   Chapter 11.11, Problem 16E

Chapter
Section
Textbook Problem

# (a) Approximate f by a Taylor polynomial with degree n at the number a. (b) Use Taylor’s Inequality to estimate the accuracy of the approximation f(x) ≈ Tn(x) when x lies in the given interval. (c) Check your result in part (b) by graphing | Rn(x) |. 16. f(x) = sin x, a = π/6, n = 4, 0 ≤ x ≤ π/3

(a)

To determine

To approximate: The function f(x)=sinx by a Taylor polynomial with degree 4 at a=π6.

Explanation

Formula used:

Taylor polynomial:

Let nth degree Taylor polynomial of f at a is denoted by Tn(x) and is defined as,

Tn(x)=i=0nf(i)(a)i!(xa)i=f(a)+f(a)1!(xa)+f(a)2!(xa)2++f(n)(a)n!(xa)n

And f is the sum of its Taylor series, f(x)=n=0f(n)(a)n!(xa)n.

Calculation:

The given function is f(x)=sinx, centered at a=π6.

Then, f(π6)=12.

The first derivative of f(x) is f(x)=cosx and the corresponding value at a=π6 is f(π6)=32.

The second derivative is f(x)=sinx and the corresponding value at a=π6 is f(π6)=12.

The third derivative is f(x)=cosx and the corresponding value at a=π6 is f(π6)=32.

The fourth derivative f(4)(x)=sinx and the corresponding value at a=π6 is f(π6)=12.

Substitute these values in the formula of Taylor series and approximate f as follows

(b)

To determine

To estimate: The accuracy of the approximation f(x)Tn(x) when x lies in the interval 0xπ3 by using Taylor’s inequality.

(c)

To determine

To check: The result in part (b) by graphing |Rn(x)|.

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