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Chapter 10: Problem 21

a. Find the nth-order Taylor polynomials of the given function centered at \(0,\) for \(n=0,1,\) and 2 b. Graph the Taylor polynomials and the function. $$f(x)=(1+x)^{-3}$$

Short Answer

Expert verified
Based on the step-by-step solution provided, write the nth-order Taylor polynomial for the function f(x) = (1+x)^(-3) centered at x = 0 for n = 0, 1, and 2.

Step by step solution

01

Identify the function and compute derivatives

Given function is: $$ f(x)=(1+x)^{-3} $$ Now, compute the first few derivatives of the function: $$ f'(x)=-3(1+x)^{-4}(1)=-3(1+x)^{-4} $$ $$ f''(x)=12(1+x)^{-5}(1)=12(1+x)^{-5} $$
02

Evaluate the derivatives at x = 0

Now evaluate the derivatives at x = 0: $$ f(0)=(1+0)^{-3}=1 $$ $$ f'(0)=-3(1+0)^{-4}=-3 $$ $$ f''(0)=12(1+0)^{-5}=12 $$
03

Find Taylor polynomials for n=0,1, and 2

Use the Taylor polynomial formula stated in the Analysis section for each n: For n=0: $$ T_{0}(x)=\frac{f(0)}{0!}(x)^{0}=1 $$ For n=1: $$ T_{1}(x)=\frac{f(0)}{0!}(x)^{0}+\frac{f'(0)}{1!}(x)^{1}=1-3x $$ For n=2: $$ T_{2}(x)=\frac{f(0)}{0!}(x)^{0}+\frac{f'(0)}{1!}(x)^{1}+\frac{f''(0)}{2!}(x)^{2}=1-3x+\frac{12}{2}x^{2}=1-3x+6x^{2} $$
04

Graph the function and the Taylor polynomials

Now graph the function and the Taylor polynomials \(T_{0}(x), T_{1}(x),\) and \(T_{2}(x)\) on the same plot. The student can use graphing software or calculator to plot them. In summary, the nth-order Taylor polynomials for given function are: - For n=0: \(T_{0}(x)=1\) - For n=1: \(T_{1}(x)=1-3x\) - For n=2: \(T_{2}(x)=1-3x+6x^{2}\) Now graph the Taylor polynomials and the function to visualize the approximations.

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Most popular questions from this chapter

Consider the following common approximations when \(x\) is near zero. a. Estimate \(f(0.1)\) and give the maximum error in the approximation. b. Estimate \(f(0.2)\) and give the maximum error in the approximation. $$f(x)=\tan ^{-1} x \approx x$$

Determine whether the following statements are true and give an explanation or counterexample. a. The function \(f(x)=\sqrt{x}\) has a Taylor series centered at 0 . b. The function \(f(x)=\csc x\) has a Taylor series centered at \(\pi / 2\) c. If \(f\) has a Taylor series that converges only on \((-2,2),\) then \(f\left(x^{2}\right)\) has a Taylor series that also converges only on (-2,2) d. If \(p(x)\) is the Taylor series for \(f\) centered at \(0,\) then \(p(x-1)\) is the Taylor series for \(f\) centered at 1 e. The Taylor series for an even function about 0 has only even powers of \(x\)

Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than \(10^{-4}\). $$\int_{0}^{0.2} \frac{\ln (1+t)}{t} d t$$

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=\frac{e^{x}+e^{-x}}{2}$$

The inverse hyperbolic sine is defined in several ways; among them are $$\sinh ^{-1} x=\ln (x+\sqrt{x^{2}+1})=\int_{0}^{x} \frac{d t}{\sqrt{1+t^{2}}}$$ Find the first four terms of the Taylor series for \(\sinh ^{-1} x\) using these two definitions (and be sure they agree).
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