Domanda del libro di testo
A differential equation Find a power series solution of the differential equation y'(x) - 4y + 12 = 0, subject to the condition y(0) = 4. Identify the solution in terms of known functions.
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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.42
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
Guida verificata passo dopo passo1
Identify the function and the center point: here, the function is \(f(x) = \frac{1}{4 + x^{2}}\) and the center is \(a = 0\).
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 \(f(x)\) and evaluate each at \(x=0\):
- Compute \(f(0)\),
- Compute \(f'(0)\),
- Compute \(f''(0)\),
- Compute \(f'''(0)\) if needed,
until you have enough terms to write the first three nonzero terms.
Write out the first three nonzero terms explicitly using the formula:
\[f(x) \approx f(0) + f'(0) x + \frac{f''(0)}{2!} x^{2} + \cdots\]
Include only the terms with nonzero coefficients.
Express the Taylor series in summation notation centered at \(a=0\):
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}\]
where the derivatives \(f^{(n)}(0)\) are as 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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08:42
Taylor Series
Derivatives of the Function
To find the Taylor series terms, you must compute the function's derivatives at the center point. These derivatives determine the coefficients of each term in the series. For rational functions like 1/(4 + x²), derivatives can be found using the chain and quotient rules.
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06:30Derivatives of Other Trig Functions
Summation Notation for Series
Summation notation concisely expresses infinite series using the sigma symbol (∑). After finding the general term of the Taylor series, it can be written as a sum from n=0 to infinity, showing the pattern of coefficients and powers of (x - a). This notation simplifies representation and analysis.
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Percorso guidato
06:00Geometric Series
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Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = sin 2x, n = 3, a = 0
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Domanda del libro di testo
Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. There is more than one way to choose the center of the series.
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Definite integrals by power series Use a Taylor series to approximate the following definite integrals. Retain as many terms as necessary to ensure the error is less than 10⁻³.
∫₀1/2 tan⁻¹ x dx
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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) = cos x, a = π/2
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Domanda del libro di testo
Convergence Write the remainder term Rₙ(x) for the Taylor series for the following functions centered at the given point a. Then show that lim ₙ → ∞ |Rₙ(x)| = 0, for all x in the given interval.
ƒ(x) = sinh x + cosh x, a = 0, - ∞ < x < ∞
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