   Chapter 11.11, Problem 2E

Chapter
Section
Textbook Problem

(a) Find the Taylor polynomials up to degree 3 for f(x) = tan x centered a = 0. Graph f and these polynomials on a common screen. (b) Evaluate f and these polynomials at x = π/6, π/4, and π/3. (c) Comment on how the Taylor polynomials converge to f(x).

(a)

To determine

To find: The Taylor polynomials up to degree 3 for f(x)=tanx centered at a=0 and graph f and the polynomials.

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 as f(x)=n=0f(n)(a)n!(xa)n.

Calculation:

The given function is f(x)=tanx and centered at a=0.

The first derivative of f(x) is f(x)=sec2x and the corresponding value at a=0 is f(0)=sec20=1.

The second derivative is f(x)=2sec2xtanx and the corresponding value at a=0 is f(0)=2sec20tan0=0.

The third derivative is f(x)=4sec2xtan2x+2sec4x and the corresponding value at a=0 is f(0)=2

(b)

To determine

To evaluate: The function f and its polynomials at x=π6,π4 and π3.

(c)

To determine

To explain: The reason on how the Taylor polynomials converges to f(x).

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