Textbook Question
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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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 11.3.23a
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
Verified step by step guidance1
Recall the definition of the Maclaurin series for a function \(f(x)\) centered at \(a=0\):
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n,\]
where \(f^{(n)}(0)\) is the \(n\)-th derivative of \(f\) evaluated at 0.
Identify the function: \(f(x) = \cosh(3x)\). We will need to find the derivatives of \(f(x)\) and evaluate them at \(x=0\).
Compute the first few derivatives of \(f(x)\):
- \(f(x) = \cosh(3x)\)
- \(f'(x) = 3 \sinh(3x)\)
- \(f''(x) = 9 \cosh(3x)\)
- \(f^{(3)}(x) = 27 \sinh(3x)\)
- \(f^{(4)}(x) = 81 \cosh(3x)\)
and so on.
Evaluate these derivatives at \(x=0\):
- \(f(0) = \cosh(0) = 1\)
- \(f'(0) = 3 \sinh(0) = 0\)
- \(f''(0) = 9 \cosh(0) = 9\)
- \(f^{(3)}(0) = 27 \sinh(0) = 0\)
- \(f^{(4)}(0) = 81 \cosh(0) = 81\)
Write the first four nonzero terms of the Maclaurin series using the formula:
\[f(x) \approx \sum_{n=0}^{3} \frac{f^{(n)}(0)}{n!} x^n,\]
but since odd derivatives are zero, only even terms contribute:
\[f(x) \approx f(0) + \frac{f''(0)}{2!} x^2 + \frac{f^{(4)}(0)}{4!} x^4 + \ldots\]
Substitute the values found to express these terms explicitly.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor and Maclaurin Series
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. When centered at zero, it is called a Maclaurin series. Each term involves the nth derivative evaluated at the center, multiplied by (x - a)^n and divided by n!.
Recommended video:
08:26
Convergence of Taylor & Maclaurin Series
Derivatives of Hyperbolic Functions
Understanding the derivatives of hyperbolic functions like cosh(x) is essential. For cosh(3x), derivatives alternate between cosh(3x) and sinh(3x), scaled by powers of 3 due to the chain rule. This pattern helps compute the terms of the Taylor series efficiently.
Recommended video:
5:50Asymptotes of Hyperbolas
Interval of Convergence
The interval of convergence is the range of x-values for which the Taylor series converges to the function. For entire functions like cosh(3x), the series converges for all real x, meaning the interval of convergence is (-∞, ∞). This ensures the series accurately represents the function everywhere.
Recommended video:
08:44Interval of Convergence
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.
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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) = 1/x, a = 1
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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)=sin x, a = π/2
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Textbook Question
{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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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
a. Compute S′(x) and C′(x).
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