   Chapter 11.10, Problem 5E

Chapter
Section
Textbook Problem

# Use the definition of a Taylor series to find the first four nonzero terms of the series for f(x) centered at the given value of a.5. f(x) = xex, a = 0

To determine

To find: The first four nonzero terms of the series for f(x) centered at 0.

Explanation

Result used:

If f has a power series expansion at a , f(x)=n=0f(n)(a)n!(xa)n , f(x)=f(a)+f(a)1!(xa)+f(a)2!(xa)2+f(a)3!(xa)3+

Calculation:

Consider the function f(x)=xex centered at a=0 .

Obtain the first four nonzero terms of the series as follows,

The function f(x)=xex at a=0 is f(0)=0 .

The first derivative of f(x) at a=0 is computed as follows,

f(x)=ddx(xex)=exddx(x)+xddx(ex)=ex(1)+x(ex)

f(x)=ex(1+x) (1)

Substitute 0 for x,

f(0)=e0(1+0)=1(1)=1

The second derivative of f(x) at a=0 is computed as follows,

f(2)(x)=d2dx2(f(x))=ddx(f(x))=ddx(ex(1+x))    (by equation (1))=exddx(1+x)+(1+x)ddx(ex)

Simplify further and obtain f(2)(x) ,

f(2)(x)=ex(1)+(1+x)(ex)=ex+ex+xex=xex+2ex

f(2)(x)=ex(x+2) (2)

Substitute 0 for x,

f(2)(0)=e0(0+2)=2               [e0=1]

The third derivative of f(x) at a=0 is computed as follows,

f(3)(x)=d3dx3(f(x))=ddx(f(x))=ddx(ex(x+2))    (by equation (2))=exddx(x+2)+(x+2)ddx(ex)

Simplify further and obtain f(3)(x) ,

f(3)(x)=ex(<

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