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 11.3.3

Chapter 11, Problem 11.3.3

Find a Taylor series for f centered at 2 given that f⁽ᵏ⁾(2)=1, for all nonnegative integers k.

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Recall the general formula for the Taylor series of a function \(f\) centered at \(a\): \[f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!} (x - a)^k,\] where \(f^{(k)}(a)\) is the \(k\)-th derivative of \(f\) evaluated at \(a\).

In this problem, the center is \(a = 2\), and we are given that for all nonnegative integers \(k\), \[f^{(k)}(2) = 1.\]

Substitute the given derivative values into the Taylor series formula: \[f(x) = \sum_{k=0}^{\infty} \frac{1}{k!} (x - 2)^k.\]

Recognize that this series matches the expansion of the exponential function \(e^y = \sum_{k=0}^{\infty} \frac{y^k}{k!}\), where \(y = x - 2\).

Therefore, the Taylor series for \(f\) centered at 2 can be expressed as \[f(x) = \sum_{k=0}^{\infty} \frac{(x - 2)^k}{k!}.\] This is the series representation of \(e^{x-2}\), but since the problem only asks for the Taylor series, this summation form is the key result.

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Key Concepts

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

Taylor Series Expansion

A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. Centered at a point a, it approximates the function near a using powers of (x - a) weighted by the function's derivatives at a.

Derivatives and Their Role in Taylor Series

The coefficients of a Taylor series are determined by the function's derivatives at the center point. Specifically, the k-th term involves the k-th derivative evaluated at the center, divided by k factorial, multiplied by (x - a)^k.

Factorials and Series Coefficients

Factorials (k!) appear in the denominator of each Taylor series term to normalize the contribution of the k-th derivative. This ensures the series converges properly and accurately represents the function near the center.

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Related Practice

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) = 8x^(3/2), a=1; approximate 8 ⋅ 1.1^(3/2)

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

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 sin x² dx

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 polynomials p₁, …, p₅ centered at a=0 for f(x)=e⁻ˣ

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.

√(1+x) ≈ 1 + x/2 on [−0.1,0.1]

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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