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Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | You get 4 is equal to x squared. And if we were just purely solving this equation, we would get x is equal to plus or minus 2. Now we're saying that the function is only defined over this interval, so negative 2 isn't part of its domain, so we're only going to focus on x is equal to 2. This right over here is definitel... |
Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | This right over here is definitely a critical number. Now have we found all of the critical numbers? Well, this is the only number other than negative 2, the only number in the interval that will make f prime of x equal to zero. What about where it's undefined? Well, f prime of x would be undefined. The only place it w... |
Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | What about where it's undefined? Well, f prime of x would be undefined. The only place it would be undefined is if you stuck a zero right over here in the denominator, but zero is not in the interval. So the only critical number in the interval is x equals 2. Now we just have to test f at the different endpoints and at... |
Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | So the only critical number in the interval is x equals 2. Now we just have to test f at the different endpoints and at the critical number and see which of those is the highest. We're going to test f of 1, which is equal to 8 times the natural log of 1 minus 1 squared. We'll test f of 4, which is equal to 8 times the ... |
Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | We'll test f of 4, which is equal to 8 times the natural log of 4 minus 4 squared, which is, of course, 16. And we're going to test f of 2. So these are the endpoints, and this is a critical number, 8 times the natural log of 2 minus minus 2 squared. Now which of these is going to be the largest? And it might be tempti... |
Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | Now which of these is going to be the largest? And it might be tempting to get a calculator out, but actually let's see if we can get a little intuition here. So this is the natural log of 1 is 0. e to the 0 power is equal to 1, so 8 times 0 is 0. So this evaluates to negative 1. Now let's see, what does this evaluate ... |
Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | So this evaluates to negative 1. Now let's see, what does this evaluate to? The natural log of 4, e is 2.7, on and on and on. So this number is going to be between 1 and 2. So it's going to be between 1 and 2. And it's actually going to be, well, between 1 and 2, you multiply that times 8, you're going to be between 8 ... |
Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | So this number is going to be between 1 and 2. So it's going to be between 1 and 2. And it's actually going to be, well, between 1 and 2, you multiply that times 8, you're going to be between 8 and 16. And then you subtract 16, so that means you're going to be between 0 and negative 8. So it's not clear, at least witho... |
Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | And then you subtract 16, so that means you're going to be between 0 and negative 8. So it's not clear, at least without using a calculator in this very rough way, which of these is large. Both of these are negative numbers, though. Now what about this, natural log of 2? Natural log of 2 is going to be some fraction. I... |
Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | Now what about this, natural log of 2? Natural log of 2 is going to be some fraction. It's going to be more than 1 half. And since it's more than 1 half, this whole thing is going to be more than 4, which means this whole thing is going to be positive. So this is negative, this is negative, this is positive. And these ... |
Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | And since it's more than 1 half, this whole thing is going to be more than 4, which means this whole thing is going to be positive. So this is negative, this is negative, this is positive. And these are our only candidates for our maximum value. So I would go with this one. Our maximum value happens when x is equal to ... |
Finding absolute extrema on a closed interval AP Calculus AB Khan Academy.mp3 | So I would go with this one. Our maximum value happens when x is equal to 2, and that maximum value is 8 natural log of 2 minus 4. That is the absolute maximum value over the interval, I guess we could say over the domain that this function is defined. If we want to verify it with a calculator, we of course could. So w... |
Creating a slope field First order differential equations Khan Academy.mp3 | Let's say that we have the differential equation dy dx, or the derivative of y with respect to x, is equal to negative x over y. Let's say we don't know how to find the solutions to this, but we at least want to get a sense of what the solutions might look like. And to do that, what we could do is we could look at a co... |
Creating a slope field First order differential equations Khan Academy.mp3 | So, let me draw some axes here. So, let me draw a relatively straight line. Alright, so that's my y-axis, and this is my x-axis. Let me do draw, let me mark this as one, that's two, that's negative one, negative two, one, two, negative one, and negative two. And what I could do is, since this differential equation is j... |
Creating a slope field First order differential equations Khan Academy.mp3 | Let me do draw, let me mark this as one, that's two, that's negative one, negative two, one, two, negative one, and negative two. And what I could do is, since this differential equation is just in terms of x's and y's and first derivatives of y with respect to x, I could go, I could sample points on the coordinate pla... |
Creating a slope field First order differential equations Khan Academy.mp3 | Let me set up a little table here. Let me do a little table here to do a bunch of x and y values. Once again, I'm just sampling some points on the coordinate plane to be able to visualize. So, x, y, and this is dy, dx. So, let's say when x is, let's say when x is zero and y is one, what is the derivative of y with resp... |
Creating a slope field First order differential equations Khan Academy.mp3 | So, x, y, and this is dy, dx. So, let's say when x is, let's say when x is zero and y is one, what is the derivative of y with respect to x? It's going to be negative zero over one, so it's just going to be zero. And so, at the point zero, one, if a solution goes through this point, its slope is going to be zero. And s... |
Creating a slope field First order differential equations Khan Academy.mp3 | And so, at the point zero, one, if a solution goes through this point, its slope is going to be zero. And so, we can visualize that by doing a little horizontal line segment right there. So, let's keep going. What about when x is one and y is one? Well then, dy, dx, the derivative of y with respect to x is negative one... |
Creating a slope field First order differential equations Khan Academy.mp3 | What about when x is one and y is one? Well then, dy, dx, the derivative of y with respect to x is negative one over one. So, it's going to be negative one. So, at the point one, comma, one, if a solution goes through that point, it would have a slope of negative one. And so, I draw a little line segment that has a slo... |
Creating a slope field First order differential equations Khan Academy.mp3 | So, at the point one, comma, one, if a solution goes through that point, it would have a slope of negative one. And so, I draw a little line segment that has a slope of negative one. What about when x is, let me do this in a new color, what about when x is one and y is zero? Well then, it's negative one over zero. So, ... |
Creating a slope field First order differential equations Khan Academy.mp3 | Well then, it's negative one over zero. So, this is actually undefined. But, it's a clue that maybe, maybe the slope there, I guess if you had a tangent line at that point, maybe it's vertical. So, I'll put that as a question mark. Vertical there. And so, maybe it's something like that if you actually did have, I guess... |
Creating a slope field First order differential equations Khan Academy.mp3 | So, I'll put that as a question mark. Vertical there. And so, maybe it's something like that if you actually did have, I guess it wouldn't be a function if you had some kind of relation that went through it. But, let's not draw that just yet, but let's try some other points. Let's say that we had, let's try the point n... |
Creating a slope field First order differential equations Khan Academy.mp3 | But, let's not draw that just yet, but let's try some other points. Let's say that we had, let's try the point negative one, negative one. So, now we have negative negative one, which is one over negative one. Well, you would have a slope of negative one here. So, negative one, negative one. You would have a slope of, ... |
Creating a slope field First order differential equations Khan Academy.mp3 | Well, you would have a slope of negative one here. So, negative one, negative one. You would have a slope of, you would have a slope of negative one. What about if you had one negative one? Well, now it's negative one over negative one. Your slope is now one. So, one negative one. |
Creating a slope field First order differential equations Khan Academy.mp3 | What about if you had one negative one? Well, now it's negative one over negative one. Your slope is now one. So, one negative one. If your solution, if a solution goes through this, its slope would look like that. And we could keep going. We could even do two negative two. |
Creating a slope field First order differential equations Khan Academy.mp3 | So, one negative one. If your solution, if a solution goes through this, its slope would look like that. And we could keep going. We could even do two negative two. That's going to have a slope of one as well. If you did positive two, positive two, that'd be negative two over two. You'd have a slope of negative one rig... |
Creating a slope field First order differential equations Khan Academy.mp3 | We could even do two negative two. That's going to have a slope of one as well. If you did positive two, positive two, that'd be negative two over two. You'd have a slope of negative one right over here. And so, we could do a bunch of points. Just keep going. I'm now just doing them in my head. |
Creating a slope field First order differential equations Khan Academy.mp3 | You'd have a slope of negative one right over here. And so, we could do a bunch of points. Just keep going. I'm now just doing them in my head. I'm not going on the table. But, you get a sense of what's going on here. Here, your slope, what if it was negative one, one. |
Creating a slope field First order differential equations Khan Academy.mp3 | I'm now just doing them in my head. I'm not going on the table. But, you get a sense of what's going on here. Here, your slope, what if it was negative one, one. It's going to have a slope of one. So, at this point, your slope, negative one, one. So, negative negative one is one over one. |
Creating a slope field First order differential equations Khan Academy.mp3 | Here, your slope, what if it was negative one, one. It's going to have a slope of one. So, at this point, your slope, negative one, one. So, negative negative one is one over one. So, you're going to have a slope like that. At negative two, two, same exact idea. It would look like that. |
Creating a slope field First order differential equations Khan Academy.mp3 | So, negative negative one is one over one. So, you're going to have a slope like that. At negative two, two, same exact idea. It would look like that. And so, you get a, when you keep drawing these line segments over these kind of, these sampled points in the Cartesian or in the X-Y plane, you start to get a sense of, ... |
Creating a slope field First order differential equations Khan Academy.mp3 | It would look like that. And so, you get a, when you keep drawing these line segments over these kind of, these sampled points in the Cartesian or in the X-Y plane, you start to get a sense of, well, what would a solution have to do? And you can start to visualize that, hey, maybe a solution, a solution would have to d... |
Creating a slope field First order differential equations Khan Academy.mp3 | This would be a solution. So, maybe it would have to do something like this. Or, if we're looking, if we're looking only at functions and not relations, I'll only, I'll make it so it's a very clear, so maybe it would have to do something like this. Or, if the function started like here, based on what we've seen so far,... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | You've already spent a lot of your mathematical lives talking about functions. The basic idea is give a valid input into a function, so a member of that function's domain, and then the function is going to tell you for that input what is going to be the corresponding output. And we call that corresponding output f of x... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | So for example, there's many ways of defining functions. You could say something like f of x is equal to x squared. So that means that whatever x, whatever you input into the function, the output is going to be that input squared. You could have something defined like this. F of x is equal to x squared if x odd, and yo... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | You could have something defined like this. F of x is equal to x squared if x odd, and you could say it's equal to x to the third otherwise. So if it's an odd integer, it's an odd integer, you just square it, but otherwise for any other real number, you take it to the third power. This is a valid way of defining a func... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | This is a valid way of defining a function. What we're going to do in this video is explore a new way, or a potentially new way for you, of defining a function, and that's by using a definite integral, but it's the same general idea. So what we have graphed here, this is the t-axis, this is the y-axis, and we have the ... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | If t is four, f of t is three. But I'm now going to define a new function based on a definite integral of f of t. Let's define our new function. Let's say g, let's call it g of x. Let's make it equal to the definite integral from negative two to x of f of t dt. Now pause this video, really take a look at it. This might... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | Let's make it equal to the definite integral from negative two to x of f of t dt. Now pause this video, really take a look at it. This might look really fancy, but what's happening here is given an input x, g of x is going to be based on what the definite integral here would be for that x. And so we can set up a little... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | And so we can set up a little table here to think about some potential values. So let's say x, and let's say g of x right over here. So if x is one, what is g of x going to be equal to? All right, so g of one is going to be equal to the definite integral going from negative two. Now x is going to be equal to one in thi... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | All right, so g of one is going to be equal to the definite integral going from negative two. Now x is going to be equal to one in this situation. That's what we're inputting into the function. So one is our upper bound of f of t dt. And what is that equal to? Well, that's going to be the area under the curve and above... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | So one is our upper bound of f of t dt. And what is that equal to? Well, that's going to be the area under the curve and above the t-axis between t equals negative two and t is equal to one. So it's gonna be this area here. And since it's on a grid, we can actually figure this out. We can actually break this up into tw... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | So it's gonna be this area here. And since it's on a grid, we can actually figure this out. We can actually break this up into two sections. This rectangular section is three wide and five high, so it has an area of 15 square units. And this little triangular section up here is two wide and one high, two times one time... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | This rectangular section is three wide and five high, so it has an area of 15 square units. And this little triangular section up here is two wide and one high, two times one times 1 1⁄2, area of a triangle. This is going to be another one. So that area is going to be equal to 16. What if x is equal to two? What is g o... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | So that area is going to be equal to 16. What if x is equal to two? What is g of two going to be equal to? Pause this video and try to figure that out. Well, g of two is going to be equal to the definite integral from negative two. And now our upper bound's going to be our input into the function to two of f of t dt. S... |
Functions defined by definite integrals (accumulation functions) AP Calculus AB Khan Academy.mp3 | Pause this video and try to figure that out. Well, g of two is going to be equal to the definite integral from negative two. And now our upper bound's going to be our input into the function to two of f of t dt. So that's going to be going from here all the way now to here. And so it's the area we just calculated. It's... |
Theorem for limits of composite functions Limits and contiuity AP Calculus Khan Academy.mp3 | In this video, we're gonna try to understand limits of composite functions, or at least a way of thinking about limits of composite functions. And in particular, we're gonna think about the case where we're trying to find the limit as x approaches a of f of g of x. And we're going to see under certain circumstances, th... |
Theorem for limits of composite functions Limits and contiuity AP Calculus Khan Academy.mp3 | And what are those circumstances you are asking? Well, this is going to be true if and only if two things are true. First of all, this limit needs to exist. So the limit as x approaches a of g of x needs to exist. So that needs to exist. And then on top of that, the function f needs to be continuous at this point, and ... |
Theorem for limits of composite functions Limits and contiuity AP Calculus Khan Academy.mp3 | So the limit as x approaches a of g of x needs to exist. So that needs to exist. And then on top of that, the function f needs to be continuous at this point, and f continuous at L. So let's look at some examples and see if we can apply this idea, or see if we can't apply it. So here I have two functions that are graph... |
Theorem for limits of composite functions Limits and contiuity AP Calculus Khan Academy.mp3 | So here I have two functions that are graphically represented right over here. Let me make sure I have enough space for them. And what we see on the left-hand side is our function f, and what we see on the right-hand side is our function g. So first, let's figure out what is the limit as x approaches negative three of ... |
Theorem for limits of composite functions Limits and contiuity AP Calculus Khan Academy.mp3 | Pause this video and see, first of all, does this theorem apply? And if it does apply, what is this limit? So the first thing we need to see is does this theorem apply? So first of all, if we were to find the limit as x approaches negative three of g of x, what is that? Well, when we're approaching negative three from ... |
Theorem for limits of composite functions Limits and contiuity AP Calculus Khan Academy.mp3 | So first of all, if we were to find the limit as x approaches negative three of g of x, what is that? Well, when we're approaching negative three from the right, it looks like our function is actually at three. And it looks like when we're approaching negative three from the left, it looks like our function is at three... |
Theorem for limits of composite functions Limits and contiuity AP Calculus Khan Academy.mp3 | So it looks like this limit is three, even though the value g of negative three is negative two, but it's a point discontinuity. As we approach it from either side, the value of the function is at three. So this thing is going to be three. So it exists, so we meet that first condition. And then the second question is, ... |
Theorem for limits of composite functions Limits and contiuity AP Calculus Khan Academy.mp3 | So it exists, so we meet that first condition. And then the second question is, is our function f continuous at this limit, continuous at three? So when x equals three, yeah, it looks like at that point, our function is definitely continuous. And so we could say that this limit is going to be the same thing as this equ... |
2015 AP Calculus AB 5c AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | So this is true if and only if F prime prime of x goes from, goes from positive, positive to negative, negative, or vice versa, or vice versa. So where do we see F prime prime of x going from positive to negative? Well that's going to be true, that's going to be true if and only if F prime of x goes from being increasi... |
2015 AP Calculus AB 5c AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | I'm using a lot of vice versa here. So now let's, and I wanted to think of it in terms of F prime because we have the graph of F prime. So F prime goes from increasing to decreasing, or vice versa, or we could go from decreasing to increasing. Well let's think about it. Let's see, over here, F prime is, F prime is decr... |
2015 AP Calculus AB 5c AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | Well let's think about it. Let's see, over here, F prime is, F prime is decreasing, decreasing, decreasing, decreasing, and then it increases. So we have a point of inflection right over here, right when F prime of x is zero. So, and that's because F prime is differentiable, so the derivative is definitely, the derivat... |
2015 AP Calculus AB 5c AP Calculus AB solved exams AP Calculus AB Khan Academy.mp3 | So, and that's because F prime is differentiable, so the derivative is definitely, the derivative is zero right at that point of inflection right over here, so if that happens at x equals negative one and over here, then F prime starts increasing, but then it, right at x equals one, then it starts decreasing. So at x e... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | And what we want to do without having to graph g, we want to figure out at what x values does g have a relative maximum. And just to remind us what's going on at a relative maximum, so let me draw a hypothetical function right over here. So a relative maximum is going to happen, so you can visually inspect this, okay, ... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | These all look like relative maxima. And what's in common? Well, the graph, the function, is going from increasing to decreasing at each of those points. It's going from increasing to decreasing. Increasing to decreasing at either of the points. Or you could say that the first derivative is going from positive to negat... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | It's going from increasing to decreasing. Increasing to decreasing at either of the points. Or you could say that the first derivative is going from positive to negative. So if you look at this interval right over here, g prime is greater than zero, and then over the next interval, when you're decreasing, g prime would... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So if you look at this interval right over here, g prime is greater than zero, and then over the next interval, when you're decreasing, g prime would be less than zero. So what we really need to think about is when does g prime, so let me see, relative, we care about relative maximum point, and so that's essentially as... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | And the values that we could look at, or the points, are our critical points. And critical points are where g prime is either zero or it is undefined. So let's think about it. Where is g prime of x equal to zero? G prime of x is equal to zero when, let's just take g prime of x. We're gonna leverage the power rule right... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | Where is g prime of x equal to zero? G prime of x is equal to zero when, let's just take g prime of x. We're gonna leverage the power rule right here. Four x to the third power, four x to the third minus five x to the fourth, minus five x to the fourth is equal to zero. Let's see, we can factor out an x to the third. W... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | Four x to the third power, four x to the third minus five x to the fourth, minus five x to the fourth is equal to zero. Let's see, we can factor out an x to the third. We have x to the third times four minus five x is equal to zero. So this is going to happen when x is equal to zero. That I could, let me not skip steps... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So this is going to happen when x is equal to zero. That I could, let me not skip steps. So this is going to happen when x to the third is equal to zero, or four minus five x is equal to zero. For x to the third equaling zero, well that's only gonna happen when x is equal to zero. And four minus five x equaling zero, w... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | For x to the third equaling zero, well that's only gonna happen when x is equal to zero. And four minus five x equaling zero, we'll add five x to both sides. You get four is equal to five x. Divide both sides by five, you get four fifths is equal to x. So here, these are the two places where our derivative is equal to ... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So here, these are the two places where our derivative is equal to zero. Now are there any places where our derivative is undefined? Well, our function right over here is just a straight up polynomial. Our derivative is another polynomial. It is defined for all real numbers. So these are our two critical, our critical ... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | Our derivative is another polynomial. It is defined for all real numbers. So these are our two critical, our critical points, or we could even say critical values. So let's think about what g prime is doing on either side of these critical values. And I'll draw a little number line here to help us visualize this. And s... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So let's think about what g prime is doing on either side of these critical values. And I'll draw a little number line here to help us visualize this. And so, so there we go, a little bit of a number line. And let's see, we care about zero, and we care about four fifths. So let's say this is negative one, this is zero,... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | And let's see, we care about zero, and we care about four fifths. So let's say this is negative one, this is zero, this is one. And so we have one critical point at, let me do this in magenta. We have one critical point here at x equals zero. And then we have another critical point, I will do this at x equals four fift... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | We have one critical point here at x equals zero. And then we have another critical point, I will do this at x equals four fifths. So four fifths is right around there. So that is four fifths. And let's just think about what g, what g prime is doing in these intervals. And these critical points are the only places wher... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So that is four fifths. And let's just think about what g, what g prime is doing in these intervals. And these critical points are the only places where g prime might switch sides, switch signs. So let's first think about this, let me pick some colors I haven't used yet. So let's think about the interval from negative ... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So let's first think about this, let me pick some colors I haven't used yet. So let's think about the interval from negative infinity to zero. So this is the open interval from negative infinity to zero. And we could just plug in a value, we could, let's try negative one. Negative one is pretty straightforward to evalu... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | And we could just plug in a value, we could, let's try negative one. Negative one is pretty straightforward to evaluate. So let's see, you have four, you're gonna have four times negative one to the third power so that's gonna be four times negative one, minus five times negative one to the fourth power. So that's just... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So that's just gonna be one. So let's see, this is going to be negative four minus five, which is negative nine. So right over here, g prime is equal to negative nine. And so we know over this whole interval, since it's to the left of this critical point, we know that g prime of x is less than zero. And so our function... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | And so we know over this whole interval, since it's to the left of this critical point, we know that g prime of x is less than zero. And so our function itself is decreasing over this interval. And so we know we need to go from increasing to decreasing. So you can already say, well we can't go from increasing to decrea... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So you can already say, well we can't go from increasing to decreasing at this critical point because we're already decreasing to the left of it. But anyway, let's just think about what's happening in the other intervals. So in the interval between zero and 4 5ths, so that interval right over there, so it's between zer... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | Well let's just sample a number there. Let's say the number, I don't know, 1 1 2 might be really straightforward. So we can evaluate g prime of 1 1 2. G prime of 1 1 2 is equal to four times 1 1 2 to the third power. 1 1 2 to the third power is 1 8th. So it's 4 8ths, or it's just 1 1 2, minus five times 1 1 2 to the fo... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | G prime of 1 1 2 is equal to four times 1 1 2 to the third power. 1 1 2 to the third power is 1 8th. So it's 4 8ths, or it's just 1 1 2, minus five times 1 1 2 to the fourth. So that's 5 1 6ths. Minus 5 1 6ths. And so this is equal to 8 1 6ths minus 5 1 6ths, which is equal to 3 1 6ths, but the important thing is it's ... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So that's 5 1 6ths. Minus 5 1 6ths. And so this is equal to 8 1 6ths minus 5 1 6ths, which is equal to 3 1 6ths, but the important thing is it's equal to a positive value. So in this blue interval right over there, and actually let me put 4 5ths in a different color so we see that it's not part of that interval. So in ... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So in this blue interval right over there, and actually let me put 4 5ths in a different color so we see that it's not part of that interval. So in this light blue interval right here between zero and 4 5ths, g prime of x is greater than zero, so we know our function is increasing. And so let's see what's happening to ... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | And the easiest value to try out would just be 1. So let's try out x equals 1. It's in that interval. So when x equals 1, I'll just write g prime of 1 is equal to 4 minus 5, which is equal to negative 1. So g prime of x is less than zero. g prime of x is less than zero. So our function, so we could say g is increasing ... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So when x equals 1, I'll just write g prime of 1 is equal to 4 minus 5, which is equal to negative 1. So g prime of x is less than zero. g prime of x is less than zero. So our function, so we could say g is increasing here, it is decreasing, oh sorry, let me be careful. G is decreasing here, the function itself is decr... |
Worked example finding relative extrema AP Calculus AB Khan Academy.mp3 | So our function, so we could say g is increasing here, it is decreasing, oh sorry, let me be careful. G is decreasing here, the function itself is decreasing because our derivative is negative, then our function is increasing here because our derivative is positive, and then our function is decreasing here. So at what ... |
Finding derivative with fundamental theorem of calculus chain rule AP®︎ Calculus Khan Academy.mp3 | Let's say that we have the function capital F of x, which we're going to define as the definite integral from one to sine of x, so that's an interesting upper bound right over there, of two t minus one, and of course, dt. And what we are curious about is trying to figure out what is F prime of x going to be equal to? S... |
Finding derivative with fundamental theorem of calculus chain rule AP®︎ Calculus Khan Academy.mp3 | All right, so some of you might have been a little bit challenged by this notion of, hey, instead of an x on this upper bound, I now have a sine of x. If it was just an x, I could have used the fundamental theorem of calculus. Just to review that, if I had a function, let me call it h of x, if I have h of x that was de... |
Finding derivative with fundamental theorem of calculus chain rule AP®︎ Calculus Khan Academy.mp3 | It would just be two x minus one. Pretty straightforward. But this one isn't quite as straightforward. Instead of having an x up here, our upper bound is a sine of x. So one way to think about it is if we were to define g of x as being equal to sine of x, equal to sine of x, our capital F of x can be expressed as capit... |
Finding derivative with fundamental theorem of calculus chain rule AP®︎ Calculus Khan Academy.mp3 | Instead of having an x up here, our upper bound is a sine of x. So one way to think about it is if we were to define g of x as being equal to sine of x, equal to sine of x, our capital F of x can be expressed as capital F of x is the same thing as h of, h of, instead of an x, everywhere we see an x, we're replacing it ... |
Finding derivative with fundamental theorem of calculus chain rule AP®︎ Calculus Khan Academy.mp3 | You can see the g of x right over there. So you replace x with g of x for where in this expression, you get h of g of x, and that is capital F of x. Now why am I doing all of that? Well, this might start making you think about the chain rule, because if this is true, then that means that capital F prime of x is going t... |
Finding derivative with fundamental theorem of calculus chain rule AP®︎ Calculus Khan Academy.mp3 | Well, this might start making you think about the chain rule, because if this is true, then that means that capital F prime of x is going to be equal to h prime of g of x, h prime of g of x, times g prime of x. And so what would that be? Well, we already know what h prime of x is. So, let me do this in another color. T... |
Finding derivative with fundamental theorem of calculus chain rule AP®︎ Calculus Khan Academy.mp3 | So, let me do this in another color. This part right over here is going to be equal to, everywhere we see an x here, we'll replace with a g of x. So it's going to be two, two times sine of x, two sine of x, and then minus one, minus one. This is this right over here. And then what's g prime of x? G prime of x, well, g ... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | We've already thought about what a definite integral means. If I'm taking the definite integral from a to b of f of x dx, I can just view that as the area below my function f, so if this is my y axis, this is my x axis, and y is equal to f of x, so something like that, y is equal to f of x, and if this is a and if this... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | So these are going to be equivalent. Let's say, let me just draw that scenario, so let me draw a scenario where it's my x axis, that is my y axis, and let's say I have, let's say I have a function that looks like that, so that is y is equal to g of x, and let's say that this right over here is a, and this right over he... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | Well, you might be tempted to say, hey, well, it's just the area again between my curve and the x axis. You might be tempted to say, hey, this is just going to be equal to five, but you have to be very careful, because if you're looking at the area above your curve and below your x axis versus below your curve and abov... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | So if I, in my horizontal axis, that is time. My vertical axis, this is velocity, and velocity is going to be measured in meters per second. Time is going to be measured in seconds. Time is measured in seconds, and actually I'm gonna do two scenarios here. So let's say that I have a first velocity time graph. Let's jus... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | Time is measured in seconds, and actually I'm gonna do two scenarios here. So let's say that I have a first velocity time graph. Let's just call it v one of t, which is equal to three, and it would be three meters per second. So one, two, three. So it would look like that. That is v one of t. And if I were to look at t... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | So one, two, three. So it would look like that. That is v one of t. And if I were to look at the definite integral going from time equals one to time equals five of v sub one of t dt, what would this be equal to? Well, here my function is above my t axis, so I'll just go from one to five, which will be around there, an... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | Well, here my function is above my t axis, so I'll just go from one to five, which will be around there, and I could just think about the area here, and this area is pretty easy to calculate. It's going to be three meters per second times four seconds. That's my change in time. And so this is going to be 12 meters. And... |
Negative definite integrals Integration and accumulation of change AP Calculus AB Khan Academy.mp3 | And so this is going to be 12 meters. And so this is going to be equal to 12. And one way to conceptualize this is this gives us our change in position. 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... |
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