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And we see that in each of these cases. Derivative is 0. Derivative is 0. Derivative is undefined. And we have a word for these points where the derivative is either 0 or the derivative is undefined. We call them critical points. So for the sake of this function, the critical points are, if we could include x sub 0, we... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
Derivative is undefined. And we have a word for these points where the derivative is either 0 or the derivative is undefined. We call them critical points. So for the sake of this function, the critical points are, if we could include x sub 0, we could include x sub 1, at x sub 0 and x sub 1 the derivative is 0, and x ... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
So for the sake of this function, the critical points are, if we could include x sub 0, we could include x sub 1, at x sub 0 and x sub 1 the derivative is 0, and x sub 2 where the function is undefined. Now, so if we have a non-endpoint minimum or maximum point, then it's going to be a critical point. But can we say it... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
If we find a critical point where the derivative is 0 or the derivative is undefined, is that going to be a maximum or minimum point? And to think about that, let's imagine this point right over here. So let's call this x sub 3. If we look at the tangent line right over here, if we look at the slope right over here, it... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
If we look at the tangent line right over here, if we look at the slope right over here, it looks like f prime of x sub 3 is equal to 0. So based on our definition of a critical point, x sub 3 would also be a critical point. But it does not appear to be a minimum or a maximum point. So a minimum or a maximum point, tha... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
So a minimum or a maximum point, that's not an endpoint. It's definitely going to be a critical point. But being a critical point by itself does not mean you're at a minimum or maximum point. So just to be clear, at all of these points, we're at a minimum or maximum point. This, we're at a critical point. All of these ... | Critical points introduction AP Calculus AB Khan Academy.mp3 |
And we saw it's really just the difference between the function and our approximation of the function. So for example, this distance right over here, that is our error at x is equal to b. And what we really care about is the absolute value of it, because at some points, f of x might be larger than the polynomial. Somet... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
Sometimes the polynomial might be larger than f of x. What we care is the absolute distance between them. And so what I want to do in this video is try to bound our error at some b. Try to bound our error. So say it's less than or equal to some constant value. Try to bound it at b for some b is greater than a. We're ju... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
Try to bound our error. So say it's less than or equal to some constant value. Try to bound it at b for some b is greater than a. We're just going to assume that b is greater than a. And we saw some tantalizing. We got to a bit of a tantalizing result that seems like we might be able to bound it in the last video. We s... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
We're just going to assume that b is greater than a. And we saw some tantalizing. We got to a bit of a tantalizing result that seems like we might be able to bound it in the last video. We saw that the n plus 1th derivative of our error function is equal to the n plus 1th derivative of our function, or that their absol... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
We saw that the n plus 1th derivative of our error function is equal to the n plus 1th derivative of our function, or that their absolute values would also be equal to. So if we can somehow bound the n plus 1th derivative of our function over some interval, an interval that matters to us, an interval that maybe has b i... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So let's see if we can do that. Well, let's just assume that we're in a reality where we do know something about the n plus 1th derivative of f of x. Let's say we do know that this, we do it in a color that I haven't used yet. I'll do it in white. So let's say that thing right over there looks something like that. So t... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
I'll do it in white. So let's say that thing right over there looks something like that. So that is f, the n plus 1th derivative. And I only care about it over this interval right over here. Who cares what it does later? I just want to bound it over the interval because at the end of the day, I just want to bound b rig... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
And I only care about it over this interval right over here. Who cares what it does later? I just want to bound it over the interval because at the end of the day, I just want to bound b right over here. So let's say that the absolute value of this, let's say that we know, let me write it over here. Let's say that we k... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So let's say that the absolute value of this, let's say that we know, let me write it over here. Let's say that we know that the absolute value of the n plus 1th derivative, the n plus 1th, and I apologize, I actually switched between the capital n and the lowercase n, and I did that in the last video. I shouldn't have... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
n plus 1th. So let's say we know that the n plus 1th derivative of f of x, the absolute value of it, let's say it's bounded, let's say it's less than or equal to some m over the interval, because we only care about the interval. It might not be bounded in general, but all we care is that it takes some maximum value ove... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So over the interval x, I could write it this way, over the interval x is a member between a and b. And this includes both of them. It's a closed interval. x could be a, x could be b, or x could be anything in between. And we can say this generally, that this derivative will have some maximum value. So this is the abso... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
x could be a, x could be b, or x could be anything in between. And we can say this generally, that this derivative will have some maximum value. So this is the absolute value is maximum value, max value, m for max. We know that it will have a maximum value if this thing is continuous. So once again, we're going to assu... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
We know that it will have a maximum value if this thing is continuous. So once again, we're going to assume that it is continuous, that it has some maximum value over this interval right over here. Well, this thing, this thing right over here, we know is the same thing as the n plus 1th derivative of the error function... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So then we know, so then that implies, that implies that the, that's a new color. Let me do that blue or that green. That implies that the n plus 1th derivative of the error function, the absolute value of it, because these are the same thing, is also bounded by m. So that's a little bit of an interesting result, but i... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
It might look similar, but this is the n plus 1th derivative of the error function. And we'll have to think about how we can get an m in the future. We're assuming that we somehow know it, and maybe we'll do some example problems where we figure that out. But this is the n plus 1th derivative. We bounded its absolute v... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
But this is the n plus 1th derivative. We bounded its absolute value, but we really want to bound the actual error function, the 0th derivative, you could say, the actual function itself. Well, we could try to integrate both sides of this and see if we can eventually get to e, to get to e of x, to get to our error func... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So let's do that. Let's take the integral, let's take the integral of both sides of this. Now the integral on this left-hand side, it's a little interesting. We take the integral of the absolute value. It would be easier if we were taking the absolute value of the integral. And lucky for us, the way it's set up, so let... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
We take the integral of the absolute value. It would be easier if we were taking the absolute value of the integral. And lucky for us, the way it's set up, so let me just write a little aside here. We know generally that if I take, and it's something for you to think about. If I take, so if I have two options, this opt... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
We know generally that if I take, and it's something for you to think about. If I take, so if I have two options, this option versus, and I know they look the same right now. So over here I'm going to have the integral of the absolute value, and over here I'm going to have the absolute value of the integral. Which of t... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
Which of these is going to be, which of these can be larger? Well, you just have to think about the scenarios. If f of x is always positive over the interval that you're taking the integration, then they're going to be the same thing. So you're going to get positive values, take the absolute value of a positive value, ... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So you're going to get positive values, take the absolute value of a positive value, it doesn't make a difference. What matters is if f of x is negative. If f of x is negative the entire time, so if this is our x-axis, that is our y-axis. If f of x is, well we saw if it's positive the entire time, you're taking the abs... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
If f of x is, well we saw if it's positive the entire time, you're taking the absolute value of a positive, absolute value of a positive, it's not going to matter, these two things are going to be equal. If f of x is negative the whole time, then you're going to get, then this integral is going to evaluate to a negativ... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So you can imagine a situation like this. If f of x looks something like that, then this right over here, the integral, you'd have positive, this would be positive, and then this would be negative right over here, and so they would cancel each other out. So this would be a smaller value than if you took the integral of... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So the integral, the absolute value of f would look something like this. So all of the areas are going to be, if you view the integral, if you view this as maybe a definite integral, all of the areas would be positive. So you're going to get a bigger value when you take the integral of the absolute value, then you will... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
Because once again, if you took the integral first for something like this, you would get a low value because this stuff would cancel out with this stuff right over here, and then you would take the absolute value of just a lower magnitude number. And so in general, the absolute value of the integral is going to be les... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
And the reason why this is useful is that we can still keep the inequality that this is less than or equal to this, but now this is a pretty straightforward integral to evaluate. The anti-derivative of the n plus 1th derivative is going to be the nth derivative. So this business right over here is just going to be the ... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
The absolute value of the nth derivative of our error function. Did I say expected value? I shouldn't. See, it even confuses me. This is the error function. I should have used r, r for remainder, but this is all error. Nothing about probability or expected value in this video. | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
See, it even confuses me. This is the error function. I should have used r, r for remainder, but this is all error. Nothing about probability or expected value in this video. This is e for error. So anyway, this is going to be the nth derivative of our error function, which is going to be less than or equal to this, wh... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
Nothing about probability or expected value in this video. This is e for error. So anyway, this is going to be the nth derivative of our error function, which is going to be less than or equal to this, which is less than or equal to the anti-derivative of m. Well, that's a constant, so that's going to be mx. And since ... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
And since we're just taking indefinite integrals, we can't forget the idea that we have a constant over here. And in general, when you're trying to create an upper bound, you want as low of an upper bound as possible. So we want to minimize what this constant is. And lucky for us, we do know what value this function ta... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
And lucky for us, we do know what value this function takes on at a point. We know that the nth derivative of our error function at a is equal to 0. I think we wrote it over here. The nth derivative at a is equal to 0, and that's because the nth derivative of the function and the approximation at a are going to be the ... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
The nth derivative at a is equal to 0, and that's because the nth derivative of the function and the approximation at a are going to be the same exact thing. And so if we evaluate both sides of this at a, and I'll do it over here on the side, we know that the absolute value of the nth derivative at a, we know that this... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
That is the lowest possible c that will meet these constraints that we know are true. So we will actually pick c to be negative ma. And then we can rewrite this whole thing as the absolute value of the nth derivative of the error function, not the expected value. I have a strange suspicion I might have said expected va... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
I have a strange suspicion I might have said expected value, but this is the error function. The absolute value of the nth derivative of the error function is less than or equal to m times x minus a. And once again, all of the constraints hold. This is for x as part of the interval, the closed interval between a and b.... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
This is for x as part of the interval, the closed interval between a and b. But it looks like we're making progress. We at least went from the n plus 1th derivative to the nth derivative. Let's see if we can keep going. So same general idea. If we know this, then we know that we can take the integral of both sides of t... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
Let's see if we can keep going. So same general idea. If we know this, then we know that we can take the integral of both sides of this. So we can take the integral of both sides of this, the antiderivative of both sides. And we know from what we figured out up here that something that's even smaller than this right ov... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So we can take the integral of both sides of this, the antiderivative of both sides. And we know from what we figured out up here that something that's even smaller than this right over here is the absolute value of the integral of the expected value. See, I said it. Of the error function, not the expected value. Of th... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
Of the error function, not the expected value. Of the error function, the nth derivative of the error function of x dx. So we know that this is less than or equal to, based on the exact same logic there. And this is useful because this is just going to be the n minus 1th derivative of our error function of x. And of co... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
And this is useful because this is just going to be the n minus 1th derivative of our error function of x. And of course, we have the absolute value outside of it. And now this is going to be less than or equal to. It's less than or equal to this, which is less than or equal to this right over here. The antiderivative ... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
It's less than or equal to this, which is less than or equal to this right over here. The antiderivative of this right over here is going to be m times x minus a squared over 2. You can do u substitution if you want. Or you can just say, hey, look, I have a little expression here. Its derivative is 1, so it's implicitl... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
Or you can just say, hey, look, I have a little expression here. Its derivative is 1, so it's implicitly there. So I can just treat it as kind of a u. So raise it to an exponent and then divide that exponent. But once again, I'm taking indefinite integrals. So I'm going to say a plus c over here. But let's use that sam... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So raise it to an exponent and then divide that exponent. But once again, I'm taking indefinite integrals. So I'm going to say a plus c over here. But let's use that same exact logic. If we evaluate this at a, you're going to have it. If we evaluate this whole, let's evaluate both sides of this at a. The left side eval... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
But let's use that same exact logic. If we evaluate this at a, you're going to have it. If we evaluate this whole, let's evaluate both sides of this at a. The left side evaluated at a we know is going to be 0. We figured that out up here in the last video. So you get 0 when you evaluate the left side at a. The right si... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
The left side evaluated at a we know is going to be 0. We figured that out up here in the last video. So you get 0 when you evaluate the left side at a. The right side evaluated at a, you get m times a minus a squared over 2. So you're going to get 0 plus c. So you're going to get 0 is less than or equal to c. Once aga... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
The right side evaluated at a, you get m times a minus a squared over 2. So you're going to get 0 plus c. So you're going to get 0 is less than or equal to c. Once again, we want to minimize our constant. We want to minimize our upper bound over here. So we want to pick the lowest possible c that meets our constraints.... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So we want to pick the lowest possible c that meets our constraints. So the lowest possible c that meets our constraint is 0. So the general idea here is that we can keep doing this. We can keep doing exactly what we're doing all the way until we keep integrating it. The exact same way that I've done it, all the way th... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
We can keep doing exactly what we're doing all the way until we keep integrating it. The exact same way that I've done it, all the way that we get, and using this exact same property here, all the way until we get the bound on the error function of x. So you could view this as the 0th derivative. You're going all the w... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
You're going all the way to the 0th derivative, which is really just the error function. The bound on the error function of x is going to be less than or equal to. What's it going to be? You can already see the pattern here. It's going to be m times x minus a. The one way to think about it, this exponent plus this deri... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
You can already see the pattern here. It's going to be m times x minus a. The one way to think about it, this exponent plus this derivative is going to be equal to n plus 1. Now this derivative is 0, so this exponent is going to be n plus 1. Whatever that exponent is, you're going to have, and maybe I should have done ... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
Now this derivative is 0, so this exponent is going to be n plus 1. Whatever that exponent is, you're going to have, and maybe I should have done it, you're going to have n plus 1 factorial over here. If you say, wait, where does this n plus 1 factorial come from? I just had a 2 here. Well, think about what happens whe... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
I just had a 2 here. Well, think about what happens when we integrate this again. You're going to raise this to the 3rd power and then divide by 3. So your denominator is going to have 2 times 3. Then when you integrate it again, you're going to raise it to the 4th power and then divide by 4. So then your denominator i... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
So your denominator is going to have 2 times 3. Then when you integrate it again, you're going to raise it to the 4th power and then divide by 4. So then your denominator is going to be 2 times 3 times 4, or 4 factorial. So whatever power you're raising to, the denominator is going to be that power factorial. What's re... | Taylor polynomial remainder (part 2) Series AP Calculus BC Khan Academy.mp3 |
Let's say we have the repeating decimal 0.4008, where the digits 4, 0, 0, 8 keep on repeating. So if we were to write it out, it would look something like this, 0.4008, 4, 0, 0, 8, 4, 0, 0, 8, and keeps on going forever. What I want you to do right now is pause the video and think about whether you can represent this r... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
So I'm assuming you've given a go at it, so let's think about it. So for each term of my infinite series, I'm going to represent one of these repeating patterns of 4008. So for example, I will make this 4008 my first term. So this could be viewed as 0, and this 4008 represents 0.4008. And then I could make this 4008 my... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
So this could be viewed as 0, and this 4008 represents 0.4008. And then I could make this 4008 my next term, or my next term will represent this 4008. And that means this 4008 is the same thing as 0.00004008. And then this next 4008, well, that represents 0. And we have eight 0's, 1, 2, 3, 4, 5, 6, 7, 8, 4008. And then... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
And then this next 4008, well, that represents 0. And we have eight 0's, 1, 2, 3, 4, 5, 6, 7, 8, 4008. And then we would just keep on going like that forever. So we're just going to keep on going like that forever. So hopefully there's a pattern here. We're essentially throwing four 0's before the decimal every time, a... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
So we're just going to keep on going like that forever. So hopefully there's a pattern here. We're essentially throwing four 0's before the decimal every time, and we could just keep on going like that forever. So this is an infinite sum. It's an infinite series. The next question is, is this a geometric series? Well, ... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
So this is an infinite sum. It's an infinite series. The next question is, is this a geometric series? Well, in order for it to be a geometric series, to go from one term to the next, you must be multiplying by the same value, by the same common ratio. So what are we multiplying when we go from 0.4008 to this one right... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
Well, in order for it to be a geometric series, to go from one term to the next, you must be multiplying by the same value, by the same common ratio. So what are we multiplying when we go from 0.4008 to this one right over here, where we add four 0's before the 4008? What are we multiplying? Well, we move the decimal f... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
Well, we move the decimal four spots to the left, so we're multiplying by 10 to the negative fourth. Or you could view it as we're multiplying by 0.001. 10 to the negative 1, 2, 3, 4. To go from here to here, well, same thing. Move the decimal four places to the left. So once again, we're multiplying by 0.0001. And so ... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
To go from here to here, well, same thing. Move the decimal four places to the left. So once again, we're multiplying by 0.0001. And so it looks pretty clear that we have a common ratio of 10 to the negative fourth power. So we can rewrite all of this business as 0.4008 times our common ratio for this first term, times... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
And so it looks pretty clear that we have a common ratio of 10 to the negative fourth power. So we can rewrite all of this business as 0.4008 times our common ratio for this first term, times our common ratio of 10 to the negative fourth to the zeroth power. So that gives us that right over there. Plus 0.4008 times 10 ... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
Plus 0.4008 times 10 to the negative fourth to the first power. And that gives us that value right over there. Plus 0.4008 times 10 to the negative fourth to the second power. And we keep on going. And so in this form, it looks a little bit clearer like a geometric series, an infinite geometric series. And if we wanted... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
And we keep on going. And so in this form, it looks a little bit clearer like a geometric series, an infinite geometric series. And if we wanted to write that out with sigma notation, we could write this as the sum from k equals 0 to infinity. To infinity of, well, what's our first term going to be? It's going to be 0.... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
To infinity of, well, what's our first term going to be? It's going to be 0.4008 times our common ratio, which we could write out as either 10 to the negative fourth or 0.0001. I'll just write it as 10 to the negative fourth to the k-th power. So the next interesting question, this clearly can be represented as a geome... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
So the next interesting question, this clearly can be represented as a geometric series, is, well, what is this sum? You might say, oh, that's just going to be 4008 repeating over and over. But I want to express it as a fraction. And so I want you to pause this video. Use what you already know about finding the sum of ... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
And so I want you to pause this video. Use what you already know about finding the sum of an infinite geometric series to try to express this thing right over here as a fraction. So I'm assuming you've had a go at it. So let's think about it. We've already seen, we've already derived in previous videos, that the sum of... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
So let's think about it. We've already seen, we've already derived in previous videos, that the sum of an infinite geometric series, now let me do this in a neutral color. If I have a series like this, k equals 0 to infinity of ar to the k power, that this sum is going to be equal to a over 1 minus r. We've derived thi... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
So in this case, this is going to be, well, our a here is 0.4008. And it's going to be that over 1 minus our common ratio. Minus, and I'll write it like this, 0.0001, 1 ten thousandth. So what's this going to be? Well, this is going to be the same thing as 0.4008. If you take 1 minus 1 ten thousandth, or you could use ... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
So what's this going to be? Well, this is going to be the same thing as 0.4008. If you take 1 minus 1 ten thousandth, or you could use this 10,000 ten thousandths, minus 1 ten thousandth, you're going to have 9,999 over, or 9,999 ten thousandths. Once again, let me write this out, just so this doesn't look confusing. 1... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
Once again, let me write this out, just so this doesn't look confusing. 1 is the same thing as 10,000 over 10,000. And you're subtracting 1 over 10,000. And so you're going to get 9,999 over 10,000. And so this is going to be the same thing as 0.4008 times 10,000 over 9,999. Well, what's this top number times 10,000? W... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
And so you're going to get 9,999 over 10,000. And so this is going to be the same thing as 0.4008 times 10,000 over 9,999. Well, what's this top number times 10,000? Well, that's just going to give us 4,008 over 9,999. And we've just expressed that repeating decimal as a fraction. So we have succeeded. And you might sa... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
Well, that's just going to give us 4,008 over 9,999. And we've just expressed that repeating decimal as a fraction. So we have succeeded. And you might say, well, maybe we can simplify this thing. And so let's see. This is already a fraction, so we've already kind of achieved it. But if we want to get a little bit simp... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
And you might say, well, maybe we can simplify this thing. And so let's see. This is already a fraction, so we've already kind of achieved it. But if we want to get a little bit simpler, if we add the digits up here, 4 plus 8 is 12. So 1 plus 2 is 3. So this up here is divisible by 3. And this down here is clearly divi... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
But if we want to get a little bit simpler, if we add the digits up here, 4 plus 8 is 12. So 1 plus 2 is 3. So this up here is divisible by 3. And this down here is clearly divisible by 3. So let's divide both of them by 3. So 3 goes into 4,008. Let's see. | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
And this down here is clearly divisible by 3. So let's divide both of them by 3. So 3 goes into 4,008. Let's see. It goes into 4 one time. Subtract. You get a 10. | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
Let's see. It goes into 4 one time. Subtract. You get a 10. 3 times 3 is 9. Subtract. You get another 10. | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
You get a 10. 3 times 3 is 9. Subtract. You get another 10. Goes into 3 times. 3 times 3 is 9. Subtract. | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
You get another 10. Goes into 3 times. 3 times 3 is 9. Subtract. Bring down an 8. 3 goes into 18 exactly 6 times. So our numerator is 1,336. | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
Subtract. Bring down an 8. 3 goes into 18 exactly 6 times. So our numerator is 1,336. This is no longer divisible by 3. The sum of the digits is not divisible by 3. It's not a multiple of 3. | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
So our numerator is 1,336. This is no longer divisible by 3. The sum of the digits is not divisible by 3. It's not a multiple of 3. And if you divide this bottom number by 3, you get 3,333. And I think we have simplified it. I think we have simplified it about as well as we can. | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
It's not a multiple of 3. And if you divide this bottom number by 3, you get 3,333. And I think we have simplified it. I think we have simplified it about as well as we can. Although we could check more. Let me know if I didn't. But either way, we have now written this. | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
I think we have simplified it about as well as we can. Although we could check more. Let me know if I didn't. But either way, we have now written this. This was pretty neat. We saw that a repeating decimal can be represented not just as an infinite series, but as an infinite geometric series. And then we were able to u... | Repeating decimal as infinite geometric series Precalculus Khan Academy.mp3 |
And I encourage you to pause this video and give it a go on your own. And I will give you a hint. The key to this is to figure out, well, what function is this the power series for, and then use that function to evaluate this. And there's another clue here is that, hey, this is kind of a mysterious or suspicious-lookin... | Worked example cosine function from power series Series AP Calculus BC Khan Academy.mp3 |
And there's another clue here is that, hey, this is kind of a mysterious or suspicious-looking number here, pi over two, that looks like something I would use a trig function to evaluate. That might be a little bit more straightforward. So I'll let you have a go at it. So I'm assuming you have tried, so let's try to wo... | Worked example cosine function from power series Series AP Calculus BC Khan Academy.mp3 |
So I'm assuming you have tried, so let's try to work through this together. And any of these types of problems, I like to at least expand out this power series so I get a better sense of what it's like. So this right over here, if I were to expand it out, this is going to be equal to, when n is zero, this is one. Actua... | Worked example cosine function from power series Series AP Calculus BC Khan Academy.mp3 |
Actually, all of these are one, so it's just gonna be one. When n is one, it's gonna be negative one x to the sixth, x to the sixth over two factorial. When n is two, it's going to be positive. Negative one squared is positive one times x to the 12th over four factorial. And then let's just do one more. When x is equal... | Worked example cosine function from power series Series AP Calculus BC Khan Academy.mp3 |
Negative one squared is positive one times x to the 12th over four factorial. And then let's just do one more. When x is equal to three, it's going to be negative x to the 18th, x to the 18th over six, over six factorial. And you just keep going on and on forever. Now, offhand, I don't know a function, especially a tri... | Worked example cosine function from power series Series AP Calculus BC Khan Academy.mp3 |
And you just keep going on and on forever. Now, offhand, I don't know a function, especially a trigonometric function, because that was kind of our clue here. This pi over two makes me feel like I might, this might be a trigonometric function right over here. Nothing jumps out at me offhand, but this does look suspicio... | Worked example cosine function from power series Series AP Calculus BC Khan Academy.mp3 |
Nothing jumps out at me offhand, but this does look suspiciously familiar. This looks awfully close to the power series, or the Maclaurin series for cosine of x, which we have seen multiple times. Let's just remind ourselves what that is. And if this doesn't look familiar, the previous video where I do the Maclaurin se... | Worked example cosine function from power series Series AP Calculus BC Khan Academy.mp3 |
And if this doesn't look familiar, the previous video where I do the Maclaurin series for cosine of x goes into detail on how I get this. The Maclaurin series for cosine of x is equal to, so I'll just write a few terms. I'll write approximately equal to one minus x squared over two factorial plus x to the fourth, plus ... | Worked example cosine function from power series Series AP Calculus BC Khan Academy.mp3 |
And just like that, you're probably seeing the similarities. Well, the first term's the same, the sine negative, positive, negative, positive, negative, positive, negative, positive, two factorial, four factorial, six factorial. The difference is the powers, the exponents on the x's. This is x squared, this is x to the... | Worked example cosine function from power series Series AP Calculus BC Khan Academy.mp3 |
This is x squared, this is x to the sixth, this is x to the fourth, that's x to the twelfth. This is x to the sixth, that's x to the eighteenth. Well, what if we, I guess something for you to think about is, well, how can we replace x with something here? Because anything that I change, if I take cosine of, if I change... | Worked example cosine function from power series Series AP Calculus BC Khan Academy.mp3 |
Because anything that I change, if I take cosine of, if I change x to, I don't know, a plus b, everywhere we see an x, you would replace it with an a plus b. Can we put a power of x here so that these things end up like that? Well, this x to the sixth is the same thing, x to the sixth is the same thing as x to the thir... | Worked example cosine function from power series Series AP Calculus BC Khan Academy.mp3 |
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