Domanda del libro di testo
Approximating ln 2 Consider the following three ways to approximate
ln 2.
d. At what value of x should the series in part (c) be evaluated to approximate ln 2? Write the resulting infinite series for ln 2.
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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.R.37
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = cos x, a = π/2
Guida verificata passo dopo passo1
Identify the function and the center point: here, the function is \(f(x) = \cos x\) and the center is \(a = \frac{\pi}{2}\).
Recall the Taylor series formula 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 \(x = a\).
Calculate the first few derivatives of \(f(x) = \cos x\) and evaluate them at \(x = \frac{\pi}{2}\):
- \(f(x) = \cos x\) so \(f\left(\frac{\pi}{2}\right) = 0\)
- \(f'(x) = -\sin x\) so \(f'\left(\frac{\pi}{2}\right) = -1\)
- \(f''(x) = -\cos x\) so \(f''\left(\frac{\pi}{2}\right) = 0\)
- \(f'''(x) = \sin x\) so \(f'''\left(\frac{\pi}{2}\right) = 1\)
Write out the first three nonzero terms of the Taylor series using the derivatives found:
\[f(x) \approx f(a) + \frac{f'(a)}{1!}(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 + \cdots\]
Substitute the values and keep only the first three nonzero terms.
Express the Taylor series in summation notation:
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}\left(\frac{\pi}{2}\right)}{n!} (x - \frac{\pi}{2})^n,\]
where you can identify the pattern of nonzero terms from the derivatives calculated.
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Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. Each term involves the nth derivative evaluated at the center point, multiplied by (x - a)^n and divided by n!. This allows approximation of functions near the point a.
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Taylor Series
Derivatives of Trigonometric Functions
Understanding the derivatives of cosine is essential, as they follow a repeating cycle: cos x, -sin x, -cos x, sin x, then back to cos x. Evaluating these derivatives at the center point helps determine the coefficients of the Taylor series terms.
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Summation Notation for Series
Summation notation concisely expresses infinite or finite sums using the sigma symbol (∑). Writing the Taylor series in summation form involves identifying the general term formula, including factorial denominators and powers of (x - a), to represent the series compactly.
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06:00Geometric Series
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Approximating ln 2 Consider the following three ways to approximate
ln 2.
c. Use the property ln a/b = ln a - ln b and the series of parts (a) and (b) to find the Taylor series for ƒ(x) = ln (1 + x)/(1 - x) b centered at 0.
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Domanda del libro di testo
Taylor series Write out the first three nonzero terms of the Taylor series for the following functions centered at the given point a. Then write the series using summation notation.
ƒ(x) = 1/(4 + x²), a = 0
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Domanda del libro di testo
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ƒ(x) = sinh x + cosh x, a = 0, - ∞ < x < ∞
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