Textbook Question
{Use of Tech} Binomial series
a. Find the first four nonzero terms of the binomial series centered at 0 for the given function.
f(x) = (1+x)⁻²/³; approximate 1.18⁻²/³.
Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 11, Problem 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).
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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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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Related Practice
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) = 1/x, a = 1
1
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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) = ln (x − 2), a = 3
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) = e²ˣ, a = 0
Textbook Question
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
a. Estimate f(0.1) and give a bound on the error in the approximation.
f(x) = eˣ ≈ 1 + x
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) = cosh 3x, a = 0