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 11.1.3

Chapter 11, Problem 11.1.3

The first three Taylor polynomials for f(x)=√(1+x) centered at 0 are p₀ = 1, p₁ = 1+x/2, and p₂ = 1 + x/2 − x²/8. Find three approximations to √1.1.

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Recall that the Taylor polynomials for \(f(x) = \sqrt{1+x}\) centered at 0 are given as \(p_0 = 1\), \(p_1 = 1 + \frac{x}{2}\), and \(p_2 = 1 + \frac{x}{2} - \frac{x^2}{8}\).

To approximate \(\sqrt{1.1}\), rewrite it as \(f(0.1)\) since \(1.1 = 1 + 0.1\) and the polynomials are centered at 0.

Substitute \(x = 0.1\) into each polynomial to get the approximations: \(p_0(0.1)\), \(p_1(0.1)\), and \(p_2(0.1)\).

Calculate \(p_0(0.1) = 1\) as the simplest approximation.

Calculate \(p_1(0.1) = 1 + \frac{0.1}{2}\) and \(p_2(0.1) = 1 + \frac{0.1}{2} - \frac{(0.1)^2}{8}\) to get more accurate approximations.

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Key Concepts

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

Taylor Polynomials

Taylor polynomials approximate a function near a specific point using a finite sum of derivatives at that point. Each polynomial of degree n matches the function's value and its first n derivatives at the center, providing increasingly accurate approximations as n increases.

Function Approximation Using Polynomials

Approximating a function value at a point involves substituting the point into the Taylor polynomial. Lower-degree polynomials give rough estimates, while higher-degree polynomials improve accuracy by including more terms that capture the function's behavior.

Evaluating Square Root Functions Near 1

The function f(x) = √(1+x) is smooth near x=0, making it suitable for Taylor expansion. Approximating √1.1 means evaluating the polynomial at x=0.1, which simplifies calculations and provides a close estimate without directly computing the square root.

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Related Practice

Textbook Question

Suppose the power series ∑ₖ₌₀∞ cₖ(x−a)ᵏ has an interval of convergence of (−3,7]. Find the center a and the radius of convergence R.

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Textbook Question

Functions to power series Find power series representations centered at 0 for the following functions using known power series. Give the interval of convergence for the resulting series.

f(x) = ln √(1 − x²)

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Textbook Question

Combining power series Use the power series representation

f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,

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

f(3x) = ln (1 − 3x)

Textbook Question

Differential equations

a. Find a power series for the solution of the following differential equations, subject to the given initial condition

b. Identify the function represented by the power series.

y′(t) − 3y = 10, y(0) = 2

Textbook Question

{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.

sin x ≈ x − x³/6 on [π/4, π/4]

Textbook Question

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

∑ₖ₌₁∞ (xᵏ/kᵏ)