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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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Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 11, Problema 11.3.29a
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
Guida verificata passo dopo passo1
Recall the definition of the Taylor series of a function \(f(x)\) centered at \(a\):
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n,\]
where \(f^{(n)}(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(x = a\).
Identify the function and center: here, \(f(x) = \frac{1}{x}\) and \(a = 1\). We will need to find the derivatives of \(f(x)\) evaluated at \(x=1\).
Compute the first four derivatives of \(f(x)\):
- \(f(x) = x^{-1}\)
- \(f'(x) = -x^{-2}\)
- \(f''(x) = 2x^{-3}\)
- \(f^{(3)}(x) = -6x^{-4}\)
- \(f^{(4)}(x) = 24x^{-5}\)
Evaluate each derivative at \(x = 1\):
- \(f(1) = 1\)
- \(f'(1) = -1\)
- \(f''(1) = 2\)
- \(f^{(3)}(1) = -6\)
- \(f^{(4)}(1) = 24\)
Write the first four nonzero terms of the Taylor series using the formula:
\[f(x) \approx f(1) + \frac{f'(1)}{1!}(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f^{(3)}(1)}{3!}(x-1)^3,\]
substituting the values found for each derivative.
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Taylor Series Definition
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point, called the center. Each term involves the nth derivative evaluated at the center, multiplied by (x - a)^n and divided by n!. This series approximates the function near the center point.
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Taylor Series
Derivatives of the Function
To construct the Taylor series, you need to compute successive derivatives of the function at the center point. For f(x) = 1/x, derivatives involve powers of x with alternating signs. Evaluating these derivatives at a = 1 provides the coefficients for the series terms.
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06:30Derivatives of Other Trig Functions
Constructing the Series Terms
Each term of the Taylor series is formed by dividing the nth derivative at the center by n! and multiplying by (x - a)^n. Identifying the first four nonzero terms requires calculating derivatives up to the third order and substituting into this formula to write the polynomial approximation.
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06:00Geometric Series
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