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 11.1.1

Chapter 11, Problem 11.1.1

Suppose you use a second-order Taylor polynomial centered at 0 to approximate a function f. What matching conditions are satisfied by the polynomial?

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Recall that a second-order Taylor polynomial of a function \(f\) centered at 0 is given by: \[P_2(x) = f(0) + f'(0) x + \frac{f''(0)}{2} x^2,\] where \(f(0)\), \(f'(0)\), and \(f''(0)\) are the function's value, first derivative, and second derivative evaluated at 0.

The purpose of the Taylor polynomial is to approximate the function near the center point (here, 0) by matching the function's behavior up to the second derivative at that point.

Therefore, the polynomial \(P_2(x)\) satisfies the following matching conditions at \(x=0\): - \(P_2(0) = f(0)\) (the polynomial and function have the same value), - \(P_2'(0) = f'(0)\) (the polynomial and function have the same first derivative), - \(P_2''(0) = f''(0)\) (the polynomial and function have the same second derivative).

These conditions ensure that the polynomial not only touches the function at \(x=0\) but also has the same slope and curvature there, providing a good local approximation.

In summary, the second-order Taylor polynomial centered at 0 matches the function's value, first derivative, and second derivative at \(x=0\).

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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 of the function at that point. The nth-order Taylor polynomial uses derivatives up to order n to create a polynomial that closely matches the function's behavior near the center.

Matching Conditions for Taylor Polynomials

The Taylor polynomial of order n matches the function and its first n derivatives at the center point. This means the polynomial and the function have the same value, slope, curvature, etc., up to the nth derivative at that point, ensuring a good local approximation.

Second-Order Taylor Polynomial Centered at 0

A second-order Taylor polynomial centered at 0 matches the function's value, first derivative, and second derivative at 0. This polynomial captures the function's value, slope, and concavity at zero, providing a quadratic approximation near that point.

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Textbook Question

{Use of Tech} Approximating definite integrals 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.

∫₀⁰ᐧ² (ln (1 + t))/t dt

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.

Find the Taylor polynomial p₃ centered at a=e for f(x)=ln x.

Textbook Question

Combining power series Use the geometric series

f(x) = 1/(1-x) = ∑ₖ₌₀∞ xᵏ, for |x| < 1,

to find the power series representation for the following functions (centered at 0). Give the interval of convergence of the new series.

f(x³) = 1/(1 − x³)

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/(3 + 4x)²

Textbook Question

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

∑ₖ₌₁∞ ((−1)ᵏ⁺¹(x−1)ᵏ)/k

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Textbook Question

Approximating definite integrals 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.∫₀⁰ᐧ²⁵ e⁻ˣ² dx

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