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Ch. 11 - Power Series
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 11 - Power SeriesProblema 11.4.79a
Capitolo 11, Problema 11.4.79a

{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
a. Compute S′(x) and C′(x).

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Identify the given functions: \( S(x) = \int_0^x \sin(t^2) \, dt \) and \( C(x) = \int_0^x \cos(t^2) \, dt \). These are defined as definite integrals with variable upper limits.
Recall the Fundamental Theorem of Calculus Part 1, which states that if \( F(x) = \int_a^x f(t) \, dt \), then \( F'(x) = f(x) \), provided \( f \) is continuous.
Apply the theorem to \( S(x) \): since \( S(x) = \int_0^x \sin(t^2) \, dt \), its derivative is \( S'(x) = \sin(x^2) \).
Similarly, apply the theorem to \( C(x) \): since \( C(x) = \int_0^x \cos(t^2) \, dt \), its derivative is \( C'(x) = \cos(x^2) \).
Summarize the results: \( S'(x) = \sin(x^2) \) and \( C'(x) = \cos(x^2) \). These derivatives express the rate of change of the Fresnel integrals with respect to \( x \).

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Fundamental Theorem of Calculus

This theorem connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, its derivative is the integrand evaluated at that limit. For example, if F(x) = ∫₀ˣ f(t) dt, then F'(x) = f(x). This principle is essential for finding derivatives of integral-defined functions like Fresnel integrals.
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Fundamental Theorem of Calculus Part 1

Fresnel Integrals

Fresnel integrals S(x) and C(x) are special functions defined as integrals of sine and cosine of t squared, respectively. They arise in optics and wave theory, representing integrals that cannot be expressed in elementary functions. Understanding their definitions is key to differentiating them correctly.
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Integration by Parts for Definite Integrals

Differentiation of Integral Functions with Variable Limits

When differentiating an integral with a variable upper limit, the derivative is the integrand evaluated at that upper limit, assuming continuity. This concept allows direct computation of S'(x) and C'(x) by substituting x into sin(t²) and cos(t²), respectively.
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Limits of Rational Functions: Denominator = 0
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Sine integral function The function Si(x) = ∫₀ˣ f(t) dt, where f(t) = {(sin t)/t if t ≠ 0, 1 if t = 0, is called the sine integral function.

a. Expand the integrand in a Taylor series centered at 0.

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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.

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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) = 1/x, a = 1

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{Use of Tech} Bessel functions Bessel functions arise in the study of wave propagation in circular geometries (for example, waves on a circular drum head). They are conveniently defined as power series. One of an infinite family of Bessel functions is

J₀(x) = ∑ₖ₌₀∞ (−1)ᵏ/(2²ᵏ(k!)²) x²ᵏ

a. Write out the first four terms of J₀.

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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) = cosh 3x, a = 0

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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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