   Chapter 11.11, Problem 9E

Chapter
Section
Textbook Problem

# Find the Taylor polynomial T3(x) for the function f centered at the number a. Graph f and T3 on the same screen.9. f(x) = xe−2x, a = 0

To determine

To find: The Taylor polynomial T3(x) for f(x)=xe2x centered at a=0 and then graph f and the polynomial.

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)=xe2x, centered at a=0.

The first derivative of f(x) is f(x)=e2x2e2xx and the corresponding value at a=0 is f(0)=1.

The second derivative is f(x)=4e2x+4e2xx and the corresponding value at a=0 is f(0)=4.

The third derivative is f(x)=12e2x8e2xx and the corresponding value at a=0 is f(0)=12

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