   Chapter 11, Problem 46RE

Chapter
Section
Textbook Problem

# Find the Taylor series of f(x) = cos x at a = π/3.

To determine

To find: The Taylor series for f(x)=cosx centered at π3 .

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)=cosx centered at a=π3 .

The function f(x)=cosx at π3 is f(π3)=12 .

Find the first derivative of f(x) at a=π3 .

f(x)=ddx(cosx)=sinx

f(x)=sinx (1)

Obtain f(π3) .

f(π3)=sin(π3)=32

Find the second derivative of f(x) at a=π3 .

f(2)(x)=d2dx2(cosx)=ddx(f(x))     [ddx(cosx)=f(x)]=ddx(sinx)    (by equation(1))=cosx

That is, f(2)(x)=cosx (2)

Compute f(2)(π3) .

f(2)(π3)=cos(π3)=12

That is, f(2)(π3)=12

Find the third derivative of f(x) at a=π3 as follows.

f(3)(x)=d3dx3(cosx)=ddx(f(2)(x))=ddx(cosx)    (by equation(2))=(sinx)

That is, f(3)(x)=sinx (3)

Compute f(3)(π3) .

f(3)(π3)=sin(π3)=32

Find the fourth derivative of f(x) at a=π3 .

f(4)(x)=d4dx4(cosx)=ddx(f(3)(x))=ddx(sinx)    (by equation(3))=cosx

That is, f(4)(x)=cosx (4)

Compute f(4)(π3)

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