/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 The first three Taylor polynomia... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 3

The first three Taylor polynomials for \(f(x)=\sqrt{1+x}\) centered at 0 are \(p_{0}(x)=1, p_{1}(x)=1+\frac{x}{2},\) and \(p_{2}(x)=1+\frac{x}{2}-\frac{x^{2}}{8}\) Find three approximations to \(\sqrt{1.1}\)

Short Answer

Expert verified
The three approximations of 鈭�1.1 using the given Taylor polynomials are 1, 1.05, and 1.04875.

Step by step solution

01

Find approximation using \(p_{0}(x)\)

To find the first approximation, plug \(x=0.1\) into the first Taylor polynomial, \(p_{0}(x) = 1\): $$ p_{0}(0.1) = 1 $$ The first approximation to \(\sqrt{1.1}\) is 1.
02

Find approximation using \(p_{1}(x)\)

To find the second approximation, plug \(x=0.1\) into the second Taylor polynomial, \(p_{1}(x) = 1 + \frac{x}{2}\): $$ p_{1}(0.1) = 1 + \frac{0.1}{2} = 1 + 0.05 = 1.05 $$ The second approximation to \(\sqrt{1.1}\) is 1.05.
03

Find approximation using \(p_{2}(x)\)

To find the third approximation, plug \(x=0.1\) into the third Taylor polynomial, \(p_{2}(x) = 1 + \frac{x}{2} - \frac{x^2}{8}\): $$ p_{2}(0.1) = 1 + \frac{0.1}{2} - \frac{(0.1)^2}{8} = 1 + 0.05 - 0.00125 = 1.04875 $$ The third approximation to \(\sqrt{1.1}\) is 1.04875. In conclusion, the three approximations to \(\sqrt{1.1}\) using the given Taylor polynomials are 1, 1.05, and 1.04875.

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Most popular questions from this chapter

The theory of optics gives rise to the two Fresnel integrals $$S(x)=\int_{0}^{x} \sin t^{2} d t \text { and } C(x)=\int_{0}^{x} \cos t^{2} d t$$ a. Compute \(S^{\prime}(x)\) and \(C^{\prime}(x)\) b. Expand \(\sin t^{2}\) and \(\cos t^{2}\) in a Maclaurin series and then integrate to find the first four nonzero terms of the Maclaurin series for \(S\) and \(C\) c. Use the polynomials in part (b) to approximate \(S(0.05)\) and \(C(-0.25)\) d. How many terms of the Maclaurin series are required to approximate \(S(0.05)\) with an error no greater than \(10^{-4} ?\) e. How many terms of the Maclaurin series are required to approximate \(C(-0.25)\) with an error no greater than \(10^{-6} ?\)

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. $$f(x)=\tan x$$

By comparing the first four terms, show that the Maclaurin series for \(\cos ^{2} x\) can be found (a) by squaring the Maclaurin series for \(\cos x,\) (b) by using the identity \(\cos ^{2} x=(1+\cos 2 x) / 2,\) or \((\mathrm{c})\) by computing the coefficients using the definition.

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 \(x\)

The function \(\operatorname{Si}(x)=\int_{0}^{x} \frac{\sin t}{t} d t\) is called the sine integral function. a. Expand the integrand in a Taylor series about 0 . b. Integrate the series to find a Taylor series for Si. c. Approximate Si(0.5) and Si(1). Use enough terms of the series so the error in the approximation does not exceed \(10^{-3}\).
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