , the pressure will start to fall , right ? the pressure goes down . so it goes something like that . and i can do this again , and i could see if the pressure falls . and , oh , it does . and i could do this again . in fact , i could take all the blood out of the left ventricle , and i could see that basically my pressure will go down to zero . so i basically have now a few dots . i can connect my dots . and i can see that i create this basically kind of a line , right ? and so this line is what you would get if you just keep reducing , and reducing , and reducing the volume in the heart . now , what if i did the opposite ? what if i -- instead of reducing the blood , what if i actually added blood ? and of course , it might be kind of tricky to think about adding blood . but just remember you can always add air to a balloon if you try hard enough . and similarly , you could actually push blood into a full left ventricle if you have enough pressure pushing down . so i could actually do this . i could actually try to do this . in fact , i 'm going to add , let 's say , a little bit of volume here , right ? and i 'm going to notice that the pressure goes up . in fact , it goes up even more than it ever did before , right ? it actually rises above the line that i had drawn in blue . and in fact , i might even do it again . i could say , well , let 's just add some more volume . let 's just see what happens . and the pressure goes up even higher . so i could connect these lines . i could say , ok , well , let 's see what these lines look like . and basically , it 's forming a nice straight line . in fact , to see it a little bit easier , let me just get rid of this blue stuff . let 's get ride of all this stuff . and you can see that you get this nice straight line that relates volume to pressure . and so this relationship between volume and pressure or pressure and volume is happening with the muscle cells contracted . remember , all this time my muscle cells are bathing in calcium . they 're completely contracted down . they have not been able to relax . they 're contracted . and so you could even say , well , this is basically the relationship between pressure and volume at the end of systole . so you could say this is the end systole pressure volume relationship . and this is kind of the long way of saying it . people actually shorten all of this down . they do n't say all these words . they usually say espvr -- end systolic pressure volume relationship . and all it means is that if you could get a situation where your heart cells are completely contracted , completely contracted down -- and that 's why i kind of made up the bit about filling it up with calcium , because i guess that 's one way to do it , but completely contracted , and you just kind of added volume or took it away , what would the pressure do ? how would it change ? and this line tells you that . in fact , one final point i want to make about this is if you remember , there 's a relationship now between elastance -- remember the term elastance -- and pressure and volume . in fact , if you take pressure divided by volume , that gives you elastance . so with this line that we 've drawn , you can actually say , well , there 's a slope to this line , right ? i tried to draw a straight line so imagine there 's a straight line between all these points , right ? well , the slope of that line -- slope is elastance , right ? that is the elastance . in fact , you might even see the term -- wherever the pressure volume loop falls on this espvr , sometimes you even see the line actually labeled , and you see it called e0 . in fact , you 'll often see that . so e0 refers to the slope of the line that is formed at the end of systole .
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it 's just going to continue to be contracted if i can somehow fill this with calcium and not allow it to go into relaxation . so my heart cell 's completely contracted . that 's the key , right ?
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if the heart muscle is contracted , would n't the aorta have the blood and not the ventricle ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x .
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all of the optimization videos are cool and everything , but i have one question : how do you find the equation for profit , cost , or stuff like that using real world data ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example .
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without plugging in random numbers ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit .
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why does sal write the first critical point to the thousandths but the second one to the ten-thousandths ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one .
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can we use calculus to optimize a relation between workforce and profit ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here .
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like for example with 100 workers how much shoes need to be manufactured for maximum profit ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ?
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should n't sal have checked the end behaviour of the graph first to see if there even was a maximum profit ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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yep . all right . now these are all we know about these , or these are both critical points .
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if the graph tended towards infinity this method could have given an incorrect result right ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business .
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while i agree with the solution derived in the video , why does n't setting r ( x ) = c ( x ) work ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 .
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do optimization questions often use the quadratic equation ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example .
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in reality would there ever be a cost equation be that perfect for differentiation ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x .
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in this problem will it also be correct if the amount of quantity to be produced that is x is found by equating marginal revenue ( mr ) and marginal cost ( mc ) functions ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 .
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when he checks the second derivative , ca n't you just plug the values back into the original equation and see which result is greater ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him .
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do n't you have to plug the x-value into the revenue and cost equations and find the distance ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business .
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only one thing i 'd like to know , how did you get the equation p ( x ) = r ( x ) c ( x ) = x^3 - 6x^2 + 15x ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 .
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is n't it much easier ( to remember and apply ) to use the first derivative test and find out whether the critical point is maximum or minimum ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that .
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when do we substitue the profit and when we do n't need to substitute ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one .
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how can i use optimization in regard to the relationship between distance , rate , and time or cost , distance , and rate ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced .
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why is a shoe factory making pears ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards .
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why change -x^3 to x^3 ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit .
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does any cost actually exist as a polynomial function ?
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you 've opened up a shoe factory and you 're trying to figure out how many thousands of pairs of shoes to produce in order to optimize your profit . and so let 's let x equal the thousands of pairs produced . now let 's think about how much money you 're going to make per pair . actually , let me say how much revenue , which is how much you actually get to sell those shoes for . so let 's write a function right here . revenue as a function of x . well , you have a wholesaler who 's willing to pay you $ 10 per pair for as many pairs as you 're willing to give him . so your revenue as a function of x is going to be 10 times x . and since x is in thousands of pairs produced , if x is 1 , that means 1,000 pairs produced times 10 , which means $ 10,000 . but this will just give you 10 . so this right over here is in thousands of dollars . so if x is 1 , that means 1,000 pairs produced . 10 times 1 says r is equal to 10 , but that really means $ 10,000 . now , it would be a nice business if all you had was revenue and no costs . but you do have costs . you have materials , you have to build your factory , have to pay your employees , you have to pay the electricity bill . and so you hire a bunch of consultants to come up with what your cost is as a function of x . and they come up with a function . they say it is the number of the thousands of pairs you produce cubed minus 6 times the thousands of pairs you produce squared plus 15 times the thousands of pairs that you produce . and once again , this is also going to be in thousands of dollars . now , given these functions of x for revenue and cost , what is profit as a function of x going to be ? well , your profit as a function of x is just going to be equal to your revenue as a function of x minus your cost as a function of x . if you produce a certain amount and let 's say you bring in , i do n't know , $ 10,000 of revenue and it costs you $ 5,000 to produce those shoes , you 'll have $ 5,000 in profit . those numbers are n't the ones that would actually you would get from this right here . i 'm just giving you an example . so this is what you want to optimize . you want to optimize p as a function of x . so what is it ? i 've just said it here in abstract terms , but we know what r of x is and what 's c of x . this is 10x minus all of this business . so minus x to the third plus 6x squared minus 15x . i just subtracted x squared , you subtract 6x squared it becomes positive , you subtract a 15x it becomes negative 15x , and then we can simplify this as -- let 's see , we have negative x to the third plus 6x squared minus 15x plus 10x , so that is minus 5x . now if we want to optimize this profit function analytically , the easiest way is to think about what are the critical points of this profit function and are any of those critical points minimum points or maximum points ? and if one of them is a maximum point , then we can say , well , let 's produce that many . that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points . so p prime of x is going to be equal to negative 3x squared plus 12x minus 5 . and so this thing is going to be defined for all x . so the only critical points we 're going to have is when the first derivative right over here is equal to 0 . so negative 3x squared plus 12x minus 5 needs to be equal to 0 in order for x to be a critical point . so now we just have to solve for x . and so we just are essentially solving a quadratic equation . just so that i do n't have as many negatives , let 's multiply both sides by negative 1 . i just like to have a clean first coefficient . so if we multiply both sides by negative 1 , we get 3x squared minus 12x plus 5 is equal to 0 . and now we can use the quadratic formula to solve for x . so x is going to be equal to negative b , which is 12 , plus or minus the square root . i always need to make my radical signs wide enough . the square root of b squared , which is 144 , minus 4 times a , which is 3 , times c , which is 5 . all of that over 2a . so 2 times 3 is 6 . so x is equal to 12 plus or minus the square root of , let 's see , 4 times 3 is 12 times 5 is 60 . 144 minus 60 is 84 . all of that over 6 . so x could be equal to 12 plus the square root of 84 over 6 or x could be equal to 12 minus the square root of 84 over 6 . so let 's figure out what these two are . and i 'll use a calculator . i 'll use the calculator for this one . so i get , let 's see , 12 plus the square root of 84 divided by 6 gives me 3.5 -- i 'll just say 3.53 . so approximately 3. -- actually , let me go one more digit , because i 'm talking about thousands . so let me say 3.528 . so this would literally be 3,528 shoes , because this is in thousands , or pairs of shoes . and then let 's do the situation where we subtract . and actually we can look at our previous entry and just change this to a subtraction . change that to not a negative sign , a subtraction . there you go . and we get 0.4725 . let me remember that . 0.4725 . approximately equal to 0.4725 . i have a horrible memory , so let me review that i wrote the same thing . 4725 . yep . all right . now these are all we know about these , or these are both critical points . these are points at which our derivative is equal to 0 . but we do n't know whether they 're minimum points , they 're points at which the function takes on a minimum value , a maximum value , or neither . to do that , i 'll use the second derivative test to figure out if our function is concave upwards or concave downwards or neither at one of these points . so let 's look at the second derivative . so p prime prime of x is going to be equal to negative 6x plus 12 . and so if we look at -- let me make sure i have enough space . so if we look at p prime prime of 3.528 . so let 's see if i can think this through . so this is between 3 and 4 . so if we take the lower value , 3 times negative 6 is negative 18 plus 12 is going to be less than 0 . and if this was 4 it 'd be even more negative , so this thing is going to be less than 0 . do n't even have to use my calculator to evaluate it . now what about this thing right over here ? 0.47 . well , 0.47 , that 's roughly 0.5 . so negative 6 times 0.5 is negative 3 . this is going to be nowhere close to being negative . this is definitely going to be positive . so p prime prime of 0.4725 is greater than 0 . so the fact that the second derivative is less than 0 , that means that my derivative is decreasing . my first derivative is decreasing when x is equal to this value , which means that our graph , our function , is concave downwards here . and concave downwards means it looks something like this . and so you can see what it looks something like that , the slope is constantly decreasing . so if you have an interval where the slope is decreasing and you know the point where the slope is exactly 0 , which is where x is equal to 3.528 , it must be a maximum . so we actually do take on a maximum value when x is 3.528 . on the other side we see that over here we 're concave upwards . the graph will look something like this over here . and if the slope is 0 where the graph looks like that , we see that that is a local minimum . and so we definitely do n't want to do this . we would produce 472 and 1/2 units if we were looking to minimize our profit , maximize our loss . so we definitely do n't want to do this . but let 's actually think about what our profit is going to be if we produce 3.528 thousands of shoes , or 3,528 shoes . well , to do that we just have to input it back into our original profit function right over here . so let 's do that . so i get my calculator out . so my original profit function is right over there . so i want to be able to see that and that . so i get negative 3.528 to the third power plus 6 times 3.528 squared minus 5 times 3.528 gives me -- and we get a drum roll now -- gives me a profit of 13.128 . so let me write this down . the profit when i produce 3,528 shoes is approximately equal to or it is equal to , if i produce exactly that many shoes , it 's equal to 13.128 . or actually it 's approximately , because i 'm still rounding 13.128 . so if i produce 3,528 shoes in a given period , i 'm going to have a profit of $ 13,128 . remember , this right over here is in thousands , this right over here is 13.128 thousands of dollars in profit , which is $ 13,128 . anyway , we are now going to be rich shoe manufacturers .
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that is going to be -- we will have optimized or we will figure out the quantity we need to produce in order to optimize our profit . so to figure out critical points , we essentially have to find the derivative of our function and figure out when does that derivative equal 0 or when is that derivative undefined ? that 's the definition of critical points .
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can we take the 3rd derivative here instead to find the shape of the concave ?
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man in glasses : hello , there . here at crash course hq we like to start out each day with a nice healthy dose of water in all its 3 forms . it 's the only substance on all of our planet earth that occurs naturally in solid , liquid and gas forms . to celebrate this magical bond between two hydrogen atoms and one oxygen atom , here today we are going to be celebrating the wonderful life-sustaining properties of water , but we 're going to do it slightly more clothed . ( boppy music ) ah , much better . when we left off here at the biology crash course we were talking about life , and the rather important fact that all life as we know it is dependent upon there being water around . scientists and astronomers are always looking out into the universe trying to figure out whether there is life elsewhere because that is kind of the most important question that we have right now . they 're always getting really excited when they find water someplace , particularly liquid water . this is one reason why i and so many other people geeked out so hard last december when mars ' seven-year-old rover opportunity found a 20-inch long vein of gypsum that was almost certainly deposited by long term liquid water on the surface of mars . this was probably billions of years ago , and so it 's going to be hard to tell whether or not the water that was there resulted in some life . maybe we can figure that out and that would be really exciting . why ? why do we think that water is necessary for life ? why does water on other planets get us so freaking excited ? let 's start out by investigating some of the amazing properties of water . in order to do that , we 're going to have to start out with this . the world 's most popular molecule , or at least the world 's most memorized molecule . we all know about it . good old h20 . two hydrogens , one oxygen , the hydrogens each sharing an electron with oxygen in what we call a covalent bond . as you can see , i 've drawn my water molecule in a particular way . this is actually the way that it appears . it is v-shaped . because this big old oxygen atom is a little bit more greedy for electrons , it has a slight negative charge ; whereas , this area here with the hydrogen atoms has a slight positive charge . thanks to this polarity , all water molecules are attracted to one another ; so much so that they actually stick together and these are called hydrogen bonds . we talked about them last time , but essentially what happens is that the positive pole around those hydrogen atoms bonds to the negative pole around the oxygen atoms of a different water molecule . it 's a weak bond , but look , they 're bonding ! seriously , i can not overstate the importance of this hydrogen bond . when your teacher asks you , `` what 's important about water ? '' start out with the hydrogen bonds and you should put it in all caps and maybe some sparkles around it . one of the cool properties that results from these hydrogen bonds is a high cohesion for water which results in high surface tension . cohesion is the attraction between two like things , like attraction between one molecule of water and another molecule of water . water has the highest cohesion of any nonmetalic liquid . you can see this if you put some water on some wax paper or some teflon or something where the water beads up like this . some leaves of plants do it really well ; it 's quite cool . since water adheres weakly to the wax paper or to the plant , but strongly to itself , the water molecules are holding those droplets together in a configuration that creates the least amount of surface area . this is high surface tension that allows some bugs and even i think one lizard and also one jesus to be able to walk on water . the cohesive force of water does have its limits , of course . there are other substances that water quite likes to stick to . take glass , for example . this is called adhesion . the water is spreading out here instead of beading up because the adhesive forces between the water and the glass are stronger than the cohesive forces of the individual water molecules in the bead of water . adhesion is attraction between two different substances , so in this case the water molecules and the glass molecules . these properties lead to one of my favorite things about water ; the fact that it can defy gravity . that really cool thing that just happened is called capillary action . explaining it can be easily done with what we now know about cohesion and adhesion . thanks to adhesion , the water molecules are attracted to the molecules in the straw . as the water molecules adhere to the straw , other molecules are drawn in by cohesion following those fellow water molecules . thank you cohesion . the surface tension created here causes the water to climb up the straw . it will continue to climb until eventually gravity pulling down on the weight of the water in the straw overpowers the surface tension . the fact that water 's a polar molecule also makes it really good at dissolving things , which we call it 's a good solvent then . scratch that . water is n't a good solvent . it 's an amazing solvent . there are more substances that can be dissolved in water than in any other liquid on earth . yes , that includes the strongest acid that we have ever created . these substances that dissolve in water , sugar or salt being ones that we 're familiar with , are called hydrophilic , and they are hydrophilic because they are polar . their polarity is stronger than the cohesive forces of the water . when you get one of these polar substances in water , it 's strong enough that it breaks all the little cohesive forces , all those little hydrogen bonds . instead of hydrogen bonding to each other , the water will hydrogen bond around these polar substances . table salt is ionic , and right now it 's being separated into ions as the poles of our water molecules interact with it . what happens when there is a molecule that can not break the cohesive forces of water ? it ca n't penetrate and come into it . basically , what happens when that substance ca n't overcome the strong cohesive forces of water , ca n't get inside of the water ? that 's when we get what we call a hydrophobic substance , or something that is fearful of water . these molecules lack charged poles . they are non-polar and are not dissolving in water because essentially they 're being pushed out of the water by water 's cohesive forces . water , we may call it the universal solvent , but that does not mean that it dissolves everything . ( boppy music ) there have been a lot of eccentric scientists throughout history , but all this talk about water got me thinking about perhaps the most eccentric of the eccentrics , a man named henry cavendish . he communicated with his female servants only via notes , and added a staircase to the back of his house to avoid contact with his housekeeper . some believe he may have suffered from a form of autism , but just about everyone will admit that he was a scientific genius . he 's best remembered as the first person to recognize hydrogen gas as a distinct substance and to determine the composition of water . in the 1700s , most people thought that water itself was an element , but cavendish observed that hydrogen , which he called inflammable air , reacted with oxygen , known then by the awesome name , dephlogisticated air , to form water . cavendish did n't totally understand what he 'd discovered here , in part because he did n't believe in chemical compounds . he explained his experiments with hydrogen in terms of a fire-like element called phlagiston . nevertheless , his experiments were groundbreaking . like his work in determining the specific gravity basically the comparative density of hydrogen and other gases with reference to common air . it 's especially impressive when you consider the crude instruments he was working with . this , for example , is what he made his hydrogen gas with . he went on not only to establish an accurate composition of the atmosphere , but also discovered the density of the earth . not bad for a guy who was so painfully shy that the only existing portrait of him was sketched without his knowledge . for all of his decades of experiments , cavendish only published about 20 papers . in the years after his death , researchers figured out that cavendish had actually pre-discovered richter 's law , ohm 's law , coulomb 's law , several other laws . that 's a lot of freaking laws . if he had gotten credit for them all , we would have had to deal with like cavendish 's 8th law and cavendish 's 4th law , so i , for one , am glad that he did n't actually get credit . we 're going to do some pretty amazing science right now . you guys are not going to believe this . okay , you ready ? it floats ! yeah , i know you 're not surprised by this , but you should be because everything else , when it 's solid is much more dense than when it 's liquid ; just like gases are much less dense than liquids are . but that simple characteristic of water that its solid form floats is one of the reasons why life on this planet , as we know it , is possible . why is it that solid water is less dense than liquid water while everything else is the exact opposite of that ? well , you can thank your hydrogen bonds once again . at around 32 degrees fahrenheit or 0 degrees celsius if you 're a scientist or from a part of the world where things make sense , water molecules start to solidify and the hydrogen bonds in those water molecules form crystalline structures that space molecules apart more evenly , in turn making frozen water less dense than its liquid form . so in almost every circumstance a floating water ice is a really good thing . if ice were denser than water it would freeze and then sink , and then freeze and then sink , and then freeze and then sink , so just trust me on this one . you do n't want to live on a world where ice sinks . not only would it totally wreak havoc on aquatic eco systems which are basically how life formed on the earth in the first place . also , all the ice in the north pole would sink and then all of the water everywhere else would rise and we would n't have any land . that would be annoying . there 's one more amazing property of water i 'm forgetting . why is it so hot in here ? heat capacity ! yes , water has a very high heat capacity and probably that means nothing to you . basically it means that water is really good at holding on to heat , which is why we like to put hot water bottles in our bed and cuddle with them when we 're lonely . aside from artificially warming your bed , it 's also very important that it 's hard to heat up and cool down the oceans significantly . they become giant heat sinks that regulate the temperature and the climate of our planet , which is why , for example , it 's so much nicer in los angeles where the ocean is constantly keeping the temperatures the same than it is in , say , nebraska . on a smaller scale , we can see water 's high heat capacity really easily and visually by putting a pot with no water in it on a stove and seeing how badly that goes . but then you put a little bit of water in it and it takes forever to freaking boil ! and if you have n't already noticed this , when water evaporates from your skin , it cools you down . that 's the principle behind sweating , which is an extremely effective , though somewhat embarrassing , part of life . this is an example of another incredibly cool thing about water . when my body gets hot and it sweats , that heat excites some of the water molecules on my skin to the point where they break those hydrogen bonds and they evaporate away . when they escape , they take that heat energy with them , leaving me cooler . lovely . this was n't exercise , though . i do n't know why i 'm sweating so much . it could be the spray bottle that i keep spraying myself with , or maybe it 's just because this is such a high stress enterprise trying to teach you people things .
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when you get one of these polar substances in water , it 's strong enough that it breaks all the little cohesive forces , all those little hydrogen bonds . instead of hydrogen bonding to each other , the water will hydrogen bond around these polar substances . table salt is ionic , and right now it 's being separated into ions as the poles of our water molecules interact with it .
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what exactly happens when oxygen and hydrogen bond to make water so dense ( compared to the two gasses that form it ) ?
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man in glasses : hello , there . here at crash course hq we like to start out each day with a nice healthy dose of water in all its 3 forms . it 's the only substance on all of our planet earth that occurs naturally in solid , liquid and gas forms . to celebrate this magical bond between two hydrogen atoms and one oxygen atom , here today we are going to be celebrating the wonderful life-sustaining properties of water , but we 're going to do it slightly more clothed . ( boppy music ) ah , much better . when we left off here at the biology crash course we were talking about life , and the rather important fact that all life as we know it is dependent upon there being water around . scientists and astronomers are always looking out into the universe trying to figure out whether there is life elsewhere because that is kind of the most important question that we have right now . they 're always getting really excited when they find water someplace , particularly liquid water . this is one reason why i and so many other people geeked out so hard last december when mars ' seven-year-old rover opportunity found a 20-inch long vein of gypsum that was almost certainly deposited by long term liquid water on the surface of mars . this was probably billions of years ago , and so it 's going to be hard to tell whether or not the water that was there resulted in some life . maybe we can figure that out and that would be really exciting . why ? why do we think that water is necessary for life ? why does water on other planets get us so freaking excited ? let 's start out by investigating some of the amazing properties of water . in order to do that , we 're going to have to start out with this . the world 's most popular molecule , or at least the world 's most memorized molecule . we all know about it . good old h20 . two hydrogens , one oxygen , the hydrogens each sharing an electron with oxygen in what we call a covalent bond . as you can see , i 've drawn my water molecule in a particular way . this is actually the way that it appears . it is v-shaped . because this big old oxygen atom is a little bit more greedy for electrons , it has a slight negative charge ; whereas , this area here with the hydrogen atoms has a slight positive charge . thanks to this polarity , all water molecules are attracted to one another ; so much so that they actually stick together and these are called hydrogen bonds . we talked about them last time , but essentially what happens is that the positive pole around those hydrogen atoms bonds to the negative pole around the oxygen atoms of a different water molecule . it 's a weak bond , but look , they 're bonding ! seriously , i can not overstate the importance of this hydrogen bond . when your teacher asks you , `` what 's important about water ? '' start out with the hydrogen bonds and you should put it in all caps and maybe some sparkles around it . one of the cool properties that results from these hydrogen bonds is a high cohesion for water which results in high surface tension . cohesion is the attraction between two like things , like attraction between one molecule of water and another molecule of water . water has the highest cohesion of any nonmetalic liquid . you can see this if you put some water on some wax paper or some teflon or something where the water beads up like this . some leaves of plants do it really well ; it 's quite cool . since water adheres weakly to the wax paper or to the plant , but strongly to itself , the water molecules are holding those droplets together in a configuration that creates the least amount of surface area . this is high surface tension that allows some bugs and even i think one lizard and also one jesus to be able to walk on water . the cohesive force of water does have its limits , of course . there are other substances that water quite likes to stick to . take glass , for example . this is called adhesion . the water is spreading out here instead of beading up because the adhesive forces between the water and the glass are stronger than the cohesive forces of the individual water molecules in the bead of water . adhesion is attraction between two different substances , so in this case the water molecules and the glass molecules . these properties lead to one of my favorite things about water ; the fact that it can defy gravity . that really cool thing that just happened is called capillary action . explaining it can be easily done with what we now know about cohesion and adhesion . thanks to adhesion , the water molecules are attracted to the molecules in the straw . as the water molecules adhere to the straw , other molecules are drawn in by cohesion following those fellow water molecules . thank you cohesion . the surface tension created here causes the water to climb up the straw . it will continue to climb until eventually gravity pulling down on the weight of the water in the straw overpowers the surface tension . the fact that water 's a polar molecule also makes it really good at dissolving things , which we call it 's a good solvent then . scratch that . water is n't a good solvent . it 's an amazing solvent . there are more substances that can be dissolved in water than in any other liquid on earth . yes , that includes the strongest acid that we have ever created . these substances that dissolve in water , sugar or salt being ones that we 're familiar with , are called hydrophilic , and they are hydrophilic because they are polar . their polarity is stronger than the cohesive forces of the water . when you get one of these polar substances in water , it 's strong enough that it breaks all the little cohesive forces , all those little hydrogen bonds . instead of hydrogen bonding to each other , the water will hydrogen bond around these polar substances . table salt is ionic , and right now it 's being separated into ions as the poles of our water molecules interact with it . what happens when there is a molecule that can not break the cohesive forces of water ? it ca n't penetrate and come into it . basically , what happens when that substance ca n't overcome the strong cohesive forces of water , ca n't get inside of the water ? that 's when we get what we call a hydrophobic substance , or something that is fearful of water . these molecules lack charged poles . they are non-polar and are not dissolving in water because essentially they 're being pushed out of the water by water 's cohesive forces . water , we may call it the universal solvent , but that does not mean that it dissolves everything . ( boppy music ) there have been a lot of eccentric scientists throughout history , but all this talk about water got me thinking about perhaps the most eccentric of the eccentrics , a man named henry cavendish . he communicated with his female servants only via notes , and added a staircase to the back of his house to avoid contact with his housekeeper . some believe he may have suffered from a form of autism , but just about everyone will admit that he was a scientific genius . he 's best remembered as the first person to recognize hydrogen gas as a distinct substance and to determine the composition of water . in the 1700s , most people thought that water itself was an element , but cavendish observed that hydrogen , which he called inflammable air , reacted with oxygen , known then by the awesome name , dephlogisticated air , to form water . cavendish did n't totally understand what he 'd discovered here , in part because he did n't believe in chemical compounds . he explained his experiments with hydrogen in terms of a fire-like element called phlagiston . nevertheless , his experiments were groundbreaking . like his work in determining the specific gravity basically the comparative density of hydrogen and other gases with reference to common air . it 's especially impressive when you consider the crude instruments he was working with . this , for example , is what he made his hydrogen gas with . he went on not only to establish an accurate composition of the atmosphere , but also discovered the density of the earth . not bad for a guy who was so painfully shy that the only existing portrait of him was sketched without his knowledge . for all of his decades of experiments , cavendish only published about 20 papers . in the years after his death , researchers figured out that cavendish had actually pre-discovered richter 's law , ohm 's law , coulomb 's law , several other laws . that 's a lot of freaking laws . if he had gotten credit for them all , we would have had to deal with like cavendish 's 8th law and cavendish 's 4th law , so i , for one , am glad that he did n't actually get credit . we 're going to do some pretty amazing science right now . you guys are not going to believe this . okay , you ready ? it floats ! yeah , i know you 're not surprised by this , but you should be because everything else , when it 's solid is much more dense than when it 's liquid ; just like gases are much less dense than liquids are . but that simple characteristic of water that its solid form floats is one of the reasons why life on this planet , as we know it , is possible . why is it that solid water is less dense than liquid water while everything else is the exact opposite of that ? well , you can thank your hydrogen bonds once again . at around 32 degrees fahrenheit or 0 degrees celsius if you 're a scientist or from a part of the world where things make sense , water molecules start to solidify and the hydrogen bonds in those water molecules form crystalline structures that space molecules apart more evenly , in turn making frozen water less dense than its liquid form . so in almost every circumstance a floating water ice is a really good thing . if ice were denser than water it would freeze and then sink , and then freeze and then sink , and then freeze and then sink , so just trust me on this one . you do n't want to live on a world where ice sinks . not only would it totally wreak havoc on aquatic eco systems which are basically how life formed on the earth in the first place . also , all the ice in the north pole would sink and then all of the water everywhere else would rise and we would n't have any land . that would be annoying . there 's one more amazing property of water i 'm forgetting . why is it so hot in here ? heat capacity ! yes , water has a very high heat capacity and probably that means nothing to you . basically it means that water is really good at holding on to heat , which is why we like to put hot water bottles in our bed and cuddle with them when we 're lonely . aside from artificially warming your bed , it 's also very important that it 's hard to heat up and cool down the oceans significantly . they become giant heat sinks that regulate the temperature and the climate of our planet , which is why , for example , it 's so much nicer in los angeles where the ocean is constantly keeping the temperatures the same than it is in , say , nebraska . on a smaller scale , we can see water 's high heat capacity really easily and visually by putting a pot with no water in it on a stove and seeing how badly that goes . but then you put a little bit of water in it and it takes forever to freaking boil ! and if you have n't already noticed this , when water evaporates from your skin , it cools you down . that 's the principle behind sweating , which is an extremely effective , though somewhat embarrassing , part of life . this is an example of another incredibly cool thing about water . when my body gets hot and it sweats , that heat excites some of the water molecules on my skin to the point where they break those hydrogen bonds and they evaporate away . when they escape , they take that heat energy with them , leaving me cooler . lovely . this was n't exercise , though . i do n't know why i 'm sweating so much . it could be the spray bottle that i keep spraying myself with , or maybe it 's just because this is such a high stress enterprise trying to teach you people things .
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these molecules lack charged poles . they are non-polar and are not dissolving in water because essentially they 're being pushed out of the water by water 's cohesive forces . water , we may call it the universal solvent , but that does not mean that it dissolves everything .
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why does water `` bead up '' on certain wax papers or plants ?
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man in glasses : hello , there . here at crash course hq we like to start out each day with a nice healthy dose of water in all its 3 forms . it 's the only substance on all of our planet earth that occurs naturally in solid , liquid and gas forms . to celebrate this magical bond between two hydrogen atoms and one oxygen atom , here today we are going to be celebrating the wonderful life-sustaining properties of water , but we 're going to do it slightly more clothed . ( boppy music ) ah , much better . when we left off here at the biology crash course we were talking about life , and the rather important fact that all life as we know it is dependent upon there being water around . scientists and astronomers are always looking out into the universe trying to figure out whether there is life elsewhere because that is kind of the most important question that we have right now . they 're always getting really excited when they find water someplace , particularly liquid water . this is one reason why i and so many other people geeked out so hard last december when mars ' seven-year-old rover opportunity found a 20-inch long vein of gypsum that was almost certainly deposited by long term liquid water on the surface of mars . this was probably billions of years ago , and so it 's going to be hard to tell whether or not the water that was there resulted in some life . maybe we can figure that out and that would be really exciting . why ? why do we think that water is necessary for life ? why does water on other planets get us so freaking excited ? let 's start out by investigating some of the amazing properties of water . in order to do that , we 're going to have to start out with this . the world 's most popular molecule , or at least the world 's most memorized molecule . we all know about it . good old h20 . two hydrogens , one oxygen , the hydrogens each sharing an electron with oxygen in what we call a covalent bond . as you can see , i 've drawn my water molecule in a particular way . this is actually the way that it appears . it is v-shaped . because this big old oxygen atom is a little bit more greedy for electrons , it has a slight negative charge ; whereas , this area here with the hydrogen atoms has a slight positive charge . thanks to this polarity , all water molecules are attracted to one another ; so much so that they actually stick together and these are called hydrogen bonds . we talked about them last time , but essentially what happens is that the positive pole around those hydrogen atoms bonds to the negative pole around the oxygen atoms of a different water molecule . it 's a weak bond , but look , they 're bonding ! seriously , i can not overstate the importance of this hydrogen bond . when your teacher asks you , `` what 's important about water ? '' start out with the hydrogen bonds and you should put it in all caps and maybe some sparkles around it . one of the cool properties that results from these hydrogen bonds is a high cohesion for water which results in high surface tension . cohesion is the attraction between two like things , like attraction between one molecule of water and another molecule of water . water has the highest cohesion of any nonmetalic liquid . you can see this if you put some water on some wax paper or some teflon or something where the water beads up like this . some leaves of plants do it really well ; it 's quite cool . since water adheres weakly to the wax paper or to the plant , but strongly to itself , the water molecules are holding those droplets together in a configuration that creates the least amount of surface area . this is high surface tension that allows some bugs and even i think one lizard and also one jesus to be able to walk on water . the cohesive force of water does have its limits , of course . there are other substances that water quite likes to stick to . take glass , for example . this is called adhesion . the water is spreading out here instead of beading up because the adhesive forces between the water and the glass are stronger than the cohesive forces of the individual water molecules in the bead of water . adhesion is attraction between two different substances , so in this case the water molecules and the glass molecules . these properties lead to one of my favorite things about water ; the fact that it can defy gravity . that really cool thing that just happened is called capillary action . explaining it can be easily done with what we now know about cohesion and adhesion . thanks to adhesion , the water molecules are attracted to the molecules in the straw . as the water molecules adhere to the straw , other molecules are drawn in by cohesion following those fellow water molecules . thank you cohesion . the surface tension created here causes the water to climb up the straw . it will continue to climb until eventually gravity pulling down on the weight of the water in the straw overpowers the surface tension . the fact that water 's a polar molecule also makes it really good at dissolving things , which we call it 's a good solvent then . scratch that . water is n't a good solvent . it 's an amazing solvent . there are more substances that can be dissolved in water than in any other liquid on earth . yes , that includes the strongest acid that we have ever created . these substances that dissolve in water , sugar or salt being ones that we 're familiar with , are called hydrophilic , and they are hydrophilic because they are polar . their polarity is stronger than the cohesive forces of the water . when you get one of these polar substances in water , it 's strong enough that it breaks all the little cohesive forces , all those little hydrogen bonds . instead of hydrogen bonding to each other , the water will hydrogen bond around these polar substances . table salt is ionic , and right now it 's being separated into ions as the poles of our water molecules interact with it . what happens when there is a molecule that can not break the cohesive forces of water ? it ca n't penetrate and come into it . basically , what happens when that substance ca n't overcome the strong cohesive forces of water , ca n't get inside of the water ? that 's when we get what we call a hydrophobic substance , or something that is fearful of water . these molecules lack charged poles . they are non-polar and are not dissolving in water because essentially they 're being pushed out of the water by water 's cohesive forces . water , we may call it the universal solvent , but that does not mean that it dissolves everything . ( boppy music ) there have been a lot of eccentric scientists throughout history , but all this talk about water got me thinking about perhaps the most eccentric of the eccentrics , a man named henry cavendish . he communicated with his female servants only via notes , and added a staircase to the back of his house to avoid contact with his housekeeper . some believe he may have suffered from a form of autism , but just about everyone will admit that he was a scientific genius . he 's best remembered as the first person to recognize hydrogen gas as a distinct substance and to determine the composition of water . in the 1700s , most people thought that water itself was an element , but cavendish observed that hydrogen , which he called inflammable air , reacted with oxygen , known then by the awesome name , dephlogisticated air , to form water . cavendish did n't totally understand what he 'd discovered here , in part because he did n't believe in chemical compounds . he explained his experiments with hydrogen in terms of a fire-like element called phlagiston . nevertheless , his experiments were groundbreaking . like his work in determining the specific gravity basically the comparative density of hydrogen and other gases with reference to common air . it 's especially impressive when you consider the crude instruments he was working with . this , for example , is what he made his hydrogen gas with . he went on not only to establish an accurate composition of the atmosphere , but also discovered the density of the earth . not bad for a guy who was so painfully shy that the only existing portrait of him was sketched without his knowledge . for all of his decades of experiments , cavendish only published about 20 papers . in the years after his death , researchers figured out that cavendish had actually pre-discovered richter 's law , ohm 's law , coulomb 's law , several other laws . that 's a lot of freaking laws . if he had gotten credit for them all , we would have had to deal with like cavendish 's 8th law and cavendish 's 4th law , so i , for one , am glad that he did n't actually get credit . we 're going to do some pretty amazing science right now . you guys are not going to believe this . okay , you ready ? it floats ! yeah , i know you 're not surprised by this , but you should be because everything else , when it 's solid is much more dense than when it 's liquid ; just like gases are much less dense than liquids are . but that simple characteristic of water that its solid form floats is one of the reasons why life on this planet , as we know it , is possible . why is it that solid water is less dense than liquid water while everything else is the exact opposite of that ? well , you can thank your hydrogen bonds once again . at around 32 degrees fahrenheit or 0 degrees celsius if you 're a scientist or from a part of the world where things make sense , water molecules start to solidify and the hydrogen bonds in those water molecules form crystalline structures that space molecules apart more evenly , in turn making frozen water less dense than its liquid form . so in almost every circumstance a floating water ice is a really good thing . if ice were denser than water it would freeze and then sink , and then freeze and then sink , and then freeze and then sink , so just trust me on this one . you do n't want to live on a world where ice sinks . not only would it totally wreak havoc on aquatic eco systems which are basically how life formed on the earth in the first place . also , all the ice in the north pole would sink and then all of the water everywhere else would rise and we would n't have any land . that would be annoying . there 's one more amazing property of water i 'm forgetting . why is it so hot in here ? heat capacity ! yes , water has a very high heat capacity and probably that means nothing to you . basically it means that water is really good at holding on to heat , which is why we like to put hot water bottles in our bed and cuddle with them when we 're lonely . aside from artificially warming your bed , it 's also very important that it 's hard to heat up and cool down the oceans significantly . they become giant heat sinks that regulate the temperature and the climate of our planet , which is why , for example , it 's so much nicer in los angeles where the ocean is constantly keeping the temperatures the same than it is in , say , nebraska . on a smaller scale , we can see water 's high heat capacity really easily and visually by putting a pot with no water in it on a stove and seeing how badly that goes . but then you put a little bit of water in it and it takes forever to freaking boil ! and if you have n't already noticed this , when water evaporates from your skin , it cools you down . that 's the principle behind sweating , which is an extremely effective , though somewhat embarrassing , part of life . this is an example of another incredibly cool thing about water . when my body gets hot and it sweats , that heat excites some of the water molecules on my skin to the point where they break those hydrogen bonds and they evaporate away . when they escape , they take that heat energy with them , leaving me cooler . lovely . this was n't exercise , though . i do n't know why i 'm sweating so much . it could be the spray bottle that i keep spraying myself with , or maybe it 's just because this is such a high stress enterprise trying to teach you people things .
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thanks to adhesion , the water molecules are attracted to the molecules in the straw . as the water molecules adhere to the straw , other molecules are drawn in by cohesion following those fellow water molecules . thank you cohesion .
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if ice is less dense than water , and the ice molecules are spread out , would that mean that part of the reason why it floats is not only from low density , but from the molecules are spread out more , spreading the density equally , too ?
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man in glasses : hello , there . here at crash course hq we like to start out each day with a nice healthy dose of water in all its 3 forms . it 's the only substance on all of our planet earth that occurs naturally in solid , liquid and gas forms . to celebrate this magical bond between two hydrogen atoms and one oxygen atom , here today we are going to be celebrating the wonderful life-sustaining properties of water , but we 're going to do it slightly more clothed . ( boppy music ) ah , much better . when we left off here at the biology crash course we were talking about life , and the rather important fact that all life as we know it is dependent upon there being water around . scientists and astronomers are always looking out into the universe trying to figure out whether there is life elsewhere because that is kind of the most important question that we have right now . they 're always getting really excited when they find water someplace , particularly liquid water . this is one reason why i and so many other people geeked out so hard last december when mars ' seven-year-old rover opportunity found a 20-inch long vein of gypsum that was almost certainly deposited by long term liquid water on the surface of mars . this was probably billions of years ago , and so it 's going to be hard to tell whether or not the water that was there resulted in some life . maybe we can figure that out and that would be really exciting . why ? why do we think that water is necessary for life ? why does water on other planets get us so freaking excited ? let 's start out by investigating some of the amazing properties of water . in order to do that , we 're going to have to start out with this . the world 's most popular molecule , or at least the world 's most memorized molecule . we all know about it . good old h20 . two hydrogens , one oxygen , the hydrogens each sharing an electron with oxygen in what we call a covalent bond . as you can see , i 've drawn my water molecule in a particular way . this is actually the way that it appears . it is v-shaped . because this big old oxygen atom is a little bit more greedy for electrons , it has a slight negative charge ; whereas , this area here with the hydrogen atoms has a slight positive charge . thanks to this polarity , all water molecules are attracted to one another ; so much so that they actually stick together and these are called hydrogen bonds . we talked about them last time , but essentially what happens is that the positive pole around those hydrogen atoms bonds to the negative pole around the oxygen atoms of a different water molecule . it 's a weak bond , but look , they 're bonding ! seriously , i can not overstate the importance of this hydrogen bond . when your teacher asks you , `` what 's important about water ? '' start out with the hydrogen bonds and you should put it in all caps and maybe some sparkles around it . one of the cool properties that results from these hydrogen bonds is a high cohesion for water which results in high surface tension . cohesion is the attraction between two like things , like attraction between one molecule of water and another molecule of water . water has the highest cohesion of any nonmetalic liquid . you can see this if you put some water on some wax paper or some teflon or something where the water beads up like this . some leaves of plants do it really well ; it 's quite cool . since water adheres weakly to the wax paper or to the plant , but strongly to itself , the water molecules are holding those droplets together in a configuration that creates the least amount of surface area . this is high surface tension that allows some bugs and even i think one lizard and also one jesus to be able to walk on water . the cohesive force of water does have its limits , of course . there are other substances that water quite likes to stick to . take glass , for example . this is called adhesion . the water is spreading out here instead of beading up because the adhesive forces between the water and the glass are stronger than the cohesive forces of the individual water molecules in the bead of water . adhesion is attraction between two different substances , so in this case the water molecules and the glass molecules . these properties lead to one of my favorite things about water ; the fact that it can defy gravity . that really cool thing that just happened is called capillary action . explaining it can be easily done with what we now know about cohesion and adhesion . thanks to adhesion , the water molecules are attracted to the molecules in the straw . as the water molecules adhere to the straw , other molecules are drawn in by cohesion following those fellow water molecules . thank you cohesion . the surface tension created here causes the water to climb up the straw . it will continue to climb until eventually gravity pulling down on the weight of the water in the straw overpowers the surface tension . the fact that water 's a polar molecule also makes it really good at dissolving things , which we call it 's a good solvent then . scratch that . water is n't a good solvent . it 's an amazing solvent . there are more substances that can be dissolved in water than in any other liquid on earth . yes , that includes the strongest acid that we have ever created . these substances that dissolve in water , sugar or salt being ones that we 're familiar with , are called hydrophilic , and they are hydrophilic because they are polar . their polarity is stronger than the cohesive forces of the water . when you get one of these polar substances in water , it 's strong enough that it breaks all the little cohesive forces , all those little hydrogen bonds . instead of hydrogen bonding to each other , the water will hydrogen bond around these polar substances . table salt is ionic , and right now it 's being separated into ions as the poles of our water molecules interact with it . what happens when there is a molecule that can not break the cohesive forces of water ? it ca n't penetrate and come into it . basically , what happens when that substance ca n't overcome the strong cohesive forces of water , ca n't get inside of the water ? that 's when we get what we call a hydrophobic substance , or something that is fearful of water . these molecules lack charged poles . they are non-polar and are not dissolving in water because essentially they 're being pushed out of the water by water 's cohesive forces . water , we may call it the universal solvent , but that does not mean that it dissolves everything . ( boppy music ) there have been a lot of eccentric scientists throughout history , but all this talk about water got me thinking about perhaps the most eccentric of the eccentrics , a man named henry cavendish . he communicated with his female servants only via notes , and added a staircase to the back of his house to avoid contact with his housekeeper . some believe he may have suffered from a form of autism , but just about everyone will admit that he was a scientific genius . he 's best remembered as the first person to recognize hydrogen gas as a distinct substance and to determine the composition of water . in the 1700s , most people thought that water itself was an element , but cavendish observed that hydrogen , which he called inflammable air , reacted with oxygen , known then by the awesome name , dephlogisticated air , to form water . cavendish did n't totally understand what he 'd discovered here , in part because he did n't believe in chemical compounds . he explained his experiments with hydrogen in terms of a fire-like element called phlagiston . nevertheless , his experiments were groundbreaking . like his work in determining the specific gravity basically the comparative density of hydrogen and other gases with reference to common air . it 's especially impressive when you consider the crude instruments he was working with . this , for example , is what he made his hydrogen gas with . he went on not only to establish an accurate composition of the atmosphere , but also discovered the density of the earth . not bad for a guy who was so painfully shy that the only existing portrait of him was sketched without his knowledge . for all of his decades of experiments , cavendish only published about 20 papers . in the years after his death , researchers figured out that cavendish had actually pre-discovered richter 's law , ohm 's law , coulomb 's law , several other laws . that 's a lot of freaking laws . if he had gotten credit for them all , we would have had to deal with like cavendish 's 8th law and cavendish 's 4th law , so i , for one , am glad that he did n't actually get credit . we 're going to do some pretty amazing science right now . you guys are not going to believe this . okay , you ready ? it floats ! yeah , i know you 're not surprised by this , but you should be because everything else , when it 's solid is much more dense than when it 's liquid ; just like gases are much less dense than liquids are . but that simple characteristic of water that its solid form floats is one of the reasons why life on this planet , as we know it , is possible . why is it that solid water is less dense than liquid water while everything else is the exact opposite of that ? well , you can thank your hydrogen bonds once again . at around 32 degrees fahrenheit or 0 degrees celsius if you 're a scientist or from a part of the world where things make sense , water molecules start to solidify and the hydrogen bonds in those water molecules form crystalline structures that space molecules apart more evenly , in turn making frozen water less dense than its liquid form . so in almost every circumstance a floating water ice is a really good thing . if ice were denser than water it would freeze and then sink , and then freeze and then sink , and then freeze and then sink , so just trust me on this one . you do n't want to live on a world where ice sinks . not only would it totally wreak havoc on aquatic eco systems which are basically how life formed on the earth in the first place . also , all the ice in the north pole would sink and then all of the water everywhere else would rise and we would n't have any land . that would be annoying . there 's one more amazing property of water i 'm forgetting . why is it so hot in here ? heat capacity ! yes , water has a very high heat capacity and probably that means nothing to you . basically it means that water is really good at holding on to heat , which is why we like to put hot water bottles in our bed and cuddle with them when we 're lonely . aside from artificially warming your bed , it 's also very important that it 's hard to heat up and cool down the oceans significantly . they become giant heat sinks that regulate the temperature and the climate of our planet , which is why , for example , it 's so much nicer in los angeles where the ocean is constantly keeping the temperatures the same than it is in , say , nebraska . on a smaller scale , we can see water 's high heat capacity really easily and visually by putting a pot with no water in it on a stove and seeing how badly that goes . but then you put a little bit of water in it and it takes forever to freaking boil ! and if you have n't already noticed this , when water evaporates from your skin , it cools you down . that 's the principle behind sweating , which is an extremely effective , though somewhat embarrassing , part of life . this is an example of another incredibly cool thing about water . when my body gets hot and it sweats , that heat excites some of the water molecules on my skin to the point where they break those hydrogen bonds and they evaporate away . when they escape , they take that heat energy with them , leaving me cooler . lovely . this was n't exercise , though . i do n't know why i 'm sweating so much . it could be the spray bottle that i keep spraying myself with , or maybe it 's just because this is such a high stress enterprise trying to teach you people things .
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that 's the principle behind sweating , which is an extremely effective , though somewhat embarrassing , part of life . this is an example of another incredibly cool thing about water . when my body gets hot and it sweats , that heat excites some of the water molecules on my skin to the point where they break those hydrogen bonds and they evaporate away .
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why is water the only thing on earth that exists in three different forms ?
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man in glasses : hello , there . here at crash course hq we like to start out each day with a nice healthy dose of water in all its 3 forms . it 's the only substance on all of our planet earth that occurs naturally in solid , liquid and gas forms . to celebrate this magical bond between two hydrogen atoms and one oxygen atom , here today we are going to be celebrating the wonderful life-sustaining properties of water , but we 're going to do it slightly more clothed . ( boppy music ) ah , much better . when we left off here at the biology crash course we were talking about life , and the rather important fact that all life as we know it is dependent upon there being water around . scientists and astronomers are always looking out into the universe trying to figure out whether there is life elsewhere because that is kind of the most important question that we have right now . they 're always getting really excited when they find water someplace , particularly liquid water . this is one reason why i and so many other people geeked out so hard last december when mars ' seven-year-old rover opportunity found a 20-inch long vein of gypsum that was almost certainly deposited by long term liquid water on the surface of mars . this was probably billions of years ago , and so it 's going to be hard to tell whether or not the water that was there resulted in some life . maybe we can figure that out and that would be really exciting . why ? why do we think that water is necessary for life ? why does water on other planets get us so freaking excited ? let 's start out by investigating some of the amazing properties of water . in order to do that , we 're going to have to start out with this . the world 's most popular molecule , or at least the world 's most memorized molecule . we all know about it . good old h20 . two hydrogens , one oxygen , the hydrogens each sharing an electron with oxygen in what we call a covalent bond . as you can see , i 've drawn my water molecule in a particular way . this is actually the way that it appears . it is v-shaped . because this big old oxygen atom is a little bit more greedy for electrons , it has a slight negative charge ; whereas , this area here with the hydrogen atoms has a slight positive charge . thanks to this polarity , all water molecules are attracted to one another ; so much so that they actually stick together and these are called hydrogen bonds . we talked about them last time , but essentially what happens is that the positive pole around those hydrogen atoms bonds to the negative pole around the oxygen atoms of a different water molecule . it 's a weak bond , but look , they 're bonding ! seriously , i can not overstate the importance of this hydrogen bond . when your teacher asks you , `` what 's important about water ? '' start out with the hydrogen bonds and you should put it in all caps and maybe some sparkles around it . one of the cool properties that results from these hydrogen bonds is a high cohesion for water which results in high surface tension . cohesion is the attraction between two like things , like attraction between one molecule of water and another molecule of water . water has the highest cohesion of any nonmetalic liquid . you can see this if you put some water on some wax paper or some teflon or something where the water beads up like this . some leaves of plants do it really well ; it 's quite cool . since water adheres weakly to the wax paper or to the plant , but strongly to itself , the water molecules are holding those droplets together in a configuration that creates the least amount of surface area . this is high surface tension that allows some bugs and even i think one lizard and also one jesus to be able to walk on water . the cohesive force of water does have its limits , of course . there are other substances that water quite likes to stick to . take glass , for example . this is called adhesion . the water is spreading out here instead of beading up because the adhesive forces between the water and the glass are stronger than the cohesive forces of the individual water molecules in the bead of water . adhesion is attraction between two different substances , so in this case the water molecules and the glass molecules . these properties lead to one of my favorite things about water ; the fact that it can defy gravity . that really cool thing that just happened is called capillary action . explaining it can be easily done with what we now know about cohesion and adhesion . thanks to adhesion , the water molecules are attracted to the molecules in the straw . as the water molecules adhere to the straw , other molecules are drawn in by cohesion following those fellow water molecules . thank you cohesion . the surface tension created here causes the water to climb up the straw . it will continue to climb until eventually gravity pulling down on the weight of the water in the straw overpowers the surface tension . the fact that water 's a polar molecule also makes it really good at dissolving things , which we call it 's a good solvent then . scratch that . water is n't a good solvent . it 's an amazing solvent . there are more substances that can be dissolved in water than in any other liquid on earth . yes , that includes the strongest acid that we have ever created . these substances that dissolve in water , sugar or salt being ones that we 're familiar with , are called hydrophilic , and they are hydrophilic because they are polar . their polarity is stronger than the cohesive forces of the water . when you get one of these polar substances in water , it 's strong enough that it breaks all the little cohesive forces , all those little hydrogen bonds . instead of hydrogen bonding to each other , the water will hydrogen bond around these polar substances . table salt is ionic , and right now it 's being separated into ions as the poles of our water molecules interact with it . what happens when there is a molecule that can not break the cohesive forces of water ? it ca n't penetrate and come into it . basically , what happens when that substance ca n't overcome the strong cohesive forces of water , ca n't get inside of the water ? that 's when we get what we call a hydrophobic substance , or something that is fearful of water . these molecules lack charged poles . they are non-polar and are not dissolving in water because essentially they 're being pushed out of the water by water 's cohesive forces . water , we may call it the universal solvent , but that does not mean that it dissolves everything . ( boppy music ) there have been a lot of eccentric scientists throughout history , but all this talk about water got me thinking about perhaps the most eccentric of the eccentrics , a man named henry cavendish . he communicated with his female servants only via notes , and added a staircase to the back of his house to avoid contact with his housekeeper . some believe he may have suffered from a form of autism , but just about everyone will admit that he was a scientific genius . he 's best remembered as the first person to recognize hydrogen gas as a distinct substance and to determine the composition of water . in the 1700s , most people thought that water itself was an element , but cavendish observed that hydrogen , which he called inflammable air , reacted with oxygen , known then by the awesome name , dephlogisticated air , to form water . cavendish did n't totally understand what he 'd discovered here , in part because he did n't believe in chemical compounds . he explained his experiments with hydrogen in terms of a fire-like element called phlagiston . nevertheless , his experiments were groundbreaking . like his work in determining the specific gravity basically the comparative density of hydrogen and other gases with reference to common air . it 's especially impressive when you consider the crude instruments he was working with . this , for example , is what he made his hydrogen gas with . he went on not only to establish an accurate composition of the atmosphere , but also discovered the density of the earth . not bad for a guy who was so painfully shy that the only existing portrait of him was sketched without his knowledge . for all of his decades of experiments , cavendish only published about 20 papers . in the years after his death , researchers figured out that cavendish had actually pre-discovered richter 's law , ohm 's law , coulomb 's law , several other laws . that 's a lot of freaking laws . if he had gotten credit for them all , we would have had to deal with like cavendish 's 8th law and cavendish 's 4th law , so i , for one , am glad that he did n't actually get credit . we 're going to do some pretty amazing science right now . you guys are not going to believe this . okay , you ready ? it floats ! yeah , i know you 're not surprised by this , but you should be because everything else , when it 's solid is much more dense than when it 's liquid ; just like gases are much less dense than liquids are . but that simple characteristic of water that its solid form floats is one of the reasons why life on this planet , as we know it , is possible . why is it that solid water is less dense than liquid water while everything else is the exact opposite of that ? well , you can thank your hydrogen bonds once again . at around 32 degrees fahrenheit or 0 degrees celsius if you 're a scientist or from a part of the world where things make sense , water molecules start to solidify and the hydrogen bonds in those water molecules form crystalline structures that space molecules apart more evenly , in turn making frozen water less dense than its liquid form . so in almost every circumstance a floating water ice is a really good thing . if ice were denser than water it would freeze and then sink , and then freeze and then sink , and then freeze and then sink , so just trust me on this one . you do n't want to live on a world where ice sinks . not only would it totally wreak havoc on aquatic eco systems which are basically how life formed on the earth in the first place . also , all the ice in the north pole would sink and then all of the water everywhere else would rise and we would n't have any land . that would be annoying . there 's one more amazing property of water i 'm forgetting . why is it so hot in here ? heat capacity ! yes , water has a very high heat capacity and probably that means nothing to you . basically it means that water is really good at holding on to heat , which is why we like to put hot water bottles in our bed and cuddle with them when we 're lonely . aside from artificially warming your bed , it 's also very important that it 's hard to heat up and cool down the oceans significantly . they become giant heat sinks that regulate the temperature and the climate of our planet , which is why , for example , it 's so much nicer in los angeles where the ocean is constantly keeping the temperatures the same than it is in , say , nebraska . on a smaller scale , we can see water 's high heat capacity really easily and visually by putting a pot with no water in it on a stove and seeing how badly that goes . but then you put a little bit of water in it and it takes forever to freaking boil ! and if you have n't already noticed this , when water evaporates from your skin , it cools you down . that 's the principle behind sweating , which is an extremely effective , though somewhat embarrassing , part of life . this is an example of another incredibly cool thing about water . when my body gets hot and it sweats , that heat excites some of the water molecules on my skin to the point where they break those hydrogen bonds and they evaporate away . when they escape , they take that heat energy with them , leaving me cooler . lovely . this was n't exercise , though . i do n't know why i 'm sweating so much . it could be the spray bottle that i keep spraying myself with , or maybe it 's just because this is such a high stress enterprise trying to teach you people things .
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why ? why do we think that water is necessary for life ? why does water on other planets get us so freaking excited ?
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how did earth produce so much water to support life ?
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man in glasses : hello , there . here at crash course hq we like to start out each day with a nice healthy dose of water in all its 3 forms . it 's the only substance on all of our planet earth that occurs naturally in solid , liquid and gas forms . to celebrate this magical bond between two hydrogen atoms and one oxygen atom , here today we are going to be celebrating the wonderful life-sustaining properties of water , but we 're going to do it slightly more clothed . ( boppy music ) ah , much better . when we left off here at the biology crash course we were talking about life , and the rather important fact that all life as we know it is dependent upon there being water around . scientists and astronomers are always looking out into the universe trying to figure out whether there is life elsewhere because that is kind of the most important question that we have right now . they 're always getting really excited when they find water someplace , particularly liquid water . this is one reason why i and so many other people geeked out so hard last december when mars ' seven-year-old rover opportunity found a 20-inch long vein of gypsum that was almost certainly deposited by long term liquid water on the surface of mars . this was probably billions of years ago , and so it 's going to be hard to tell whether or not the water that was there resulted in some life . maybe we can figure that out and that would be really exciting . why ? why do we think that water is necessary for life ? why does water on other planets get us so freaking excited ? let 's start out by investigating some of the amazing properties of water . in order to do that , we 're going to have to start out with this . the world 's most popular molecule , or at least the world 's most memorized molecule . we all know about it . good old h20 . two hydrogens , one oxygen , the hydrogens each sharing an electron with oxygen in what we call a covalent bond . as you can see , i 've drawn my water molecule in a particular way . this is actually the way that it appears . it is v-shaped . because this big old oxygen atom is a little bit more greedy for electrons , it has a slight negative charge ; whereas , this area here with the hydrogen atoms has a slight positive charge . thanks to this polarity , all water molecules are attracted to one another ; so much so that they actually stick together and these are called hydrogen bonds . we talked about them last time , but essentially what happens is that the positive pole around those hydrogen atoms bonds to the negative pole around the oxygen atoms of a different water molecule . it 's a weak bond , but look , they 're bonding ! seriously , i can not overstate the importance of this hydrogen bond . when your teacher asks you , `` what 's important about water ? '' start out with the hydrogen bonds and you should put it in all caps and maybe some sparkles around it . one of the cool properties that results from these hydrogen bonds is a high cohesion for water which results in high surface tension . cohesion is the attraction between two like things , like attraction between one molecule of water and another molecule of water . water has the highest cohesion of any nonmetalic liquid . you can see this if you put some water on some wax paper or some teflon or something where the water beads up like this . some leaves of plants do it really well ; it 's quite cool . since water adheres weakly to the wax paper or to the plant , but strongly to itself , the water molecules are holding those droplets together in a configuration that creates the least amount of surface area . this is high surface tension that allows some bugs and even i think one lizard and also one jesus to be able to walk on water . the cohesive force of water does have its limits , of course . there are other substances that water quite likes to stick to . take glass , for example . this is called adhesion . the water is spreading out here instead of beading up because the adhesive forces between the water and the glass are stronger than the cohesive forces of the individual water molecules in the bead of water . adhesion is attraction between two different substances , so in this case the water molecules and the glass molecules . these properties lead to one of my favorite things about water ; the fact that it can defy gravity . that really cool thing that just happened is called capillary action . explaining it can be easily done with what we now know about cohesion and adhesion . thanks to adhesion , the water molecules are attracted to the molecules in the straw . as the water molecules adhere to the straw , other molecules are drawn in by cohesion following those fellow water molecules . thank you cohesion . the surface tension created here causes the water to climb up the straw . it will continue to climb until eventually gravity pulling down on the weight of the water in the straw overpowers the surface tension . the fact that water 's a polar molecule also makes it really good at dissolving things , which we call it 's a good solvent then . scratch that . water is n't a good solvent . it 's an amazing solvent . there are more substances that can be dissolved in water than in any other liquid on earth . yes , that includes the strongest acid that we have ever created . these substances that dissolve in water , sugar or salt being ones that we 're familiar with , are called hydrophilic , and they are hydrophilic because they are polar . their polarity is stronger than the cohesive forces of the water . when you get one of these polar substances in water , it 's strong enough that it breaks all the little cohesive forces , all those little hydrogen bonds . instead of hydrogen bonding to each other , the water will hydrogen bond around these polar substances . table salt is ionic , and right now it 's being separated into ions as the poles of our water molecules interact with it . what happens when there is a molecule that can not break the cohesive forces of water ? it ca n't penetrate and come into it . basically , what happens when that substance ca n't overcome the strong cohesive forces of water , ca n't get inside of the water ? that 's when we get what we call a hydrophobic substance , or something that is fearful of water . these molecules lack charged poles . they are non-polar and are not dissolving in water because essentially they 're being pushed out of the water by water 's cohesive forces . water , we may call it the universal solvent , but that does not mean that it dissolves everything . ( boppy music ) there have been a lot of eccentric scientists throughout history , but all this talk about water got me thinking about perhaps the most eccentric of the eccentrics , a man named henry cavendish . he communicated with his female servants only via notes , and added a staircase to the back of his house to avoid contact with his housekeeper . some believe he may have suffered from a form of autism , but just about everyone will admit that he was a scientific genius . he 's best remembered as the first person to recognize hydrogen gas as a distinct substance and to determine the composition of water . in the 1700s , most people thought that water itself was an element , but cavendish observed that hydrogen , which he called inflammable air , reacted with oxygen , known then by the awesome name , dephlogisticated air , to form water . cavendish did n't totally understand what he 'd discovered here , in part because he did n't believe in chemical compounds . he explained his experiments with hydrogen in terms of a fire-like element called phlagiston . nevertheless , his experiments were groundbreaking . like his work in determining the specific gravity basically the comparative density of hydrogen and other gases with reference to common air . it 's especially impressive when you consider the crude instruments he was working with . this , for example , is what he made his hydrogen gas with . he went on not only to establish an accurate composition of the atmosphere , but also discovered the density of the earth . not bad for a guy who was so painfully shy that the only existing portrait of him was sketched without his knowledge . for all of his decades of experiments , cavendish only published about 20 papers . in the years after his death , researchers figured out that cavendish had actually pre-discovered richter 's law , ohm 's law , coulomb 's law , several other laws . that 's a lot of freaking laws . if he had gotten credit for them all , we would have had to deal with like cavendish 's 8th law and cavendish 's 4th law , so i , for one , am glad that he did n't actually get credit . we 're going to do some pretty amazing science right now . you guys are not going to believe this . okay , you ready ? it floats ! yeah , i know you 're not surprised by this , but you should be because everything else , when it 's solid is much more dense than when it 's liquid ; just like gases are much less dense than liquids are . but that simple characteristic of water that its solid form floats is one of the reasons why life on this planet , as we know it , is possible . why is it that solid water is less dense than liquid water while everything else is the exact opposite of that ? well , you can thank your hydrogen bonds once again . at around 32 degrees fahrenheit or 0 degrees celsius if you 're a scientist or from a part of the world where things make sense , water molecules start to solidify and the hydrogen bonds in those water molecules form crystalline structures that space molecules apart more evenly , in turn making frozen water less dense than its liquid form . so in almost every circumstance a floating water ice is a really good thing . if ice were denser than water it would freeze and then sink , and then freeze and then sink , and then freeze and then sink , so just trust me on this one . you do n't want to live on a world where ice sinks . not only would it totally wreak havoc on aquatic eco systems which are basically how life formed on the earth in the first place . also , all the ice in the north pole would sink and then all of the water everywhere else would rise and we would n't have any land . that would be annoying . there 's one more amazing property of water i 'm forgetting . why is it so hot in here ? heat capacity ! yes , water has a very high heat capacity and probably that means nothing to you . basically it means that water is really good at holding on to heat , which is why we like to put hot water bottles in our bed and cuddle with them when we 're lonely . aside from artificially warming your bed , it 's also very important that it 's hard to heat up and cool down the oceans significantly . they become giant heat sinks that regulate the temperature and the climate of our planet , which is why , for example , it 's so much nicer in los angeles where the ocean is constantly keeping the temperatures the same than it is in , say , nebraska . on a smaller scale , we can see water 's high heat capacity really easily and visually by putting a pot with no water in it on a stove and seeing how badly that goes . but then you put a little bit of water in it and it takes forever to freaking boil ! and if you have n't already noticed this , when water evaporates from your skin , it cools you down . that 's the principle behind sweating , which is an extremely effective , though somewhat embarrassing , part of life . this is an example of another incredibly cool thing about water . when my body gets hot and it sweats , that heat excites some of the water molecules on my skin to the point where they break those hydrogen bonds and they evaporate away . when they escape , they take that heat energy with them , leaving me cooler . lovely . this was n't exercise , though . i do n't know why i 'm sweating so much . it could be the spray bottle that i keep spraying myself with , or maybe it 's just because this is such a high stress enterprise trying to teach you people things .
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why is it so hot in here ? heat capacity ! yes , water has a very high heat capacity and probably that means nothing to you .
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is heat capacity , as mentioned another term for specific heat or is there a difference between the two ?
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man in glasses : hello , there . here at crash course hq we like to start out each day with a nice healthy dose of water in all its 3 forms . it 's the only substance on all of our planet earth that occurs naturally in solid , liquid and gas forms . to celebrate this magical bond between two hydrogen atoms and one oxygen atom , here today we are going to be celebrating the wonderful life-sustaining properties of water , but we 're going to do it slightly more clothed . ( boppy music ) ah , much better . when we left off here at the biology crash course we were talking about life , and the rather important fact that all life as we know it is dependent upon there being water around . scientists and astronomers are always looking out into the universe trying to figure out whether there is life elsewhere because that is kind of the most important question that we have right now . they 're always getting really excited when they find water someplace , particularly liquid water . this is one reason why i and so many other people geeked out so hard last december when mars ' seven-year-old rover opportunity found a 20-inch long vein of gypsum that was almost certainly deposited by long term liquid water on the surface of mars . this was probably billions of years ago , and so it 's going to be hard to tell whether or not the water that was there resulted in some life . maybe we can figure that out and that would be really exciting . why ? why do we think that water is necessary for life ? why does water on other planets get us so freaking excited ? let 's start out by investigating some of the amazing properties of water . in order to do that , we 're going to have to start out with this . the world 's most popular molecule , or at least the world 's most memorized molecule . we all know about it . good old h20 . two hydrogens , one oxygen , the hydrogens each sharing an electron with oxygen in what we call a covalent bond . as you can see , i 've drawn my water molecule in a particular way . this is actually the way that it appears . it is v-shaped . because this big old oxygen atom is a little bit more greedy for electrons , it has a slight negative charge ; whereas , this area here with the hydrogen atoms has a slight positive charge . thanks to this polarity , all water molecules are attracted to one another ; so much so that they actually stick together and these are called hydrogen bonds . we talked about them last time , but essentially what happens is that the positive pole around those hydrogen atoms bonds to the negative pole around the oxygen atoms of a different water molecule . it 's a weak bond , but look , they 're bonding ! seriously , i can not overstate the importance of this hydrogen bond . when your teacher asks you , `` what 's important about water ? '' start out with the hydrogen bonds and you should put it in all caps and maybe some sparkles around it . one of the cool properties that results from these hydrogen bonds is a high cohesion for water which results in high surface tension . cohesion is the attraction between two like things , like attraction between one molecule of water and another molecule of water . water has the highest cohesion of any nonmetalic liquid . you can see this if you put some water on some wax paper or some teflon or something where the water beads up like this . some leaves of plants do it really well ; it 's quite cool . since water adheres weakly to the wax paper or to the plant , but strongly to itself , the water molecules are holding those droplets together in a configuration that creates the least amount of surface area . this is high surface tension that allows some bugs and even i think one lizard and also one jesus to be able to walk on water . the cohesive force of water does have its limits , of course . there are other substances that water quite likes to stick to . take glass , for example . this is called adhesion . the water is spreading out here instead of beading up because the adhesive forces between the water and the glass are stronger than the cohesive forces of the individual water molecules in the bead of water . adhesion is attraction between two different substances , so in this case the water molecules and the glass molecules . these properties lead to one of my favorite things about water ; the fact that it can defy gravity . that really cool thing that just happened is called capillary action . explaining it can be easily done with what we now know about cohesion and adhesion . thanks to adhesion , the water molecules are attracted to the molecules in the straw . as the water molecules adhere to the straw , other molecules are drawn in by cohesion following those fellow water molecules . thank you cohesion . the surface tension created here causes the water to climb up the straw . it will continue to climb until eventually gravity pulling down on the weight of the water in the straw overpowers the surface tension . the fact that water 's a polar molecule also makes it really good at dissolving things , which we call it 's a good solvent then . scratch that . water is n't a good solvent . it 's an amazing solvent . there are more substances that can be dissolved in water than in any other liquid on earth . yes , that includes the strongest acid that we have ever created . these substances that dissolve in water , sugar or salt being ones that we 're familiar with , are called hydrophilic , and they are hydrophilic because they are polar . their polarity is stronger than the cohesive forces of the water . when you get one of these polar substances in water , it 's strong enough that it breaks all the little cohesive forces , all those little hydrogen bonds . instead of hydrogen bonding to each other , the water will hydrogen bond around these polar substances . table salt is ionic , and right now it 's being separated into ions as the poles of our water molecules interact with it . what happens when there is a molecule that can not break the cohesive forces of water ? it ca n't penetrate and come into it . basically , what happens when that substance ca n't overcome the strong cohesive forces of water , ca n't get inside of the water ? that 's when we get what we call a hydrophobic substance , or something that is fearful of water . these molecules lack charged poles . they are non-polar and are not dissolving in water because essentially they 're being pushed out of the water by water 's cohesive forces . water , we may call it the universal solvent , but that does not mean that it dissolves everything . ( boppy music ) there have been a lot of eccentric scientists throughout history , but all this talk about water got me thinking about perhaps the most eccentric of the eccentrics , a man named henry cavendish . he communicated with his female servants only via notes , and added a staircase to the back of his house to avoid contact with his housekeeper . some believe he may have suffered from a form of autism , but just about everyone will admit that he was a scientific genius . he 's best remembered as the first person to recognize hydrogen gas as a distinct substance and to determine the composition of water . in the 1700s , most people thought that water itself was an element , but cavendish observed that hydrogen , which he called inflammable air , reacted with oxygen , known then by the awesome name , dephlogisticated air , to form water . cavendish did n't totally understand what he 'd discovered here , in part because he did n't believe in chemical compounds . he explained his experiments with hydrogen in terms of a fire-like element called phlagiston . nevertheless , his experiments were groundbreaking . like his work in determining the specific gravity basically the comparative density of hydrogen and other gases with reference to common air . it 's especially impressive when you consider the crude instruments he was working with . this , for example , is what he made his hydrogen gas with . he went on not only to establish an accurate composition of the atmosphere , but also discovered the density of the earth . not bad for a guy who was so painfully shy that the only existing portrait of him was sketched without his knowledge . for all of his decades of experiments , cavendish only published about 20 papers . in the years after his death , researchers figured out that cavendish had actually pre-discovered richter 's law , ohm 's law , coulomb 's law , several other laws . that 's a lot of freaking laws . if he had gotten credit for them all , we would have had to deal with like cavendish 's 8th law and cavendish 's 4th law , so i , for one , am glad that he did n't actually get credit . we 're going to do some pretty amazing science right now . you guys are not going to believe this . okay , you ready ? it floats ! yeah , i know you 're not surprised by this , but you should be because everything else , when it 's solid is much more dense than when it 's liquid ; just like gases are much less dense than liquids are . but that simple characteristic of water that its solid form floats is one of the reasons why life on this planet , as we know it , is possible . why is it that solid water is less dense than liquid water while everything else is the exact opposite of that ? well , you can thank your hydrogen bonds once again . at around 32 degrees fahrenheit or 0 degrees celsius if you 're a scientist or from a part of the world where things make sense , water molecules start to solidify and the hydrogen bonds in those water molecules form crystalline structures that space molecules apart more evenly , in turn making frozen water less dense than its liquid form . so in almost every circumstance a floating water ice is a really good thing . if ice were denser than water it would freeze and then sink , and then freeze and then sink , and then freeze and then sink , so just trust me on this one . you do n't want to live on a world where ice sinks . not only would it totally wreak havoc on aquatic eco systems which are basically how life formed on the earth in the first place . also , all the ice in the north pole would sink and then all of the water everywhere else would rise and we would n't have any land . that would be annoying . there 's one more amazing property of water i 'm forgetting . why is it so hot in here ? heat capacity ! yes , water has a very high heat capacity and probably that means nothing to you . basically it means that water is really good at holding on to heat , which is why we like to put hot water bottles in our bed and cuddle with them when we 're lonely . aside from artificially warming your bed , it 's also very important that it 's hard to heat up and cool down the oceans significantly . they become giant heat sinks that regulate the temperature and the climate of our planet , which is why , for example , it 's so much nicer in los angeles where the ocean is constantly keeping the temperatures the same than it is in , say , nebraska . on a smaller scale , we can see water 's high heat capacity really easily and visually by putting a pot with no water in it on a stove and seeing how badly that goes . but then you put a little bit of water in it and it takes forever to freaking boil ! and if you have n't already noticed this , when water evaporates from your skin , it cools you down . that 's the principle behind sweating , which is an extremely effective , though somewhat embarrassing , part of life . this is an example of another incredibly cool thing about water . when my body gets hot and it sweats , that heat excites some of the water molecules on my skin to the point where they break those hydrogen bonds and they evaporate away . when they escape , they take that heat energy with them , leaving me cooler . lovely . this was n't exercise , though . i do n't know why i 'm sweating so much . it could be the spray bottle that i keep spraying myself with , or maybe it 's just because this is such a high stress enterprise trying to teach you people things .
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aside from artificially warming your bed , it 's also very important that it 's hard to heat up and cool down the oceans significantly . they become giant heat sinks that regulate the temperature and the climate of our planet , which is why , for example , it 's so much nicer in los angeles where the ocean is constantly keeping the temperatures the same than it is in , say , nebraska . on a smaller scale , we can see water 's high heat capacity really easily and visually by putting a pot with no water in it on a stove and seeing how badly that goes .
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how does a boat or ship float on the ocean ?
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man in glasses : hello , there . here at crash course hq we like to start out each day with a nice healthy dose of water in all its 3 forms . it 's the only substance on all of our planet earth that occurs naturally in solid , liquid and gas forms . to celebrate this magical bond between two hydrogen atoms and one oxygen atom , here today we are going to be celebrating the wonderful life-sustaining properties of water , but we 're going to do it slightly more clothed . ( boppy music ) ah , much better . when we left off here at the biology crash course we were talking about life , and the rather important fact that all life as we know it is dependent upon there being water around . scientists and astronomers are always looking out into the universe trying to figure out whether there is life elsewhere because that is kind of the most important question that we have right now . they 're always getting really excited when they find water someplace , particularly liquid water . this is one reason why i and so many other people geeked out so hard last december when mars ' seven-year-old rover opportunity found a 20-inch long vein of gypsum that was almost certainly deposited by long term liquid water on the surface of mars . this was probably billions of years ago , and so it 's going to be hard to tell whether or not the water that was there resulted in some life . maybe we can figure that out and that would be really exciting . why ? why do we think that water is necessary for life ? why does water on other planets get us so freaking excited ? let 's start out by investigating some of the amazing properties of water . in order to do that , we 're going to have to start out with this . the world 's most popular molecule , or at least the world 's most memorized molecule . we all know about it . good old h20 . two hydrogens , one oxygen , the hydrogens each sharing an electron with oxygen in what we call a covalent bond . as you can see , i 've drawn my water molecule in a particular way . this is actually the way that it appears . it is v-shaped . because this big old oxygen atom is a little bit more greedy for electrons , it has a slight negative charge ; whereas , this area here with the hydrogen atoms has a slight positive charge . thanks to this polarity , all water molecules are attracted to one another ; so much so that they actually stick together and these are called hydrogen bonds . we talked about them last time , but essentially what happens is that the positive pole around those hydrogen atoms bonds to the negative pole around the oxygen atoms of a different water molecule . it 's a weak bond , but look , they 're bonding ! seriously , i can not overstate the importance of this hydrogen bond . when your teacher asks you , `` what 's important about water ? '' start out with the hydrogen bonds and you should put it in all caps and maybe some sparkles around it . one of the cool properties that results from these hydrogen bonds is a high cohesion for water which results in high surface tension . cohesion is the attraction between two like things , like attraction between one molecule of water and another molecule of water . water has the highest cohesion of any nonmetalic liquid . you can see this if you put some water on some wax paper or some teflon or something where the water beads up like this . some leaves of plants do it really well ; it 's quite cool . since water adheres weakly to the wax paper or to the plant , but strongly to itself , the water molecules are holding those droplets together in a configuration that creates the least amount of surface area . this is high surface tension that allows some bugs and even i think one lizard and also one jesus to be able to walk on water . the cohesive force of water does have its limits , of course . there are other substances that water quite likes to stick to . take glass , for example . this is called adhesion . the water is spreading out here instead of beading up because the adhesive forces between the water and the glass are stronger than the cohesive forces of the individual water molecules in the bead of water . adhesion is attraction between two different substances , so in this case the water molecules and the glass molecules . these properties lead to one of my favorite things about water ; the fact that it can defy gravity . that really cool thing that just happened is called capillary action . explaining it can be easily done with what we now know about cohesion and adhesion . thanks to adhesion , the water molecules are attracted to the molecules in the straw . as the water molecules adhere to the straw , other molecules are drawn in by cohesion following those fellow water molecules . thank you cohesion . the surface tension created here causes the water to climb up the straw . it will continue to climb until eventually gravity pulling down on the weight of the water in the straw overpowers the surface tension . the fact that water 's a polar molecule also makes it really good at dissolving things , which we call it 's a good solvent then . scratch that . water is n't a good solvent . it 's an amazing solvent . there are more substances that can be dissolved in water than in any other liquid on earth . yes , that includes the strongest acid that we have ever created . these substances that dissolve in water , sugar or salt being ones that we 're familiar with , are called hydrophilic , and they are hydrophilic because they are polar . their polarity is stronger than the cohesive forces of the water . when you get one of these polar substances in water , it 's strong enough that it breaks all the little cohesive forces , all those little hydrogen bonds . instead of hydrogen bonding to each other , the water will hydrogen bond around these polar substances . table salt is ionic , and right now it 's being separated into ions as the poles of our water molecules interact with it . what happens when there is a molecule that can not break the cohesive forces of water ? it ca n't penetrate and come into it . basically , what happens when that substance ca n't overcome the strong cohesive forces of water , ca n't get inside of the water ? that 's when we get what we call a hydrophobic substance , or something that is fearful of water . these molecules lack charged poles . they are non-polar and are not dissolving in water because essentially they 're being pushed out of the water by water 's cohesive forces . water , we may call it the universal solvent , but that does not mean that it dissolves everything . ( boppy music ) there have been a lot of eccentric scientists throughout history , but all this talk about water got me thinking about perhaps the most eccentric of the eccentrics , a man named henry cavendish . he communicated with his female servants only via notes , and added a staircase to the back of his house to avoid contact with his housekeeper . some believe he may have suffered from a form of autism , but just about everyone will admit that he was a scientific genius . he 's best remembered as the first person to recognize hydrogen gas as a distinct substance and to determine the composition of water . in the 1700s , most people thought that water itself was an element , but cavendish observed that hydrogen , which he called inflammable air , reacted with oxygen , known then by the awesome name , dephlogisticated air , to form water . cavendish did n't totally understand what he 'd discovered here , in part because he did n't believe in chemical compounds . he explained his experiments with hydrogen in terms of a fire-like element called phlagiston . nevertheless , his experiments were groundbreaking . like his work in determining the specific gravity basically the comparative density of hydrogen and other gases with reference to common air . it 's especially impressive when you consider the crude instruments he was working with . this , for example , is what he made his hydrogen gas with . he went on not only to establish an accurate composition of the atmosphere , but also discovered the density of the earth . not bad for a guy who was so painfully shy that the only existing portrait of him was sketched without his knowledge . for all of his decades of experiments , cavendish only published about 20 papers . in the years after his death , researchers figured out that cavendish had actually pre-discovered richter 's law , ohm 's law , coulomb 's law , several other laws . that 's a lot of freaking laws . if he had gotten credit for them all , we would have had to deal with like cavendish 's 8th law and cavendish 's 4th law , so i , for one , am glad that he did n't actually get credit . we 're going to do some pretty amazing science right now . you guys are not going to believe this . okay , you ready ? it floats ! yeah , i know you 're not surprised by this , but you should be because everything else , when it 's solid is much more dense than when it 's liquid ; just like gases are much less dense than liquids are . but that simple characteristic of water that its solid form floats is one of the reasons why life on this planet , as we know it , is possible . why is it that solid water is less dense than liquid water while everything else is the exact opposite of that ? well , you can thank your hydrogen bonds once again . at around 32 degrees fahrenheit or 0 degrees celsius if you 're a scientist or from a part of the world where things make sense , water molecules start to solidify and the hydrogen bonds in those water molecules form crystalline structures that space molecules apart more evenly , in turn making frozen water less dense than its liquid form . so in almost every circumstance a floating water ice is a really good thing . if ice were denser than water it would freeze and then sink , and then freeze and then sink , and then freeze and then sink , so just trust me on this one . you do n't want to live on a world where ice sinks . not only would it totally wreak havoc on aquatic eco systems which are basically how life formed on the earth in the first place . also , all the ice in the north pole would sink and then all of the water everywhere else would rise and we would n't have any land . that would be annoying . there 's one more amazing property of water i 'm forgetting . why is it so hot in here ? heat capacity ! yes , water has a very high heat capacity and probably that means nothing to you . basically it means that water is really good at holding on to heat , which is why we like to put hot water bottles in our bed and cuddle with them when we 're lonely . aside from artificially warming your bed , it 's also very important that it 's hard to heat up and cool down the oceans significantly . they become giant heat sinks that regulate the temperature and the climate of our planet , which is why , for example , it 's so much nicer in los angeles where the ocean is constantly keeping the temperatures the same than it is in , say , nebraska . on a smaller scale , we can see water 's high heat capacity really easily and visually by putting a pot with no water in it on a stove and seeing how badly that goes . but then you put a little bit of water in it and it takes forever to freaking boil ! and if you have n't already noticed this , when water evaporates from your skin , it cools you down . that 's the principle behind sweating , which is an extremely effective , though somewhat embarrassing , part of life . this is an example of another incredibly cool thing about water . when my body gets hot and it sweats , that heat excites some of the water molecules on my skin to the point where they break those hydrogen bonds and they evaporate away . when they escape , they take that heat energy with them , leaving me cooler . lovely . this was n't exercise , though . i do n't know why i 'm sweating so much . it could be the spray bottle that i keep spraying myself with , or maybe it 's just because this is such a high stress enterprise trying to teach you people things .
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these molecules lack charged poles . they are non-polar and are not dissolving in water because essentially they 're being pushed out of the water by water 's cohesive forces . water , we may call it the universal solvent , but that does not mean that it dissolves everything .
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where did all the water come from ?
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