Ch. 11 - Power Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 11 - Power Series

Problem 11.1.2

Chapter 11, Problem 11.1.2

Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.

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Recall that a Taylor polynomial of order \(n\) approximates a function by matching the function's value and its first \(n\) derivatives at a specific point.

Understand that increasing the order \(n\) means including higher-degree terms in the polynomial, which capture more details about the function's behavior near the point of expansion.

Recognize that, generally, as the order of the Taylor polynomial increases, the approximation becomes more accurate near the point of expansion because the polynomial better matches the function's shape.

Note that the accuracy improvement depends on the function being sufficiently smooth (infinitely differentiable) and the approximation being considered close to the expansion point.

Keep in mind that while higher-order polynomials improve accuracy locally, the approximation might not improve or could even worsen far from the expansion point due to issues like Runge's phenomenon.

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Key Concepts

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

Taylor Polynomial

A Taylor polynomial approximates a function near a specific point using a finite sum of derivatives at that point. The polynomial's degree determines how many terms are included, capturing more details of the function's behavior as the degree increases.

Order of Approximation

The order of a Taylor polynomial refers to its degree, indicating how many derivative terms are used. Higher-order polynomials generally provide better approximations because they incorporate more information about the function's curvature and higher derivatives.

Error and Accuracy in Taylor Approximations

The accuracy of a Taylor polynomial depends on the remainder term, which measures the difference between the function and its polynomial approximation. Typically, increasing the order reduces this error near the expansion point, improving accuracy within a certain interval.

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Related Practice

Textbook Question

{Use of Tech} Newton's derivation of the sine and arcsine series Newton discovered the binomial series and then used it ingeniously to obtain many more results. Here is a case in point.

a. Referring to the figure, show that x = sin s or s = sin ⁻¹ x.

b. The area of a circular sector of radius r subtended by an angle θ is 1/2r²θ. Show that the area of the circular sector APE is s/2, which implies that

s = 2 ∫₀ˣ √(1 − t²) dt − x √(1 −x²)

c. Use the binomial series for f(x) = √(1 − x²) to obtain the first few terms of the Taylor series for s=sin ⁻¹ x.

d. Newton next inverted the series in part (c) to obtain the Taylor series for x=sin s. He did this by assuming sin s = ∑ aₖ sᵏ and solving x = sin(sin ⁻¹ x) for the coefficients aₖ. Find the first few terms of the Taylor series for sin s using this idea (a computer algebra system might be helpful as well).

Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x(e¹/ˣ − 1)

Textbook Question

Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Use the Maclaurin series

(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + ⋯, for −1 < x < 1.

(1 + 4x)⁻²

Textbook Question

Suppose f(0)=1, f'(0)=0, f''(0)=2, and f⁽³⁾(0)=6. Find the third-order Taylor polynomial for f centered at 0 and use it to approximate f(0.2).

Textbook Question

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

√e

Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→₄ (x² 16)/(ln (x 3)}

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