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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 11 - Power SeriesProblem 11.1.67a
Chapter 11, Problem 11.1.67a

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


a. √(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
1
Identify the function to approximate: \(f(x) = \sqrt{1 + 2x} = (1 + 2x)^{1/2}\), and note that the Taylor polynomials are centered at 0 (Maclaurin polynomials).
Recall the general formula for the Taylor polynomial of degree 2 centered at 0: \(p_2(x) = f(0) + f'(0)x + \frac{f''(0)}{2}x^2\).
Calculate the derivatives of \(f(x)\): - First, \(f(x) = (1 + 2x)^{1/2}\). - Use the chain rule for the first derivative: \(f'(x) = \frac{1}{2}(1 + 2x)^{-1/2} \cdot 2 = (1 + 2x)^{-1/2}\). - For the second derivative, differentiate \(f'(x)\): \(f''(x) = -\frac{1}{2}(1 + 2x)^{-3/2} \cdot 2 = - (1 + 2x)^{-3/2}\).
Evaluate the function and its derivatives at \(x=0\): - \(f(0) = (1 + 0)^{1/2} = 1\), - \(f'(0) = (1 + 0)^{-1/2} = 1\), - \(f''(0) = - (1 + 0)^{-3/2} = -1\).
Substitute these values into the Taylor polynomial formula: \(p_2(x) = 1 + 1 \cdot x + \frac{-1}{2} x^2 = 1 + x - \frac{1}{2} x^2\). Compare this polynomial with the given options to find the matching polynomial.

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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.
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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, the second derivative gives the quadratic term, and so forth, reflecting how the function changes near the center.
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Matching Functions to Polynomials via Coefficients

To match a function with its Taylor polynomial, compare the polynomial's coefficients to those derived from the function's derivatives at zero. This involves calculating or recalling derivatives and evaluating them at zero to identify the correct polynomial approximation.
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Related Practice
Textbook Question

Probability: sudden−death playoff Teams A and B go into suddendeath overtime after playing to a tie. The teams alternate possession of the ball, and the first team to score wins. Assume each team has a 1/6 chance of scoring when it has the ball, and Team A has the ball first.


a. The probability that Team A ultimately wins is ∑ₖ₌₀∞ (1/6)(5/6)²ᵏ. Evaluate this series.

Textbook Question

Taylor series


a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.


f(x)=sin x, a = π/2

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Textbook Question

Taylor series


a. Use the definition of a Taylor series to find the first four nonzero terms of the Taylor series for the given function centered at a.


f(x) = ln x, a = 3

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Textbook Question

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

a. The function f(x) = √x has a Taylor series centered at 0.

Textbook Question

Taylor series and interval of convergence


a. Use the definition of a Taylor/Maclaurin series to find the first four nonzero terms of the Taylor series for the given function centered at a.


f(x)=2/(1−x)³, a=0

Textbook Question

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


a. Only even powers of x appear in the Taylor polynomials for f(x)=e⁻²ˣ centered at 0.

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