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Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | If my velocity is three meters per second, and since it's positive, you can conceptualize that as it's going to the right at three meters per second, what is my change in position? Well, I would have gone 12 meters to the right. And you don't need calculus to figure that out. Three meters per second times four seconds ... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | Three meters per second times four seconds would be 12 meters. But what if it were the other way around? What if I had another velocity function? Let's call that v sub two of t that is equal to negative two meters per second. And it's just a constant negative two meters per second. So this is v sub two of t right over ... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | Let's call that v sub two of t that is equal to negative two meters per second. And it's just a constant negative two meters per second. So this is v sub two of t right over here. What would or what should the definite integral from one to five of v sub two of t be dt be equal to? Well, it should be equal to my change ... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | What would or what should the definite integral from one to five of v sub two of t be dt be equal to? Well, it should be equal to my change in position. But if my velocity is negative, that means I'm moving to the left. That means my change in position should be to the left as opposed to to the right. And so we can jus... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | That means my change in position should be to the left as opposed to to the right. And so we can just look at this area right over here. Well, if you just look at it as the rectangle, it's gonna be two times four, which is equal to eight. But you have to be very careful. Since it is below my horizontal axis and above m... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | But you have to be very careful. Since it is below my horizontal axis and above my function, this is going to be negative. And this should make a lot of sense. If I'm going two meters per second to the left for four seconds, or another way to think about it, if I'm going negative two meters per second for four seconds ... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | And what I want to think about is what is the limit of f of x as x approaches infinity? And there are several ways that you could do this. You could actually try to plug in larger and larger numbers for x and see if it seems to be approaching some value. Or you could reason through this. And when I talk about reasoning... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | Or you could reason through this. And when I talk about reasoning through this, it's to think about the behavior of this numerator and denominator as x gets very, very, very large. And when I'm talking about that, what I'm saying is as x gets very, very large, let's just focus on the numerator, as x gets very, very lar... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | Something squaring gets large, but something being raised to the 5th power gets raised that much, much faster. Similarly, in the denominator, this term right over here, the highest degree term, 6x to the 5th, is going to grow much, much, much faster than any of these other terms. Even though this has 100 as a coefficie... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | So as x gets very, very, very large, this thing is going to approximate 4x to the 5th over 6x to the 5th for very large x. Or we could say as x approaches infinity. Now what could this be simplified to? Well, you have x to the 5th divided by x to the 5th. These are going to grow together. So you can think of them as ca... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | Well, you have x to the 5th divided by x to the 5th. These are going to grow together. So you can think of them as canceling out, and so you are left with 2 thirds. So what you could say is the limit of f of x as x approaches infinity, as x gets larger and larger and larger, all of these other terms aren't going to mat... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | So what you could say is the limit of f of x as x approaches infinity, as x gets larger and larger and larger, all of these other terms aren't going to matter that much, and so it's going to approach 2 thirds. Now let's look at the graph and see if that actually makes sense. What we're actually saying is that we have a... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | So let's look at the graph. So right here is the graph. Got it from Wolfram Alpha. We see indeed as x gets larger and larger and larger, f of x seems to be approaching this value that looks right at around 2 thirds. So it looks like we have a horizontal asymptote right over here. Let me draw that a little bit neater. W... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | We see indeed as x gets larger and larger and larger, f of x seems to be approaching this value that looks right at around 2 thirds. So it looks like we have a horizontal asymptote right over here. Let me draw that a little bit neater. We have a horizontal asymptote right at 2 thirds. So let me draw it as neatly as I c... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | We have a horizontal asymptote right at 2 thirds. So let me draw it as neatly as I can. So this right over here is y is equal to 2 thirds. The limit as x gets really, really large, as it approaches infinity, y is getting closer and closer and closer to 2 thirds. When we just look at the graph here, it seems like the sa... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | The limit as x gets really, really large, as it approaches infinity, y is getting closer and closer and closer to 2 thirds. When we just look at the graph here, it seems like the same thing is happening from the bottom direction when x approaches negative infinity. So we could say the limit of f of x as x approaches ne... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | We can use the exact same logic. When x becomes a very, very negative number, as it becomes further and further to the left on the number line, the only terms that are going to matter are going to be the 4x to the 5th and the 6x to the 5th. So this is true for very large x's. It's also true for very negative x's. So we... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | It's also true for very negative x's. So we could also say as x approaches negative infinity, this is also true. Then the x to the 5th over the x to the 5th is going to cancel out. These are the dominant terms. We're going to get it equaling 2 thirds. Once again, you see that in the graph here. We have a horizontal asy... |
Limits at infinity of quotients (Part 1) Limits and continuity AP Calculus AB Khan Academy.mp3 | These are the dominant terms. We're going to get it equaling 2 thirds. Once again, you see that in the graph here. We have a horizontal asymptote at y is equal to 2 thirds. Whether we take the limit of f of x as x approaches infinity, we get 2 thirds. And the limit of f of x as x approaches negative infinity is 2 third... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | What is the rate of change of the area, a of t, so our area's going to be a function of t, what is the rate of change of the area of the triangle at that instant? And so what we're going to do in this exercise, instead of going straight and trying to solve it, what we need to do here is to identify the various units of... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | Let's match each expression with its units, and like always, pause the video and see if you can do it on your own. All right, so the first one is b prime of t. So this is the rate of change of which the base is changing with respect to time. So if we think about it, b of t, that is the base, that is going to be in mete... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | So this is going to be in meters. If we say b prime of t, this is going to be how much our base is changing with respect to time. So this is going to be meters per, and they give us right over here, they say it's decreasing at a rate of 13 meters per hour. So the units here are meters per hour. And so b prime of t, tha... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | So the units here are meters per hour. And so b prime of t, that is going to be in meters per hour. A at time t sub zero. Remember, a is the area of our triangle, and we're measuring everything in meters, as you can tell from the information they've given us. And so area is going to be in square units, and so it's goin... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | Remember, a is the area of our triangle, and we're measuring everything in meters, as you can tell from the information they've given us. And so area is going to be in square units, and so it's going to be in square meters. Now the height at time t sub zero. Well, both the base and the height, those are lengths, they'r... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | Well, both the base and the height, those are lengths, they're gonna be measured in meters. And so our height at time t sub zero is going to be in meters. And then here we have the rate of change of our area with respect to time. So our area, we already know, is in meters squared. But we wanna know, this here, this is ... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | So our area, we already know, is in meters squared. But we wanna know, this here, this is going to be the rate of change of our area with respect to time. So it's going to be an amount of area per unit time, and time here, we're using hours, as you can see from some of the information they've given us. So this is going... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | So this is going to be area per unit time, or meters squared per hour. So it's going to be right over here. So it's area per unit time, and the length we're using in this is meters, and time is hours. All right. Now they say, match each expression with its given value. So what is the base of the triangle at time t sub ... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | All right. Now they say, match each expression with its given value. So what is the base of the triangle at time t sub zero? Do they give that to us? Well, let's see. They say at a certain time, at a certain instant, t sub zero, the base, I'm gonna underline this in a different color, the base at a certain instant, t s... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | Do they give that to us? Well, let's see. They say at a certain time, at a certain instant, t sub zero, the base, I'm gonna underline this in a different color, the base at a certain instant, t sub zero, the base is five meters. So they say the base at time t sub zero, the base is a function of time, but they tell us t... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | So they say the base at time t sub zero, the base is a function of time, but they tell us that it is five meters. So this is five meters right over here. Now what about the rate of change of the base with respect to time? Do they tell us that? Well, look right over here. That's actually the first piece of information t... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | Do they tell us that? Well, look right over here. That's actually the first piece of information they gave us. So the base, B of t of a triangle, is decreasing at a rate of 13 meters per hour. So the rate of change of the base, that is B prime of t, which is equal to dB dt, and they tell us that that is, it's decreasin... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | So the base, B of t of a triangle, is decreasing at a rate of 13 meters per hour. So the rate of change of the base, that is B prime of t, which is equal to dB dt, and they tell us that that is, it's decreasing at a rate of 13 meters per hour. So that would be negative 13 meters per hour. And so the rate of change of t... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | And so the rate of change of the base with respect to time is going to be negative 13. They gave us that. Now A prime of t, this is the rate of change of the area at time t sub zero. Did they give us this? Well, they ask us that. What is the rate of change of the area, A of t of the triangle at that instant? So this is... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | Did they give us this? Well, they ask us that. What is the rate of change of the area, A of t of the triangle at that instant? So this is what we actually need to figure out, but they haven't given it to us, otherwise there's no problem to solve. So this one right over here is not given. This is what we are trying to s... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | So this is what we actually need to figure out, but they haven't given it to us, otherwise there's no problem to solve. So this one right over here is not given. This is what we are trying to solve for. And then finally we have the first derivative of the height with respect to time. So you could view this as dh dt. Wh... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | And then finally we have the first derivative of the height with respect to time. So you could view this as dh dt. What is this going to be? Do they give it to us? Well, look right over here. They say the height of the triangle is increasing at a rate of six meters per hour. So if they're saying h of t is increasing, t... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | Do they give it to us? Well, look right over here. They say the height of the triangle is increasing at a rate of six meters per hour. So if they're saying h of t is increasing, they're telling us the rate of change of h of t with respect to time, so that's h prime of t, and they're telling us that it is increasing at ... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | So if they're saying h of t is increasing, they're telling us the rate of change of h of t with respect to time, so that's h prime of t, and they're telling us that it is increasing at six meters per hour. So it's gonna be positive six meters per hour. So they did indeed give us that. Now why is all of this a useful ex... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | Now why is all of this a useful exercise to go through? Well, now we are really ready to solve the question because in general, if we're talking about any triangle, we know that area is equal to 1 1 2 base times height. Now in this situation, area and our base and our height, they're all going to be functions of t. So ... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | This is just the product rule here, plus the first function, b of t, times the derivative of the second function with respect to time. And we need to figure out not just the general expression, they want us to know what the rate of change of the area, so a prime of t at that instant, at t sub zero. So what we wanna fig... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | Well, that's just going to be equal to 1 1 2 times b prime of t sub zero times h of t sub zero plus b of t sub zero times h prime of t sub zero. Now, this might seem daunting, except they've given us a lot of this information. What is b prime of t sub zero? Well, they tell us the rate of change of b with respect to tim... |
Analyzing related rates problems expressions AP Calculus AB Khan Academy.mp3 | Well, they tell us the rate of change of b with respect to time, and it seems like it's just gonna stay at negative 13 meters per hour, so they gave us this. And h, what is the height at time t sub zero? Well, they tell us right over here. At a certain instant, the base is five meters and the height is one meter, so th... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | And like always, I encourage you to pause the video and see if you can work through this on your own. So let's look at this first statement. So this first statement says both the limit of g of x as x approaches six from the right-hand side and the limit as x approaches six from the left-hand side of g of x exist. All r... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | All right, so let's first think about the limit of g of x as x approaches six from the right-hand side, as we approach six from values greater than six. So if we look over here, we could say, okay, when x is equal to nine, g of nine is right over there. G of eight is right over here. G of seven is right over here. It l... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | G of seven is right over here. It looks like it's between negative three and negative four. G of 6.5 looks like it's a little bit, it's still between negative three and negative four, but it's closer to negative three. G of 6.1 is even closer to negative three. G of 6.01 is even closer to negative three. So it looks li... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | G of 6.1 is even closer to negative three. G of 6.01 is even closer to negative three. So it looks like the limit from the right-hand side does exist. So it looks like this one exists. Now let's see, and I'm just looking at it graphically, and that's all they can expect you to do in an exercise like this. Now let's thi... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | So it looks like this one exists. Now let's see, and I'm just looking at it graphically, and that's all they can expect you to do in an exercise like this. Now let's think about the limit as x approaches six from the left-hand side. So I could start anywhere, but let's say when x is equal to three, g of three is a litt... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | So I could start anywhere, but let's say when x is equal to three, g of three is a little more than one. G of four looks like it's a little bit less than two. G of five looks like it's close to three. G of 5.5 looks like it's between five and six. G of 5.75 looks like it's approaching nine. And as we get closer and clo... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | G of 5.5 looks like it's between five and six. G of 5.75 looks like it's approaching nine. And as we get closer and closer, as x gets closer and closer to six from below, from values to the left of six, it looks like we're unbounded. We are approaching infinity. And so technically, we would say this limit does not exis... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | We are approaching infinity. And so technically, we would say this limit does not exist. So this one does not exist. So I won't check this one off. Some people will say the limit is approaching infinity, but that technically is, infinity is not a value that you can say it is approaching in the classical, formal definit... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | So I won't check this one off. Some people will say the limit is approaching infinity, but that technically is, infinity is not a value that you can say it is approaching in the classical, formal definition of a limit. So for these purposes, we would just say this does not exist. Now let's see. They say the limit as x ... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | Now let's see. They say the limit as x approaches six of g of x exists. Well, the only way that the limit exists is if both the left and the right limits exist, and they approach the same thing. Well, we don't even, our limit as x approaches six from the negative side, or from the left-hand side, I guess I could say, d... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | Well, we don't even, our limit as x approaches six from the negative side, or from the left-hand side, I guess I could say, does not even exist. So this cannot be true. So that's not gonna be true. The first one's not gonna be true. G is defined at x equals six. So at x equals six, it doesn't look like g is defined. An... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | The first one's not gonna be true. G is defined at x equals six. So at x equals six, it doesn't look like g is defined. And looking at this graph, I can't tell you what g of six should be. We have an open circle over here, so g of six is not equal to negative three, and this goes up to infinity, and we have a vertical ... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | And looking at this graph, I can't tell you what g of six should be. We have an open circle over here, so g of six is not equal to negative three, and this goes up to infinity, and we have a vertical asymptote actually drawn right over here at x equals six. So g is not defined at x equals six. So I'll rule that one out... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | So I'll rule that one out. G is continuous at x equals six. Well, you can see that it goes up to infinity, then it jumps down, back down here, then continues. So just, when you just think about it in common sense language, it looks very discontinuous. And if you wanna think about it more formally, in order for somethin... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | So just, when you just think about it in common sense language, it looks very discontinuous. And if you wanna think about it more formally, in order for something to be continuous, the limit needs to exist at that value. The function needs to be defined at that value, and the value of the function needs to be equal to ... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | And neither of these, the first two conditions aren't true, and so these can't even equal each other, because neither of these exist. So this is not continuous at x equals six, and so the only thing I could check here is none of the above. Let's do another one of these. So the first statement, both the right hand and t... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | So the first statement, both the right hand and the left hand limit exist as x approaches three. So let's think about it. So x equals three is where we have this little discontinuity here, this jump discontinuity. So let's approach, let's go from the positive, from values larger than three. So when x is equal to five, ... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | So let's approach, let's go from the positive, from values larger than three. So when x is equal to five, g of five is a little bit more negative than negative three. G of four is between negative two and negative three. G of 3.5 is getting a little bit closer to negative two. G of 3.1, it's getting even closer, closer... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | G of 3.5 is getting a little bit closer to negative two. G of 3.1, it's getting even closer, closer to negative two. G of 3.01 is even closer to negative two. So it looks like this limit right over here, oh, I'm circling the wrong one. It looks like this limit exists, and in fact, it looks like it is approaching negati... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | So it looks like this limit right over here, oh, I'm circling the wrong one. It looks like this limit exists, and in fact, it looks like it is approaching negative two. So this right over here is equal to negative two, the limit of g of x as x approaches three from the right hand side. And now let's think about it from... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | And now let's think about it from the left hand side. So we can start, I can start here. G of one looks like it's a little bit greater than negative one. G of two, it's less than one. G of 2.5 is between one and two. G of 2.9, looks like it's a little bit less than two. G of 2.99 is getting even closer to two. |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | G of two, it's less than one. G of 2.5 is between one and two. G of 2.9, looks like it's a little bit less than two. G of 2.99 is getting even closer to two. G of 2.999999 would be even closer to, so it looks like this thing right over here is approaching two. So both of these limits, the limit from the right and the l... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | G of 2.99 is getting even closer to two. G of 2.999999 would be even closer to, so it looks like this thing right over here is approaching two. So both of these limits, the limit from the right and the limit from the left, exist. The limit of g of x as x approaches three exists. So these are the one-sided limit. This i... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | The limit of g of x as x approaches three exists. So these are the one-sided limit. This is the actual limit. Now in order for this to exist, both the right and left-handed limits need to exist, and they need to approach the same value. Well this first statement, we saw that both of these exist, but they aren't approac... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | Now in order for this to exist, both the right and left-handed limits need to exist, and they need to approach the same value. Well this first statement, we saw that both of these exist, but they aren't approaching the same value. From the left, we are, sorry, from the right, we are approaching negative two, and from t... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | So this limit does not exist. So I will not check that out, or I will not check that box. G is defined at x equals three. Well when x equals three, we see a solid dot right over there. And so it is indeed defined. It is indeed defined there. G is continuous at x equals three. |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | Well when x equals three, we see a solid dot right over there. And so it is indeed defined. It is indeed defined there. G is continuous at x equals three. Well in order for g to be continuous at x equals three, the limit must exist there. It must be defined there, and the value of the function there needs to be equal t... |
Worked example Continuity at a point Limits and continuity AP Calculus AB Khan Academy.mp3 | G is continuous at x equals three. Well in order for g to be continuous at x equals three, the limit must exist there. It must be defined there, and the value of the function there needs to be equal to the value of the limit. Well the function is defined there, but the limit doesn't exist there. So it cannot be continu... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | We are being asked, what is the smallest, this is a little typo here, what is the smallest possible sum of squares of two numbers if their product is negative 16? So let's say that these two numbers are x and y, x and y. So how could we define the sum of the squares of the two numbers? So I'll just call that the sum of... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | So I'll just call that the sum of the squares, s for sum of the squares, and it would just be equal to x squared plus y squared. And this is what we wanna minimize. We want to minimize, minimize s. Now, right now, s is expressed as a function of x and y. We don't know how to minimize with respect to two variables, so w... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | We don't know how to minimize with respect to two variables, so we have to get this in terms of only one variable, and lucky for us, they give us another piece of information. Their product is negative 16. So x times y is equal to negative 16. So let's say we wanted this expression right over here only in terms of x. W... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | So let's say we wanted this expression right over here only in terms of x. Well, then we could solve, we can figure out what y is in terms of x, and then substitute. So let's do that right over here. If we divide both sides by x, we get y is equal to negative 16 over x. And so let's replace our y in this expression wit... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | If we divide both sides by x, we get y is equal to negative 16 over x. And so let's replace our y in this expression with negative 16 over x. So then we would get our sum of squares as a function of x is going to be equal to x squared plus y squared. Y is negative 16 over x. Negative 16 over x. And then that's what we ... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | Y is negative 16 over x. Negative 16 over x. And then that's what we will now square. So this is equal to x squared plus, what is this? 256, 256 over x squared, or we could write that as 256, 256 x to the negative 2 power. That is the sum of our squares that we now want to minimize. Well, to minimize this, we would wan... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | So this is equal to x squared plus, what is this? 256, 256 over x squared, or we could write that as 256, 256 x to the negative 2 power. That is the sum of our squares that we now want to minimize. Well, to minimize this, we would wanna look at the critical points of this, which is where the derivative is either zero o... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | Well, to minimize this, we would wanna look at the critical points of this, which is where the derivative is either zero or undefined, and see whether those critical points are possibly a minimum or a maximum point. They don't have to be, but those are the ones, if we have a minimum or a maximum point, they're going to... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | So the derivative, s prime, let me do this in a different color, s prime of x, I'll do it right over here, actually. The derivative, s prime of x, with respect to x, is going to be equal to 2x times negative 2 times 5, 2x plus 256 times negative 2, so that's minus 512 x to the negative 3 power. X to the negative 3 powe... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | Now, this is going to be undefined, this is going to be undefined when x is equal to zero, but if x is equal to zero, then y is undefined, so this whole thing breaks down. So that isn't a useful critical point, x equals zero. So let's think about any other ones. Well, it's defined everywhere else, so let's think about ... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | Well, it's defined everywhere else, so let's think about where the derivative is equal to zero. So when does this thing equal zero? So when does 2x minus 512 x to the negative 3 equal zero? Well, we can add 512 x to the negative 3 to both sides, so you get 2x is equal to 512 x to the negative 3rd power. We can multiply... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | Well, we can add 512 x to the negative 3 to both sides, so you get 2x is equal to 512 x to the negative 3rd power. We can multiply both sides times x to the 3rd power, multiply both sides times x to the 3rd, so all of the x's go away on the right-hand side. So you get 2x to the 4th is equal to 512. We can divide both s... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | We can divide both sides by 2, and you get x to the 4th power is equal to 256. And so what is the 4th root of 256? Well, we could take the square root of both sides just to help us here. So let's see, if we take, so it's going to be x squared is going to be equal to, 256 is 16 squared, so this is 16, this is going to b... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | So let's see, if we take, so it's going to be x squared is going to be equal to, 256 is 16 squared, so this is 16, this is going to be x squared is equal to 16, or x is equal to 4. Now, that's our only critical point we have, so that's probably the x value that minimizes our sum of squares right over here, but let's ma... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | So s prime prime of x is going to be equal to 2, and then we're going to have negative 3 times negative 512, so I'll just write that as plus 3 times 512, that's going to be 1536, is that right? Yeah, 3 times 500 is 1500, 3 times 12 is 36, x to the negative 4 power. So this thing is going to be, this thing right over he... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | So we are always in a concave upwards situation. Concave upwards means that our graph might look something like that. Actually, I don't want to draw a little squiggle, it might look something like that, and you see the reason why the second derivative implies concave upwards, a second derivative positive means that our... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | You see it's negative, less negative, even less negative, zero, positive, more positive, so it's increasing over the entire place. So if you have a critical point where the derivative is equal to zero, so the slope is equal to zero, and it's concave upwards, you see pretty clearly that we have minimized the function. S... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | We actually don't even have to figure out what y has to be equal to in order to minimize the sum of squares, we could just put it back into this, but just for fun, we see that y would be negative 16 over x, so y would be equal to negative 4, and we could just figure out now what our sum of squares is. Our minimum sum o... |
Optimization sum of squares Applications of derivatives AP Calculus AB Khan Academy.mp3 | I could have just tried out numbers whose product is negative 16, and I probably would have tried out four and negative four in not too much time, and then I would have been able to maybe figure out that it's lower than if I did two and negative eight, or negative two and eight, or one and 16, and that's true. You prob... |
Related rates balloon Applications of derivatives AP Calculus AB Khan Academy.mp3 | You know the spot on the ground that is directly below the hot air balloon. Let's say it took off from that point. It's just been going straight up ever since. And you know, you've measured it out, that you're 500 meters away from there. So you know that you are 500 meters away from that. And you're also able to measur... |
Related rates balloon Applications of derivatives AP Calculus AB Khan Academy.mp3 | And you know, you've measured it out, that you're 500 meters away from there. So you know that you are 500 meters away from that. And you're also able to measure the angle between the horizontal and the hot air balloon. You could do that with, I don't know, I'm not exactly a surveyor, but I guess a viewfinder or someth... |
Related rates balloon Applications of derivatives AP Calculus AB Khan Academy.mp3 | You could do that with, I don't know, I'm not exactly a surveyor, but I guess a viewfinder or something like that. So you're able to, and I'm not sure if that's the right tool, but there are tools that you can measure the angles between the horizontal and something that's not on the horizontal. So you know that this an... |
Related rates balloon Applications of derivatives AP Calculus AB Khan Academy.mp3 | We're gonna keep it pi over four, because when you take derivatives of trig functions, you assume that you're dealing with radians. So right over here, this is pi over four radians. And you also are able to measure the rate at which this angle is changing. So this is changing, changing, changing at 0.2 radians per minu... |
Related rates balloon Applications of derivatives AP Calculus AB Khan Academy.mp3 | So this is changing, changing, changing at 0.2 radians per minute. Now, my question to you, or the question that you're trying to figure out as you watch this hot air balloon is how fast is it rising right now? How fast is it rising just as the angle between the horizontal and kind of the line between you and the hot a... |
Related rates balloon Applications of derivatives AP Calculus AB Khan Academy.mp3 | So let's think about what we know and what we're trying to figure out. So we know a couple of things. We know that theta is equal to pi over four. If we call theta the angle right over here. So this is theta. We also know the rate at which theta is changing. We know d theta, let me do this in yellow. |
Related rates balloon Applications of derivatives AP Calculus AB Khan Academy.mp3 | If we call theta the angle right over here. So this is theta. We also know the rate at which theta is changing. We know d theta, let me do this in yellow. We know d theta, dt, is equal to 0.2 radians per minute. Now, what are we trying to figure out? Well, we're trying to figure out the rate at which the height of the ... |
Related rates balloon Applications of derivatives AP Calculus AB Khan Academy.mp3 | We know d theta, let me do this in yellow. We know d theta, dt, is equal to 0.2 radians per minute. Now, what are we trying to figure out? Well, we're trying to figure out the rate at which the height of the balloon is changing. So if you call this distance right over here, this distance right over here, h, what we wan... |
Related rates balloon Applications of derivatives AP Calculus AB Khan Academy.mp3 | Well, we're trying to figure out the rate at which the height of the balloon is changing. So if you call this distance right over here, this distance right over here, h, what we wanna figure out is dh dt. That's what we don't know. So what we'd wanna come up with is a relationship between dh dt, d theta dt, and maybe t... |
Related rates balloon Applications of derivatives AP Calculus AB Khan Academy.mp3 | So what we'd wanna come up with is a relationship between dh dt, d theta dt, and maybe theta if we need it. Or another way to think about it, if we can come up with a relationship between h and theta, then we could take the derivative with respect to t, and we'll probably get a relationship between all of this stuff. S... |
Related rates balloon Applications of derivatives AP Calculus AB Khan Academy.mp3 | Well, it's a little bit of trigonometry. We know, we're trying to figure out h. We already know what this length is right over here. We know opposite over adjacent, that's the definition of tangent. So let's write that down. So we know that the tangent, the tangent of theta, tangent of theta is equal to the opposite si... |
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