Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x)=2/(1−x)³, a=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 11.1.71b
{Use of Tech} Small argument approximations Consider the following common approximations when x is near zero.
a. Estimate f(0.1) and give a bound on the error in the approximation.
f(x) = tan⁻¹ x ≈ x
Verified step by step guidance1
Identify the function and the approximation given: the function is \(f(x) = \tan^{-1}(x)\) and the approximation near zero is \(f(x) \approx x\).
Recall that the approximation \(\tan^{-1}(x) \approx x\) comes from the first term of the Taylor series expansion of \(\tan^{-1}(x)\) around \(x=0\).
To estimate \(f(0.1)\) using the approximation, substitute \(x=0.1\) into the approximation: \(f(0.1) \approx 0.1\).
To find a bound on the error, use the Lagrange form of the remainder for the Taylor series. The error term after the linear approximation is given by \(R_2 = \frac{f''(c)}{2} x^2\) for some \(c\) between 0 and 0.1.
Calculate the second derivative of \(f(x) = \tan^{-1}(x)\), which is \(f''(x) = -\frac{2x}{(1+x^2)^2}\). Then find the maximum absolute value of \(f''(c)\) on the interval \([0, 0.1]\) to bound the error \(|R_2| \leq \frac{\max|f''(c)|}{2} (0.1)^2\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Small Angle Approximations
Small angle approximations simplify functions near zero by replacing them with simpler expressions, such as approximating arctan(x) by x when x is close to zero. This helps estimate function values quickly without complex calculations.
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Determining Error and Relative Error
Taylor Series Expansion
The Taylor series expresses a function as an infinite sum of terms calculated from its derivatives at a point. For arctan(x), the first term is x, and higher-order terms provide corrections, allowing approximation and error estimation near zero.
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Error Bound in Approximations
Error bounds quantify the maximum difference between the true function value and its approximation. Using the remainder term of the Taylor series, one can estimate how accurate the approximation f(x) ≈ x is for a given x, ensuring reliable results.
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04:57Determining Error and Relative Error
Related Practice
Textbook Question
Symmetry
b. Use infinite series to show that sin x is an odd function. That is, show sin (-x) = -sin x.
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Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = (1 + x²)⁻¹, a = 0
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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
b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.
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Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x) = e⁻ˣ, a=0
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Textbook Question
Taylor series and interval of convergence
b. Write the power series using summation notation.
f(x)=3ˣ, a=0
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