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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 11.4.79d
Chapter 11, Problem 11.4.79d

{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?

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Recall that the Fresnel integral \( S(x) = \int_0^x \sin(t^2) \, dt \) can be expanded into a Maclaurin series by first expanding \( \sin(t^2) \) as a power series and then integrating term-by-term.
Write the Maclaurin series for \( \sin(z) \), which is \( \sin(z) = \sum_{n=0}^\infty (-1)^n \frac{z^{2n+1}}{(2n+1)!} \). Substitute \( z = t^2 \) to get \( \sin(t^2) = \sum_{n=0}^\infty (-1)^n \frac{t^{4n+2}}{(2n+1)!} \).
Integrate the series term-by-term from 0 to \( x \) to find the Maclaurin series for \( S(x) \): \[ S(x) = \sum_{n=0}^\infty (-1)^n \frac{1}{(2n+1)!} \int_0^x t^{4n+2} \, dt = \sum_{n=0}^\infty (-1)^n \frac{x^{4n+3}}{(2n+1)! (4n+3)}. \]
To approximate \( S(0.05) \) with an error no greater than \( 10^{-4} \), use the alternating series remainder estimation. The error is less than or equal to the absolute value of the first omitted term in the series.
Calculate the terms of the series for increasing \( n \) until the absolute value of the next term is less than or equal to \( 10^{-4} \). The number of terms used up to that point is the number required for the desired accuracy.

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Key Concepts

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

Fresnel Integrals

Fresnel integrals S(x) and C(x) are defined as integrals of sine and cosine functions with quadratic arguments, specifically S(x) = ∫₀ˣ sin(t²) dt and C(x) = ∫₀ˣ cos(t²) dt. They arise in optics and wave theory, representing diffraction patterns. Understanding their integral form is essential for approximating their values using series expansions.
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Maclaurin Series Expansion

A Maclaurin series is a Taylor series expansion of a function about zero, expressing it as an infinite sum of derivatives at zero multiplied by powers of x. For Fresnel integrals, the integrand sin(t²) can be expanded into a power series, which is then integrated term-by-term to find the series for S(x). This allows approximation of S(x) by summing a finite number of terms.
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Error Estimation in Series Approximation

When approximating functions by partial sums of their series, the remainder or error term measures the difference from the exact value. To ensure the error is below a given tolerance (e.g., 10⁻⁴), one must estimate the size of the next term or use bounds on the remainder. This guides how many terms are needed for the desired accuracy.
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Related Practice
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

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

d. Suppose f'' is continuous on an interval that contains a, where f has an inflection point at a. Then the second−order Taylor polynomial for f at a is linear.

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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

e. How many terms of the Maclaurin series are required to approximate C(−0.25) with an error no greater than 10⁻⁶?

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

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

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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