   Chapter 11.11, Problem 22E

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) |. 22. f(x) = sinh 2x, a = 0, n = 5, −1 ≤ x ≤ 1

(a)

To determine

To approximate: The function f(x)=sinh2x by a Taylor polynomial with degree 5 at a=0.

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)=sinh2x, centered at a=0.

Then, f(0)=0.

The first derivative of f(x) is f(x)=2cosh2x and the corresponding value at a=0 is f(0)=2.

The second derivative is f(x)=4sinh2x and the corresponding value at a=0 is f(0)=0.

The third derivative is f(x)=8cosh2x and the corresponding value at a=0 is f(0)=8.

The fourth derivative is f(4)(x)=16sinh2x and the corresponding value at a=0 is f(4)(0)=0.

The fifth derivative is f(5)(x)=32cosh2x and the corresponding value at a=0 is f(5)(0)=32

(b)

To determine

To estimate: The accuracy of the approximation f(x)Tn(x) when x lies in the interval 1x1 by using Taylor’s inequality.

(c)

To determine

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

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