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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 11.1.5
Chapter 11, Problem 11.1.5

Suppose f(0)=1, f'(0)=0, f''(0)=2, and f⁽³⁾(0)=6. Find the third-order Taylor polynomial for f centered at 0 and use it to approximate f(0.2).

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Recall the formula for the third-order Taylor polynomial of a function \( f \) centered at \( a = 0 \): \[ P_3(x) = f(0) + f'(0) x + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3 \]
Substitute the given values into the polynomial: \[ f(0) = 1, \quad f'(0) = 0, \quad f''(0) = 2, \quad f^{(3)}(0) = 6 \] So, \[ P_3(x) = 1 + 0 \cdot x + \frac{2}{2} x^2 + \frac{6}{6} x^3 \]
Simplify the coefficients in the polynomial: \[ P_3(x) = 1 + x^2 + x^3 \]
To approximate \( f(0.2) \), substitute \( x = 0.2 \) into the polynomial: \[ P_3(0.2) = 1 + (0.2)^2 + (0.2)^3 \]
Calculate the powers of \( 0.2 \) (without final summation) to prepare for approximation: \[ (0.2)^2 = 0.04, \quad (0.2)^3 = 0.008 \] Then, sum these terms with 1 to get the approximate value of \( f(0.2) \).

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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 derivatives at that point. The nth-order Taylor polynomial includes terms up to the nth derivative, providing a polynomial that closely matches the function's behavior near the center.
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Derivatives and Their Role in Taylor Series

Derivatives at the center point determine the coefficients of the Taylor polynomial. Each term involves the function's nth derivative evaluated at the center, divided by n! and multiplied by (x - center)^n, capturing the function's local rate of change.
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Polynomial Approximation and Evaluation

Once the Taylor polynomial is constructed, it can be used to approximate function values near the center by substituting the desired x-value. This provides an estimate that becomes more accurate with higher-order polynomials and points closer to the center.
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Related Practice
Textbook Question

Limits Evaluate the following limits using Taylor series.

lim ₓ→∞ x(e¹/ˣ − 1)

Textbook Question

Manipulating Taylor series Use the Taylor series in Table 11.5 to find the first four nonzero terms of the Taylor series for the following functions centered at 0.


(1 + x⁴)⁻¹

Textbook Question

Power series for derivatives


a. Differentiate the Taylor series centered at 0 for the following functions.

b. Identify the function represented by the differentiated series.

c. Give the interval of convergence of the power series for the derivative.


f(x) = (1 − x)⁻¹

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Textbook Question

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


∑ₖ₌₁∞ (kx)ᵏ

Textbook Question

Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers.

√e

Textbook Question

Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.

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