   Chapter 11.11, Problem 7E

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.7. f(x) = ln x, a = 1

To determine

To find: The Taylor polynomial T3(x) for f(x)=lnx centered at a=1 and 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)=lnx, centered at a=1.

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

The second derivative is f(x)=1x2 and the corresponding value at a=1 is f(1)=1.

The third derivative is f(x)=2x3 and the corresponding value at a=1 is f(1)=2.

Substitute these values in the formula of Taylor series as follows

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