   Chapter 11.10, Problem 10E

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.10. f(x) = cos2 x, 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)=cos2x centered at a=0 .

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

The function f(x)=cos2x at a=0 is f(0)=1 .

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

f(x)=ddx(cos2x)=2cosxddx(cosx)=2cosx(sinx)=sin2x      [sin2x=2cosxsinx]

f(x)=sin2x (1)

Substitute 0 for x,

f(0)=sin(20)=sin(0)=0

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

f(2)(x)=d2dx2(f(x))=ddx(f(x)) =ddx(sin2x)    (by equation(1))=ddx(sin2x)

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

f(2)(x)=2cos2x (2)

Substitute 0 for x,

f(2)(0)=2cos2(0)=2(1)=2

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

f(3)(x)=d3dx3(f(x))=ddx(f(2)(x))=ddx(2cos2x)    (by equation(2))=2(2sin2x)

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

f(3)(x)=4sin2x (3)

Substitute 0 for x,

f(3)(0)=4sin2(0)=4(0)=0

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

f(4)(x)=d4dx4(f(x))=ddx(f(3)(x))=ddx(4sin2x)    (by equation(3))=4(2cos2x)

Simplify further and obtain f(4)(x) ,

f(4)(x)=8cos2x (4)

Substitute 0 for x,

f(4)(

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