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 11.1.67d

Chapter 11, Problem 11.1.67d

Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.

d. 1/(1 + 2x)

A. p₂(x)= 1 + 2x + 2x²
B. p₂(x) = 1 − 6x + 24x²
C. p₂(x) = 1 + x − x²/2
D. p₂(x) = 1 − 2x + 4x²
E. p₂(x) = 1 − x + (3/2)x²
F. p₂(x) = 1 − 2x + 2x²

Verified step by step guidance

Recall that the Taylor polynomial of a function \(f(x)\) centered at 0 (Maclaurin polynomial) up to degree 2 is given by: \[f(0) + f'(0)x + \frac{f''(0)}{2}x^2\]

Identify the function given: \[f(x) = \frac{1}{1 + 2x}\]

Calculate the value of the function at 0: \[f(0) = \frac{1}{1 + 0} = 1\]

Find the first derivative of \(f(x)\): \[f'(x) = -\frac{2}{(1 + 2x)^2}\] Then evaluate at 0: \[f'(0) = -2\]

Find the second derivative of \(f(x)\): \[f''(x) = \frac{8}{(1 + 2x)^3}\] Then evaluate at 0: \[f''(0) = 8\] Use these to write the Taylor polynomial: \[p_2(x) = 1 - 2x + \frac{8}{2}x^2 = 1 - 2x + 4x^2\] Match this with the given polynomials to find the correct one.

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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 functions near a specific point (here, 0) using derivatives. The nth-degree Taylor polynomial uses derivatives up to order n to create a polynomial that matches the function's value and slope behavior at that point, providing a local approximation.

Derivatives and Their Role in Taylor Series

Derivatives of a function at the center point determine the coefficients of the Taylor polynomial. The first derivative gives the linear term coefficient, the second derivative gives the quadratic term coefficient (divided by 2!), and so on, reflecting the function's curvature and rate of change.

Geometric Series and Rational Functions

Functions like 1/(1 + 2x) can be expressed as a geometric series when |2x| < 1, allowing expansion into a power series. Recognizing this helps find the Taylor polynomial by truncating the series to the desired degree, linking rational functions to polynomial approximations.

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Geometric Series

Related Practice

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. Only even powers of x appear in the nth−order Taylor polynomial for f(x)=√(1+x²) centered at 0.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. If f has a Taylor series that converges only on (−2,2), then f(x²) has a Taylor series that also converges only on (−2,2).

Textbook Question

Taylor series and interval of convergence

c. Determine the interval of convergence of the series.

f(x) = cosh 3x, a = 0

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Textbook Question

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals

S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt

d. How many terms of the Maclaurin series are required to approximate S(0.05) with an error no greater than 10⁻⁴?

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. If p(x) is the Taylor series for f centered at 0, then p(x−1) is the Taylor series for f centered at 1.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If f(x)=∑ₖ₌₀∞ cₖ xᵏ=0, for all x on an interval (−a, a), then cₖ = 0, for all k.

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