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so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | so once again , our slope here is a constant positive line . let me be careful here because at this point , our slope wo n't really be defined , because our slope , you could draw multiple tangent lines at this little pointy point . so let me just draw a circle right over there . | why ca n't the slope be defined at this point ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | our slope is a constant positive value . so once again , our slope here is a constant positive line . let me be careful here because at this point , our slope wo n't really be defined , because our slope , you could draw multiple tangent lines at this little pointy point . | so if a line on a function is constant ( straight ) then is the derivative going to always be a straight horizontal line as well ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | but then when we go over here , even though the value of our function has gone down , we still have a constant positive slope . in fact , the slope of this line looks identical to the slope of this line . let me do that in a different color . | why do the beginning of the pink line and the end of the orange line not end in open circles ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | and then we look at this point right over here . so right at this point , our slope is going to be undefined . there 's no way that you could find the slope over -- or this point of discontinuity . | should n't the derivative be undefined at those spots , because we only know the limit from one side ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | and so when you look at the derivative , the slope is still a positive value . but as we get larger and larger x 's up to this point , the slope is getting less and less positive , all the way to 0 . and then the slope is getting more and more negative . | 5 , why does sal put the point on the x axis ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | here the slope seems constant . our slope is a constant positive value . so once again , our slope here is a constant positive line . | why is the y value zero ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | and then we look at this point right over here . so right at this point , our slope is going to be undefined . there 's no way that you could find the slope over -- or this point of discontinuity . | why the last jump of the derivate ( ) is a defined point and not undefined ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . | how does one know what specific points to draw the derivative through ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | but the whole point of this video is to give you an intuition for thinking about what the slope of this function might look like at any point . and by doing so , we have essentially drawn the derivative over that interval . | in other words , how is the y-intercept of the derivative determined ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . | is there a difference between f ( x ) lines that in end in closed circles vis-a-vis lines that end in open circles ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | so once again , our slope here is a constant positive line . let me be careful here because at this point , our slope wo n't really be defined , because our slope , you could draw multiple tangent lines at this little pointy point . so let me just draw a circle right over there . | sal said the derivative of the point does n't exist , why ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | but the whole point of this video is to give you an intuition for thinking about what the slope of this function might look like at any point . and by doing so , we have essentially drawn the derivative over that interval . | if derivatives are functions themselves , i 'm assuming it 's possible to take the derivative of a derivative ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | but the whole point of this video is to give you an intuition for thinking about what the slope of this function might look like at any point . and by doing so , we have essentially drawn the derivative over that interval . | how does one determine a derivative exists ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . | for a derivative to exist the function has to be continuous , right ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | so we could say -- let me make it clear what interval i 'm talking about -- the slope over this interval is 0 . and then finally , in this last section -- let me do this in orange -- the slope becomes negative . but it 's a constant negative . | in the orange line ( the last section of the derivative graph ) have an open or closed hole ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | but the whole point of this video is to give you an intuition for thinking about what the slope of this function might look like at any point . and by doing so , we have essentially drawn the derivative over that interval . | can the graph of the derivative be drawn in any number of ways , since different graphs often have the same interpretation or is there a set graph that a derivative of a function has to be ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | let me be careful here because at this point , our slope wo n't really be defined , because our slope , you could draw multiple tangent lines at this little pointy point . so let me just draw a circle right over there . but then as we get right over here , the slope seems to be positive . | why does n't sal shade the circle ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | now let 's think about as we get to this point . here the slope seems constant . our slope is a constant positive value . | so the prime graph reflects only the slope ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | and it seems actually a little bit more negative than these were positive . so i would draw it right over there . so it 's a weird looking function . but the whole point of this video is to give you an intuition for thinking about what the slope of this function might look like at any point . and by doing so , we have ... | if i were told to sketch the graph of an even function f ( x ) which has a derivative at every point , how would it look like ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | notice here , for example , the slope is still positive . and so when you look at the derivative , the slope is still a positive value . but as we get larger and larger x 's up to this point , the slope is getting less and less positive , all the way to 0 . | how do we know that for a fact that wherever the turning point is , that is the same value of the x-intercept for the derivative ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | and once again , it 's undefined here at this point of discontinuity . so the slope will look something like that . and then we go up here . | can you consider then , that since the derivative of the slope is measuring the change in y with respect to x , that graphing the derivative of a function , that is measuring something like say , speed , would resulting in the graphing of the change in velocity over time when compared to the graph of the function ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | here the slope seems constant . our slope is a constant positive value . so once again , our slope here is a constant positive line . | how is the slope , the way it is drawn , positive when sal is talking exactly ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | but then when we go over here , even though the value of our function has gone down , we still have a constant positive slope . in fact , the slope of this line looks identical to the slope of this line . let me do that in a different color . | could someone explain why sal drew the lines on different levels and not one continuous line ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | and we just said we have a constant positive slope . so let 's say it looks something like that over this interval . and then we look at this point right over here . | , what i mean to say is that why are some lines above the x-axis like the yellow and red lines while the blue one is on the x-axis ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . | at what values of x is f ( x ) continuous but not differentiable ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | but then when we go over here , even though the value of our function has gone down , we still have a constant positive slope . in fact , the slope of this line looks identical to the slope of this line . let me do that in a different color . | in the example , was there a reason behind sal sketching the line at that specific point on the y-axis ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | and then we look at this point right over here . so right at this point , our slope is going to be undefined . there 's no way that you could find the slope over -- or this point of discontinuity . | so , my question is , how can you take the slope of an undefined slope ( whether because of discontinuity or not ) and thus taking the derivative of a function where that function is undefined ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . | the derivative of the parabolic portion of the function is explained/represented as a straight line , whereas in the video , `` figuring out which function is the derivative '' , what appears to be a similar figure for f ( x ) has a derivative line that is not straight ( rather , sigmoidal ) - ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | and then we go up here . the value of the function goes up , but now the function is flat . so the slope over that interval is 0 . | in first interval there is sine function then why the derivative of that function is straight line ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | so let 's say it looks something like that over this interval . and then we look at this point right over here . so right at this point , our slope is going to be undefined . | yet a unique limit for that point exists , so it should also exist a derivative right ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | but then when we go over here , even though the value of our function has gone down , we still have a constant positive slope . in fact , the slope of this line looks identical to the slope of this line . let me do that in a different color . | is the derivative of that slope the stangent line stretched from one point to another on a slope ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . | why is the derivative of y=x computed as x=1 , should n't it be y=-x to make it according to the slope signs ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | and it seems actually a little bit more negative than these were positive . so i would draw it right over there . so it 's a weird looking function . | if you took that ( red ) segment as an interval alone and evaluated it , would n't the f ' ( x ) of its left endpoint still be defined because the limit would only include evaluation from the positive side ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | but let 's say that , so let 's see , here the slope is quite positive . so let 's say the slope is right over here . and then it gets less and less and less positive . | would you say that the drawing of the slope/derivative on the second graph must be between [ -1,1 ] because the slope must always be a fraction , therefore between these two numbers ? |
so i 've got this crazy discontinuous function here , which we 'll call f of x . and my goal is to try to draw its derivative right over here . so what i 'm going to need to think about is the slope of the tangent line , or the slope at each point in this curve , and then try my best to draw that slope . so let 's try ... | so it gets less and less and less positive . notice here , for example , the slope is still positive . and so when you look at the derivative , the slope is still a positive value . but as we get larger and larger x 's up to this point , the slope is getting less and less positive , all the way to 0 . | what would be the derivative of an infinitely changing slope ? |
there is a half an apple pie left . you want to eat twice what your little brother eats , but you also need to save a slice for your mom . you can cut her a slice that is 30 degrees . what is the measure of your piece of the pie in degrees ? so to tackle this , we just have to remember a few things . we have to remembe... | let 's let x be equal to the degree measure of my brother 's pie or your little brother 's pie . so this is degree measure of brother 's pie . and then what would the amount of pie you eat be ? well , it says you eat twice what your little brother eats . | where did the idea come from that these 3 slices would consume the entire remains of the pie ? |
there is a half an apple pie left . you want to eat twice what your little brother eats , but you also need to save a slice for your mom . you can cut her a slice that is 30 degrees . what is the measure of your piece of the pie in degrees ? so to tackle this , we just have to remember a few things . we have to remembe... | and then we are left with 3x equaling 180 degrees minus 30 degrees is equal to 150 degrees . and now we can just divide both sides by 3 , and we 're left with x equaling 150 divided by 3 is 50 degrees . x is equal to 50 degrees . now , we have to be careful . | could n't you just do 180-30=150 , then divide 150 by 3 which equals 50 , and then multiply 50 times 2 to get 100 ? |
there is a half an apple pie left . you want to eat twice what your little brother eats , but you also need to save a slice for your mom . you can cut her a slice that is 30 degrees . what is the measure of your piece of the pie in degrees ? so to tackle this , we just have to remember a few things . we have to remembe... | so that is 2x . x is the measure of this angle , and then 2x is a measure of that angle . so you see that 30 degrees plus x plus 2x , or x plus 2x plus 30 degrees , is going to be equal to 180 degrees . | what is the measure of the larger angle in degrees ? |
there is a half an apple pie left . you want to eat twice what your little brother eats , but you also need to save a slice for your mom . you can cut her a slice that is 30 degrees . what is the measure of your piece of the pie in degrees ? so to tackle this , we just have to remember a few things . we have to remembe... | let 's let x be equal to the degree measure of my brother 's pie or your little brother 's pie . so this is degree measure of brother 's pie . and then what would the amount of pie you eat be ? well , it says you eat twice what your little brother eats . | where did the idea come from that these 3 slices would consume the entire remains of the pie ? |
there is a half an apple pie left . you want to eat twice what your little brother eats , but you also need to save a slice for your mom . you can cut her a slice that is 30 degrees . what is the measure of your piece of the pie in degrees ? so to tackle this , we just have to remember a few things . we have to remembe... | let 's let x be equal to the degree measure of my brother 's pie or your little brother 's pie . so this is degree measure of brother 's pie . and then what would the amount of pie you eat be ? well , it says you eat twice what your little brother eats . | does your family have to eat the whole pie ? |
there is a half an apple pie left . you want to eat twice what your little brother eats , but you also need to save a slice for your mom . you can cut her a slice that is 30 degrees . what is the measure of your piece of the pie in degrees ? so to tackle this , we just have to remember a few things . we have to remembe... | so if we had a whole pie -- so if we started here , and we had a whole pie , we went all the way around , that would be 360 degrees . but we only have a half pie . so we only have 180 degrees and let me do that in a color you 're more likely to see . | how many slices did sam cut the pie into ? |
there is a half an apple pie left . you want to eat twice what your little brother eats , but you also need to save a slice for your mom . you can cut her a slice that is 30 degrees . what is the measure of your piece of the pie in degrees ? so to tackle this , we just have to remember a few things . we have to remembe... | so that is 2x . x is the measure of this angle , and then 2x is a measure of that angle . so you see that 30 degrees plus x plus 2x , or x plus 2x plus 30 degrees , is going to be equal to 180 degrees . | what is the complement of an angle ? |
there is a half an apple pie left . you want to eat twice what your little brother eats , but you also need to save a slice for your mom . you can cut her a slice that is 30 degrees . what is the measure of your piece of the pie in degrees ? so to tackle this , we just have to remember a few things . we have to remembe... | so that is 2x . x is the measure of this angle , and then 2x is a measure of that angle . so you see that 30 degrees plus x plus 2x , or x plus 2x plus 30 degrees , is going to be equal to 180 degrees . | and what is the supplement of an angle ? |
there is a half an apple pie left . you want to eat twice what your little brother eats , but you also need to save a slice for your mom . you can cut her a slice that is 30 degrees . what is the measure of your piece of the pie in degrees ? so to tackle this , we just have to remember a few things . we have to remembe... | you want to eat twice what your little brother eats , but you also need to save a slice for your mom . you can cut her a slice that is 30 degrees . what is the measure of your piece of the pie in degrees ? | how many degrees is the smaller slice ? |
there is a half an apple pie left . you want to eat twice what your little brother eats , but you also need to save a slice for your mom . you can cut her a slice that is 30 degrees . what is the measure of your piece of the pie in degrees ? so to tackle this , we just have to remember a few things . we have to remembe... | so that is 2x . x is the measure of this angle , and then 2x is a measure of that angle . so you see that 30 degrees plus x plus 2x , or x plus 2x plus 30 degrees , is going to be equal to 180 degrees . | what is the measure of abc ? |
as we 've already seen , the 5th century bce starts off with athens and sparta and various greek city-states fighting on the same side against the persian invaders . but as we saw in the last video , as soon as the persians are dealt with , tensions start to rise between athens and sparta and their various allies . sp... | the athenian navy is getting more and more powerful . and once again , things culminate , and now this is the beginning of the actual peloponnesian war , the thing that people are referring to when they talk about the peloponnesian war . in 431 bce , the king of sparta is convinced or is , i guess you could say he is i... | why are these wars ( greco-persian , peloponnesian ) so important that they are taught to children across europe and in other societies of european origin ? |
as we 've already seen , the 5th century bce starts off with athens and sparta and various greek city-states fighting on the same side against the persian invaders . but as we saw in the last video , as soon as the persians are dealt with , tensions start to rise between athens and sparta and their various allies . sp... | and so this is the attack at syracuse . attack at syracuse , the failed attempt of the athenians to get syracuse , to get syracuse . and this is a two-year period of time , because once again , this is no joke to send your navy and to try to get at syracuse . | what was the point of syracuse ? |
as we 've already seen , the 5th century bce starts off with athens and sparta and various greek city-states fighting on the same side against the persian invaders . but as we saw in the last video , as soon as the persians are dealt with , tensions start to rise between athens and sparta and their various allies . sp... | and so this is the attack at syracuse . attack at syracuse , the failed attempt of the athenians to get syracuse , to get syracuse . and this is a two-year period of time , because once again , this is no joke to send your navy and to try to get at syracuse . | did syracuse use navel or ground or both to fend off athens ? |
as we 've already seen , the 5th century bce starts off with athens and sparta and various greek city-states fighting on the same side against the persian invaders . but as we saw in the last video , as soon as the persians are dealt with , tensions start to rise between athens and sparta and their various allies . sp... | attica 's something you 'll hear a lot about . it is this region right over here , this little out-jutting of land , that athens is on . and this first phase of the peloponnesian war is called the archidamian war , named for the king of sparta who somewhat reluctantly decides to invade attica . | when rome took over athens the romans took the statue of athena from the parthenon right ? |
as we 've already seen , the 5th century bce starts off with athens and sparta and various greek city-states fighting on the same side against the persian invaders . but as we saw in the last video , as soon as the persians are dealt with , tensions start to rise between athens and sparta and their various allies . sp... | but it 's not a great victory because as you can imagine , you have towns that have been destroyed , large parts of greece have been weakened , and it leaves the whole area open to attack from others . and as we will see in the next century , in the 4th century , we have phillip of macedon , or macedon depending on how... | so what 's the difference between `` macedon '' and `` macedonia '' ? |
as we 've already seen , the 5th century bce starts off with athens and sparta and various greek city-states fighting on the same side against the persian invaders . but as we saw in the last video , as soon as the persians are dealt with , tensions start to rise between athens and sparta and their various allies . sp... | the third phase is often called the ionian war . ionian war . ionia is this region that 's now in modern day , off the coast of modern-day turkey . | which city-states were neutral during the peloponnesian war ? |
as we 've already seen , the 5th century bce starts off with athens and sparta and various greek city-states fighting on the same side against the persian invaders . but as we saw in the last video , as soon as the persians are dealt with , tensions start to rise between athens and sparta and their various allies . sp... | and we even saw it in the last video , you have an earthquake in sparta , potentially right around the same time that sparta was planning an invasion of athens , leaving the spartans vulnerable . there 's a helot uprising , these spartan slaves . the athenians send hoplites to apparently help the spartans but the spart... | did the helot uprising continue thrueout the peloponnesian war ? |
as we 've already seen , the 5th century bce starts off with athens and sparta and various greek city-states fighting on the same side against the persian invaders . but as we saw in the last video , as soon as the persians are dealt with , tensions start to rise between athens and sparta and their various allies . sp... | the third phase is often called the ionian war . ionian war . ionia is this region that 's now in modern day , off the coast of modern-day turkey . | what happened after the peloponnesian war ? |
as we 've already seen , the 5th century bce starts off with athens and sparta and various greek city-states fighting on the same side against the persian invaders . but as we saw in the last video , as soon as the persians are dealt with , tensions start to rise between athens and sparta and their various allies . sp... | as we 've already seen , the 5th century bce starts off with athens and sparta and various greek city-states fighting on the same side against the persian invaders . but as we saw in the last video , as soon as the persians are dealt with , tensions start to rise between athens and sparta and their various allies . | do we know that from the greek historian sources as well ? |
let 's say you 're going to the grocery store , but you realize you left your keys in your room . you go up to get them and then walk into the room and look around for a second , and realize you completely forgot why you even came here in the first place . we all know how frustrating this can be , right ? but this sce... | but it is certainly not a normal part of aging . so , dementia , in very general terms , is something we use to describe when someone has troubles learning , remembering , and communicating . but where does alzheimer 's disease fit into this ? well , alzheimer 's disease is a type of dementia . | can someone be diagnosed with alzheimer 's , or do you have to guess , like schizophrenia ? |
let 's say you 're going to the grocery store , but you realize you left your keys in your room . you go up to get them and then walk into the room and look around for a second , and realize you completely forgot why you even came here in the first place . we all know how frustrating this can be , right ? but this sce... | for example somebody with dementia might have troubles with speaking or writing coherently , or understanding what was spoken or written . they also might have trouble recognizing their surroundings , especially when those surroundings should normally be very familiar to them . planning and performing tasks that requir... | also , are the symptoms universal , or do they vary from patient to patient ? |
let 's say you 're going to the grocery store , but you realize you left your keys in your room . you go up to get them and then walk into the room and look around for a second , and realize you completely forgot why you even came here in the first place . we all know how frustrating this can be , right ? but this sce... | when problems remembering or problems with your thinking skills in general become so severe and so common that they actually interfere with your daily life , it might be diagnosed as dementia . dementia , though , is not a specific disease . what do i mean by that ? | would chronic traumatic encephalopathy be considered a form of dementia ? |
let 's say you 're going to the grocery store , but you realize you left your keys in your room . you go up to get them and then walk into the room and look around for a second , and realize you completely forgot why you even came here in the first place . we all know how frustrating this can be , right ? but this sce... | when problems remembering or problems with your thinking skills in general become so severe and so common that they actually interfere with your daily life , it might be diagnosed as dementia . dementia , though , is not a specific disease . what do i mean by that ? | can dementia happen in younger people ? |
let 's say you 're going to the grocery store , but you realize you left your keys in your room . you go up to get them and then walk into the room and look around for a second , and realize you completely forgot why you even came here in the first place . we all know how frustrating this can be , right ? but this sce... | unfortunately , though , the main type of cells that alzheimer 's disease targets and affects are these precious neurons . and depending on where the affected neurons live in your brain , different functions of your brain can be affected . for example , if nerve cells in this area of your cerebrum are affected , you mi... | do all people have the same symptoms or are they different per person ? |
let 's say you 're going to the grocery store , but you realize you left your keys in your room . you go up to get them and then walk into the room and look around for a second , and realize you completely forgot why you even came here in the first place . we all know how frustrating this can be , right ? but this sce... | but where does alzheimer 's disease fit into this ? well , alzheimer 's disease is a type of dementia . specifically , it 's what 's known as a neuro degenerative disease , and it counts for about 60 to 80 % of all cases of dementia , affecting about five million people . | are infectious agents the other cause of dementia apart from alzheimer ? |
let 's say you 're going to the grocery store , but you realize you left your keys in your room . you go up to get them and then walk into the room and look around for a second , and realize you completely forgot why you even came here in the first place . we all know how frustrating this can be , right ? but this sce... | and to make matters worse , many times , patients are n't even aware that they 're experiencing any troubles or any sort of cognitive deficiencies at all . now , dementia is most common in the elderly , especially after age 65 . but it is certainly not a normal part of aging . | do you have any ideas on keeping an elderly person 's mind active to prevent the worsening of dementia ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . | is the hypotenuse the longest side of the right triangle , or the side opposite of the right angle ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . | if you square root the whole equation , should n't it be a+b=c ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and in this circumstance we 're solving for the hypotenuse . and we know that because this side over here , it is the side opposite the right angle . if we look at the pythagorean theorem , this is c. this is the longest side . | can you find the height of the triangle when you know the side lengths ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . | other than ( 3,4,5 ) , which sets of whole numbers are solutions to the equation a^2+b^2=c^2 ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | so let 's say that that is my triangle , and this is the 90 degree angle right there . and i think you know how to do this already . you go right what it opens into . | how do you know when to simplify and when not to ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | so this is going to be 108 . so that 's what b squared is , and now we want to take the principal root , or the positive root , of both sides . and you get b is equal to the square root , the principal root , of 108 . now let 's see if we can simplify this a little bit . | what is a principle root ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | let 's say a is equal to 6 . and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . | how do you know which leg is `` a '' and the other is `` b '' ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | let me tell you what the pythagorean theorem is . so if we have a triangle , and the triangle has to be a right triangle , which means that one of the three angles in the triangle have to be 90 degrees . and you specify that it 's 90 degrees by drawing that little box right there . | so the square in the middle of the triangle multiplies the outer numbers ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | you go opposite the right angle . the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . | is the hypotenuse the longest side of any triangle or just the longest side of a right triangle ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . | is there a negative square root ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . | why does the pythagorean theorem not work on other triangles like obtuse or acute ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . | is there any way to solve for the missing angle using pythagoras theorem ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . | is there a way to make the hypotenuse the same as another side ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . | why ca n't we just do the square root of 108 even if it ends up a decimal ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . | i learnt that c is the hypotenuse , however , how do you know which side is a and which is b ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . | is there a type of pythagorean theorem for other triangles ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | now let 's see if we can simplify this a little bit . the square root of 108 . and what we could do is we could take the prime factorization of 108 and see how we can simplify this radical . | at the end why would you put under a square root/ radical sign ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . | what does pythagorean theorem actually means ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | that is 16 . and 3 squared is the same thing as 3 times 3 . so that is 9 . | is 3,4,5 and its multiples the only numbers that make the pythagorean theorem work ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . | why does it have to be a right triangle ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | let 's say that our triangle looks like this . and that is our right angle . let 's say this side over here has length 12 , and let 's say that this side over here has length 6 . | why would n't the pythagorean theorem work on non-right angle triangles ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and it 's good to know , because we 'll keep referring to it . and just so we always are good at identifying the hypotenuse , let me draw a couple of more right triangles . so let 's say i have a triangle that looks like that . | does the theorem work for non-right triangles ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | 4 times 9 , this is 36 . so this is the square root of 36 times the square root of 3 . the principal root of 36 is 6 . | why does sal write the square root of the last 3 ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . | so only right triangle works ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . | if the triangle edge opposite the right angle is called the hypotenuse , do the other two sides have their own names as well ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . | is hypotenuse the opposite of the 90 degree angle of a triangle or the opposite of any angle that we want to take ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give you one more piece of terminology . | is there a way to apply the pythagorean theorem to find the other two sides if you know only one side ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . | why is the hypotenuse the longest side ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . | how many laws and theorems are there in math ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | we have the right angle here . you go opposite the right angle . the longest side , the hypotenuse , is right there . | if the triangle edge opposite the right angle is called the hypotenuse , do the other two sides have their own names as well ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and before i show you how to do that , let me give you one more piece of terminology . the longest side of a right triangle is the side opposite the 90 degree angle -- or opposite the right angle . so in this case it is this side right here . | if it says the hypotenuse is the side opposite the right angle does that mean that pythagoras ' theorem is only used to figure out the hypotenuse ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . | if a^2 + b^2 = c^2 ; does that mean a + b = c ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | what is this ? 2 times 2 is 4 . 4 times 9 , this is 36 . | if a^2 + b^2 = c^2 , does n't that mean you can make it more simple and do a + b = c ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | so 25 is equal to c squared . and we could take the positive square root of both sides . i guess , just if you look at it mathematically , it could be negative 5 as well . | how could you solve for the square root of pi ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | the longest side , the hypotenuse , is right there . so if we think about the pythagorean theorem -- that a squared plus b squared is equal to c squared -- 12 you could view as c. this is the hypotenuse . the c squared is the hypotenuse squared . so you could say 12 is equal to c. and then we could say that these sides... | could n't we just do a + b = c ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | or , we could call it a right angle . and a triangle that has a right angle in it is called a right triangle . so this is called a right triangle . now , with the pythagorean theorem , if we know two sides of a right triangle we can always figure out the third side . and before i show you how to do that , let me give y... | can a triangle have two right angles ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | so let 's say i have a triangle that looks like that . and i were to tell you that this angle right here is 90 degrees . in this situation this is the hypotenuse , because it is opposite the 90 degree angle . | 90 + 90 is 180 , and the other angle would have 0 which is no angle ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and then we say b -- this colored b -- is equal to question mark . and now we can apply the pythagorean theorem . a squared , which is 6 squared , plus the unknown b squared is equal to the hypotenuse squared -- is equal to c squared . | can the pythagorean theorem be applyed to polygons or other shapes ? |
in this video we 're going to get introduced to the pythagorean theorem , which is fun on its own . but you 'll see as you learn more and more mathematics it 's one of those cornerstone theorems of really all of math . it 's useful in geometry , it 's kind of the backbone of trigonometry . you 're also going to use it ... | and the way to figure out where that right triangle is , and kind of it opens into that longest side . that longest side is called the hypotenuse . and it 's good to know , because we 'll keep referring to it . | why is the longer line called the hypotenuse ? |
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