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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 11.1.49
Chapter 11, Problem 11.1.49

{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.


e⁰ᐧ²⁵, n=4

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Identify the function to approximate: here, it is \(f(x) = e^x\), and we want to approximate \(e^{0.25}\) using the 4th-order Taylor polynomial centered at 0.
Recall the Taylor polynomial of order \(n\) for \(f(x)\) centered at 0 is given by: \[T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(0)}{k!} x^k,\] where \(f^{(k)}(0)\) is the \(k\)th derivative of \(f\) evaluated at 0.
The remainder (error) term for the Taylor polynomial approximation is given by the Lagrange form: \[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\] for some \(c\) between 0 and \(x\).
Since \(f(x) = e^x\), all derivatives are \(e^x\), which are positive and increasing. To find an upper bound on the error, evaluate the maximum of \(|f^{(n+1)}(c)|\) on the interval between 0 and 0.25, which is \(e^{0.25}\) because \(e^x\) is increasing.
Substitute \(n=4\), \(x=0.25\), and the maximum derivative value into the remainder formula to get the error bound: \[|R_4(0.25)| \leq \frac{e^{0.25}}{5!} (0.25)^5.\] This expression gives a bound on the error when approximating \(e^{0.25}\) with the 4th-order Taylor polynomial centered at 0.

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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 centered at 0 (Maclaurin polynomial) uses derivatives up to order n to estimate the function's value close to zero.
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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 involving the (n+1)th derivative evaluated at some point in the interval.
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Taylor Series

Bounding the Error for Exponential Functions

For functions like e^x, all derivatives are e^x, which are positive and increasing for positive x. To bound the error, evaluate the maximum value of the (n+1)th derivative on the interval from 0 to the point of approximation, ensuring the error estimate is valid.
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Exponential Functions
Related Practice
Textbook Question

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.


f(x) = sin x, a = 0

Textbook Question

Remainders Find the remainder in the Taylor series centered at the point a for the following functions. Then show that lim ₙ→∞ Rₙ(x)=0, for all x in the interval of convergence.


f(x) = e⁻ˣ, a = 0

Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


∑ₖ₌₀∞ (2x)ᵏ/k!

Textbook Question

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.


ln (1 + x²)

Textbook Question

Radius and interval of convergence Determine the radius and interval of convergence of the following power series.


−x²/1 + x⁴/2! −x⁶/3! + x⁸/4! − ⋯

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Textbook Question

Use of Tech Linear and quadratic approximation


a. Find the linear approximating polynomial for the following functions centered at the given point a.


b. Find the quadratic approximating polynomial for the following functions centered at a.


c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.


f(x) = cos x, a = π/4; approximate cos (0.24π)

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