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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.1.52
Chapter 11, Problem 11.1.52

{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.


ln 1.04, n=3

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1
Identify the function and the point of expansion: here, the function is \(f(x) = \ln(1+x)\), and the Taylor polynomial is centered at \(0\) (Maclaurin series).
Write the general form of the remainder (error) term for the Taylor polynomial of order \(n\) centered at \(0\): the Lagrange form of the remainder is given by \(R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\) where \(c\) is some number between \(0\) and \(x\).
Calculate the \((n+1)\)-th derivative of \(f(x) = \ln(1+x)\): First derivative: \(f'(x) = \frac{1}{1+x}\) Second derivative: \(f''(x) = -\frac{1}{(1+x)^2}\) Third derivative: \(f^{(3)}(x) = \frac{2}{(1+x)^3}\) Fourth derivative: \(f^{(4)}(x) = -\frac{6}{(1+x)^4}\) Since \(n=3\), we need the 4th derivative for the remainder term.
Substitute \(n=3\) and \(x=0.04\) into the remainder formula: \(R_3(0.04) = \frac{f^{(4)}(c)}{4!} (0.04)^4\) where \(c\) is between \(0\) and \(0.04\).
Find an upper bound for \(|f^{(4)}(c)|\) on the interval \([0, 0.04]\): since \(f^{(4)}(x) = -\frac{6}{(1+x)^4}\), its absolute value is \(\frac{6}{(1+x)^4}\). The maximum occurs at the smallest denominator, which is at \(x=0\), so \(|f^{(4)}(c)| \leq \frac{6}{1^4} = 6\). Use this to bound the error: \(|R_3(0.04)| \leq \frac{6}{4!} (0.04)^4\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Taylor Polynomial Approximation

A Taylor polynomial approximates a function near a point using a finite sum of its derivatives at that point. The nth-order Taylor polynomial uses derivatives up to order n, providing an approximation that becomes more accurate as n increases or as the input approaches the center.
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Taylor Polynomials

Remainder (Error) Term in Taylor Series

The remainder term quantifies the difference between the actual function value and its Taylor polynomial approximation. It provides an upper bound on the error, often expressed using the Lagrange form, which involves the (n+1)th derivative evaluated at some point in the interval.
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Taylor Series

Bounding the Error for Logarithmic Functions

To estimate the error when approximating ln(1.04) with a Taylor polynomial centered at 0, one must analyze the derivatives of ln(x) and find a maximum bound for the remainder term on the interval between 0 and 0.04. This ensures the error estimate is valid and helps assess the approximation's accuracy.
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Graphs of Logarithmic Functions
Related Practice
Textbook Question

Suppose you know the Maclaurin series for f and that it converges to f(x) for |x|<1. How do you find the Maclaurin series for f(x²) and where does it converge?

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Textbook Question

{Use of Tech} A different kind of approximation When approximating a function f using a Taylor polynomial, we use information about f and its derivative at one point. An alternative approach (called interpolation) uses information about f at several different points. Suppose we wish to approximate f(x)=sin x on the interval [0, π].


a. Write the (quadratic) Taylor polynomial p₂ for f centered at π/2.


b. Now consider a quadratic interpolating polynomial q(x) = ax² + bx + c. The coefficients a, b, and c are chosen such that the following conditions are satisfied:

q(0) = f(0), q(π/2) = f(π/2), and q(π) = f(π)


Show that q(x) = −(4/π²)x² + (4/π)x.


c. Graph f, p₂, and q on [0, π].


d. Find the error in approximating f(x) = sin x at the points π/4, π/2, 3π/4, and π using p₂ and q.


e. Which function, p₂ or q, is a better approximation to f on [0, π]? Explain.

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Textbook Question

Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.

ƒ(x) = tan⁻¹(4x), a = 0

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Textbook Question

Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.


f(x) = eᶜᵒˢ ˣ

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Textbook Question

{Use of Tech} Approximating sin x Let f(x)=sin x, and let pₙ and qₙ be nth−order Taylor polynomials for f centered at 0 and π, respectively.

a. Find p₅ and q₅

b. Graph f, p₅, and q₅ on the interval [−π, 2π]. On what interval is p₅ a better approximation to f than q₅? On what interval is q₅ a better approximation to f than p₅?

c. Complete the following table showing the errors in the approximations given by p₅ and q₅ at selected points.

d. At which points in the table is p₅ a better approximation to f than q₅? At which points do p₅ and q₅ give equal approximations to f? Explain your observations.

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Textbook Question

{Use of Tech} Estimating errors Use the remainder to find a bound on the error in approximating the following quantities with the nth-order Taylor polynomial centered at 0. Estimates are not unique.


sin 0.3, n = 4

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