Understanding Taylor's Remainder Theorem

In this video, we're going to talk about how to calculate the maximum error of an approximation using Taylor's Remainder Theorem. So let's go ahead and begin. So what is the formula that we need to use? Here's the formula that you need. So the remainder is going to be the n plus 1 derivative evaluated at some number z. times x minus c raised to the n plus 1 divided by n plus 1 factorial. So that's the formula we need to use. Now, we need to determine what is x, what is c, what is n, and what is z. But let's start with n. What do you think n is equal to? What's the value of n? n is basically the last number you see here. n is 4. Now what about x and c? This number, we're trying to approximate the natural log of 1.1. That's your x value. Now, the 0.1 that you see in every term, that's equal to x minus c. So if x is 1, I mean if x is 1.1, c has to be 1, because 1.1 minus 1 is 0.1. So we have r4 of x that's equal to f of n plus 1, so that's f of 5 of z times x minus c, c is 1, n plus 1 is 5, and then over 5 factorial. Now what is z? z is a number between x and c. So x, we said it's 1.1, c is 1. So z could be 1, it could be 1.1, it could be 1.05. z is a number such that the fifth derivative will have a maximum value. So we need to find what the fifth derivative is. So let's start with the original function f of x, which is the natural log of x. The first derivative is going to be 1 over x. And then the second derivative, 1 over x is the same as x to the minus 1. So using the power rule, that's going to be negative 1 x to the minus 2. So we can rewrite that as negative 1 over x squared. And then if we take the third derivative, that's going to be negative 1 over x squared. That's going to be 2x to the minus 3. And then if we take the fourth derivative, negative 6x to the negative 4. And then the fifth derivative is going to be 24x to the negative 5. So now that we have the fifth derivative, we have it as 24 over x to the fifth power. Is z going to be 1, 1.1, or some number in between? Which of those values will give us the maximum value for the... the fifth derivative. Notice that x is in the denominator of the fraction, and as you decrease the denominator of the fraction, the value of the entire fraction goes up. So we want the lowest x value. So 1 is lower than 1.1. So we're going to choose a z value of 1. So, f, the fifth derivative of 1, that's going to be 24 over 1 to the fifth power, so that equals 24. So, that's r4 of 1.1, that's going to be, this is 24. And then plugging in 1.1 into x, that's going to be 1.1 minus 1, so simply 0.1 to the 5th power, and then divide it by 5 factorial. Go ahead and type that in. So the remainder, the maximum value, is 2 times 10 to the negative 6. So that is the maximum error for this particular approximation. Now the next thing we need to do is determine the exact value of the error. So we need to find the difference between the natural log of 1.1 and the value of the fourth degree polynomial at 1.1. So ln 1.1. That's 0.0953101798. Now let's evaluate the fourth degree polynomial when x is 1.1. So if you type in 0.1 minus 0.1 squared divided by 2 plus 0.1 cubed divided by 3 minus 0.1 to the 4th power divided by 4. That's going to give you 0.0953083333. Now, if you take the difference between these two values, So let's say the absolute value difference between ln 1.1 and the fourth degree polynomial evaluated at 1.1. That's going to give us the exact error. And so you should get 1.846 times 10 to negative 6. So this is the exact error. And this is the maximum error for this particular approximation. So we should expect a higher value for the maximum error compared to the exact error, which we do have. Now let's go ahead and work on another similar problem. So determine the maximum error of the approximation and also calculate the exact value of the error. So what do we notice in this problem? Well let's write the formula to begin with. We know that the remainder, which is going to be the maximum error, that's going to be f n plus 1 evaluated at z times x minus c raised to the n plus 1 divided by n plus 1 factorial. So what is n in this problem? Notice that n is 2. Now what else do we know? What is x and what is c? x is what we see here. So x is 1.2. Now what is the value of c? So notice that the difference between x and c, that's 0.2. So if you set x minus c equal to 0.2 and replace x with 1.2, then you need to see that c is equal to 1. 1.2 minus 1 is 0.2. So now we have n, x, and c. Now we need to determine the value of z. If n is 2, then n plus 1 is 3, which indicates that we need to find the value of the third derivative. So what's f of x? We know that x is 1.2, so if you replace 1.2 with x, we can see that x is 1.2. see that f of x is the square root of x, which is the same as x to the 1 half. So the first derivative is going to be 1 half x to the negative half. The second derivative is going to be negative. negative 1 fourth x to the negative 1 half minus 1, which is negative 3 over 2. And so the third derivative is going to be negative 1 fourth times negative 3 over 2. So that's 3 over 8. And then negative 3 over 2 minus 1, that's negative 5 over 2. So I'm going to rewrite the third derivative as 3 over 8 and then x to the 5 halves. 5 halves is basically 2.5. And 2.5 is 2 plus 0.5. So we're going to have x squared times x to the 0.5, which is basically the square root of x. So that's the third derivative. Now we know that z is between x and c, so it's between 1 and 1.2. Which one will give us a maximum third derivative value? 1 or 1.2? Now remember, if you decrease the value of the denominator of a fraction, the value of the whole fraction goes up. So since we're plugging in a number into the bottom, we want to plug in the lowest value, which is c. In this case, 1. So we're going to say that z, we're going to set it equal to 1. Our goal is to determine the maximum error. So now that we know that z is equal to 1, we can calculate the error. So n is 2, so we're going to have r sub 2 of x, and that's n plus 1, so this is the third derivative. Add z, and then times x, c is 1, n plus 1 is 3, divided by 3 factorial. So now let's plug in z and x. So x is 1.2, and this is going to be the third derivative. evaluated at 1 and then this is going to be 1.2 minus 1 to the third power over 3 factorial so if we plug in 1 into this expression it's just going to be 3 over 8 so we have 3 over 8 times 0.2 raised to the third power over 3 factorial go ahead and type that in So you should get 5 times 10 to the negative 4. So this answer represents the maximum error value. So now let's determine the exact value of the error. And in order to do that, We simply need to take the difference between these two values. So the square root of 1.2... is 1.095445115 and the second degree polynomial evaluated at 1.2 which is equal to all this that's going to be 1 plus 0.5 times 0.2 minus 0.2 squared divided by 8 and so that's just 1.095 So the error, the exact error, is going to be the difference between these two values. And so this is going to be about 4.45 times 10 to the negative 4. And so this is the exact error. As you can see, it's less than the maximum error of 5 times 10 to the negative 4. But it's still pretty close to it. And so that's it for this video. So now you know how to use Taylor's Remainder Theorem to determine the maximum error, and you know how to easily determine the exact error, which is the difference between the actual value and the approximation.

times x minus c raised to the n plus 1 divided by n plus 1 factorial. So that's the formula we need to use. Now, we need to determine what is x, what is c, what is n, and what is z.

But let's start with n. What do you think n is equal to? What's the value of n? n is basically the last number you see here.

n is 4. Now what about x and c? This number, we're trying to approximate the natural log of 1.1. That's your x value.

Now, the 0.1 that you see in every term, that's equal to x minus c. So if x is 1, I mean if x is 1.1, c has to be 1, because 1.1 minus 1 is 0.1. So we have r4 of x that's equal to f of n plus 1, so that's f of 5 of z times x minus c, c is 1, n plus 1 is 5, and then over 5 factorial.

Now what is z? z is a number between x and c. So x, we said it's 1.1, c is 1. So z could be 1, it could be 1.1, it could be 1.05. z is a number such that the fifth derivative will have a maximum value. So we need to find what the fifth derivative is.

So let's start with the original function f of x, which is the natural log of x. The first derivative is going to be 1 over x. And then the second derivative, 1 over x is the same as x to the minus 1. So using the power rule, that's going to be negative 1 x to the minus 2. So we can rewrite that as negative 1 over x squared. And then if we take the third derivative, that's going to be negative 1 over x squared. That's going to be 2x to the minus 3. And then if we take the fourth derivative, negative 6x to the negative 4. And then the fifth derivative is going to be 24x to the negative 5. So now that we have the fifth derivative, we have it as 24 over x to the fifth power.

Is z going to be 1, 1.1, or some number in between? Which of those values will give us the maximum value for the... the fifth derivative. Notice that x is in the denominator of the fraction, and as you decrease the denominator of the fraction, the value of the entire fraction goes up.

So we want the lowest x value. So 1 is lower than 1.1. So we're going to choose a z value of 1. So, f, the fifth derivative of 1, that's going to be 24 over 1 to the fifth power, so that equals 24. So, that's r4 of 1.1, that's going to be, this is 24. And then plugging in 1.1 into x, that's going to be 1.1 minus 1, so simply 0.1 to the 5th power, and then divide it by 5 factorial.

Go ahead and type that in. So the remainder, the maximum value, is 2 times 10 to the negative 6. So that is the maximum error for this particular approximation. Now the next thing we need to do is determine the exact value of the error. So we need to find the difference between the natural log of 1.1 and the value of the fourth degree polynomial at 1.1.

So ln 1.1. That's 0.0953101798. Now let's evaluate the fourth degree polynomial when x is 1.1. So if you type in 0.1 minus 0.1 squared divided by 2 plus 0.1 cubed divided by 3 minus 0.1 to the 4th power divided by 4. That's going to give you 0.0953083333. Now, if you take the difference between these two values, So let's say the absolute value difference between ln 1.1 and the fourth degree polynomial evaluated at 1.1.

That's going to give us the exact error. And so you should get 1.846 times 10 to negative 6. So this is the exact error. And this is the maximum error for this particular approximation.

So we should expect a higher value for the maximum error compared to the exact error, which we do have. Now let's go ahead and work on another similar problem. So determine the maximum error of the approximation and also calculate the exact value of the error. So what do we notice in this problem? Well let's write the formula to begin with.

We know that the remainder, which is going to be the maximum error, that's going to be f n plus 1 evaluated at z times x minus c raised to the n plus 1 divided by n plus 1 factorial. So what is n in this problem? Notice that n is 2. Now what else do we know? What is x and what is c? x is what we see here.

So x is 1.2. Now what is the value of c? So notice that the difference between x and c, that's 0.2. So if you set x minus c equal to 0.2 and replace x with 1.2, then you need to see that c is equal to 1. 1.2 minus 1 is 0.2. So now we have n, x, and c.

Now we need to determine the value of z. If n is 2, then n plus 1 is 3, which indicates that we need to find the value of the third derivative. So what's f of x?

We know that x is 1.2, so if you replace 1.2 with x, we can see that x is 1.2. see that f of x is the square root of x, which is the same as x to the 1 half. So the first derivative is going to be 1 half x to the negative half.

The second derivative is going to be negative. negative 1 fourth x to the negative 1 half minus 1, which is negative 3 over 2. And so the third derivative is going to be negative 1 fourth times negative 3 over 2. So that's 3 over 8. And then negative 3 over 2 minus 1, that's negative 5 over 2. So I'm going to rewrite the third derivative as 3 over 8 and then x to the 5 halves. 5 halves is basically 2.5. And 2.5 is 2 plus 0.5. So we're going to have x squared times x to the 0.5, which is basically the square root of x.

So that's the third derivative. Now we know that z is between x and c, so it's between 1 and 1.2. Which one will give us a maximum third derivative value?

1 or 1.2? Now remember, if you decrease the value of the denominator of a fraction, the value of the whole fraction goes up. So since we're plugging in a number into the bottom, we want to plug in the lowest value, which is c.

In this case, 1. So we're going to say that z, we're going to set it equal to 1. Our goal is to determine the maximum error. So now that we know that z is equal to 1, we can calculate the error. So n is 2, so we're going to have r sub 2 of x, and that's n plus 1, so this is the third derivative.

Add z, and then times x, c is 1, n plus 1 is 3, divided by 3 factorial. So now let's plug in z and x. So x is 1.2, and this is going to be the third derivative. evaluated at 1 and then this is going to be 1.2 minus 1 to the third power over 3 factorial so if we plug in 1 into this expression it's just going to be 3 over 8 so we have 3 over 8 times 0.2 raised to the third power over 3 factorial go ahead and type that in So you should get 5 times 10 to the negative 4. So this answer represents the maximum error value. So now let's determine the exact value of the error.

And in order to do that, We simply need to take the difference between these two values. So the square root of 1.2... is 1.095445115 and the second degree polynomial evaluated at 1.2 which is equal to all this that's going to be 1 plus 0.5 times 0.2 minus 0.2 squared divided by 8 and so that's just 1.095 So the error, the exact error, is going to be the difference between these two values.

And so this is going to be about 4.45 times 10 to the negative 4. And so this is the exact error. As you can see, it's less than the maximum error of 5 times 10 to the negative 4. But it's still pretty close to it. And so that's it for this video.

So now you know how to use Taylor's Remainder Theorem to determine the maximum error, and you know how to easily determine the exact error, which is the difference between the actual value and the approximation.

Transcript for:Understanding Taylor's Remainder Theorem

Transcript for:
Understanding Taylor's Remainder Theorem