Remainders Find the remainder Rₙ for the nth−order Taylor poly...

Question

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.

f(x) = sin x, a = 0

Accepted Answer

Welcome back, everyone. In this problem, using the Taylor polynomial of degree N for the function F X equals LN 1 + X centered at A equals 0, derive a general formula for the remainder term RN X that quantifies the error between F of X and its Taylor polynomial. Express your answer in terms of N, X, and the value C between 0 and X. Now, for starters, let's recall Taylor's theorem for the remainder. Recall that the theorem basically says that the remainder after N terms for the Taylor polynomial centered at A R X. is equal to the n + 1 derivative of F of C divided by n + 1 factorial multiplied by x minus a to the n + 1 where C is some value between A and X. Now noticing our problem, we're told that our Taylor polynomial is centered at A equals 0. So at A equals 0. Then our remainder. Is now going to become the n + 1 derivative of F of C divided by N + 1 factorial. Multiplied by x to the power of n + 1. So I could even remove all parenthesis here and simply write x to the N + 1. So we know that what this is what our remainder will be, but no, we'll need to figure out what the N +1 derivative of FFC is. So we'll need to figure out the N +1 derivative of FFX. So let's find a few of the derivatives of FF X to see if we can generalize. Now, for starters, given F of X. Given F of X equals to LN or the natural log of 1 + X, OK. Then we know that our first derivative of X is going to be equal to 1 divided by 1 + X. The second derivative. OK, of F of X will be equal to -1 divided by 1 + X squared. The 3rd derivative is going to be equal to 2 divided by 1 + X to the 3rd, OK. The 4th derivative, is going to be equal to -6 divided by 1 + X raise to the 4th. And by now you may notice a pattern, OK? By now, let me make some space here. Generally, OK, you may realize. That if we find the case derivative of F of X, it's going to be equal to -1 to the power of K + 1 multiplied by K minus 1 factorial multiplied by 1 + X to the power of negative k. That's the pattern that we can realize here. So in this case, This means then that if this is going to be the case derivative of F of X, then the N + 1 derivative of FFC if we substitute here is going to be equal to -1 to the power of N + 1 + 1 or N + 2 multiplied by N factorial multiplied by 1 + C to the power of the negative value of N + 1. So now, if we take this and put this into our remainder formula. This means that RF X. Is going to be equal to -1. OK, to the power of N + 2. OK. -1 to the point of N + 2 multiplied by N factorial, multiplied by 1 + C. So the negative N + 1 multiplied by X to the N + 1, OK? And that's divided by N + 1 factorial. Now, if we think about it here, N factorial divided by N + 1 factorial equals 1 divided by N + 1. So if we put that into our formula here, then this will simplify to -1 to the power of N + 2 divided by N + 1 multiplied by 1 + C to the negative N + 1 multiplied by X to the N + 1, where C is between 0 and X. Thus, this is the remainder for the nthor Taylor polynomial of F of X equals LN 1 + X centered at A equals 0. Thanks a lot for watching everyone. I hope this video helped.

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.

f(x) = sin x, a = 0

Key Concepts

Taylor Polynomial and Taylor Series

Remainder Term (Lagrange Form)

Derivatives of sin x

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