Domanda del libro di testo
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)=3ˣ, a=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 11.3.13a
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)=2/(1−x)³, a=0
Guida verificata passo dopo passo1
Recall the definition of the Taylor series of a function \(f(x)\) centered at \(a=0\) (Maclaurin series):
\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n,\]
where \(f^{(n)}(0)\) is the \(n\)-th derivative of \(f\) evaluated at 0.
Start by finding the function and its first few derivatives:
- \(f(x) = \frac{2}{(1-x)^3}\)
Calculate \(f'(x)\), \(f''(x)\), and \(f'''(x)\) using the chain rule and power rule.
Evaluate each derivative at \(x=0\):
- Compute \(f(0)\), \(f'(0)\), \(f''(0)\), and \(f'''(0)\) to find the coefficients for the Taylor series terms.
Write the first four nonzero terms of the Taylor series using the formula:
\[\frac{f^{(n)}(0)}{n!} x^n\]
for \(n=0,1,2,3\).
Express the partial sum of the Taylor series with these four terms explicitly, which approximates \(f(x)\) near \(x=0\).
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Taylor and Maclaurin Series
A Taylor series represents a function as an infinite sum of terms calculated from the derivatives at a single point. When centered at zero, it is called a Maclaurin series. Each term involves the nth derivative of the function evaluated at the center, multiplied by (x - a)^n and divided by n!.
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08:26
Convergence of Taylor & Maclaurin Series
Derivatives of the Function
To find the Taylor series terms, you must compute successive derivatives of the function at the center point. These derivatives determine the coefficients of the series. For f(x) = 2/(1−x)^3, calculating the first few derivatives at x=0 is essential to find the first four nonzero terms.
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06:30Derivatives of Other Trig Functions
Interval of Convergence
The interval of convergence is the range of x-values for which the Taylor series converges to the function. It depends on the function's singularities and can be found using tests like the ratio test. For rational functions like 2/(1−x)^3, the interval is typically related to the distance from the center to the nearest singularity.
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08:44Interval of Convergence
Pratica correlata
Domanda del libro di testo
Matching functions with polynomials Match functions a–f with Taylor polynomials A–F (all centered at 0). Give reasons for your choices.
a. √(1 + 2x)
A. p₂(x)= 1 + 2x + 2x²
B. p₂(x) = 1 − 6x + 24x²
C. p₂(x) = 1 + x − x²/2
D. p₂(x) = 1 − 2x + 4x²
E. p₂(x) = 1 − x + (3/2)x²
F. p₂(x) = 1 − 2x + 2x²
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Domanda del libro di testo
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function f(x) = √x has a Taylor series centered at 0.
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Domanda del libro di testo
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) = 2ˣ, a = 1
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Domanda del libro di testo
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) = e⁻ˣ, a=0
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Domanda del libro di testo
ƒ(x) = eˣ, a = 0; e-0.08
a. Find the Taylor polynomials of order n = 1 and n = 2 for the given functions centered at the given point a.
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