Domanda del libro di testo
Taylor series
b. Write the power series using summation notation.
f(x) = 1/x, a = 1
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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 11.1.71b
{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
Guida verificata passo dopo passo1
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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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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Percorso guidato
04:57Determining Error and Relative Error
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