교과서 질문
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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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
11장, 문제 11.R.56
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
검증된 단계별 안내1
Recall that the Taylor series expansion of \( \tan^{-1} x \) centered at 0 is given by the alternating series \( \tan^{-1} x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} \). Write out the first few terms explicitly: \( x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \).
Set up the integral of the Taylor series term-by-term from 0 to \( \frac{1}{2} \): \( \int_0^{1/2} \tan^{-1} x \, dx = \int_0^{1/2} \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{2n+1} \, dx = \sum_{n=0}^{\infty} (-1)^n \frac{1}{2n+1} \int_0^{1/2} x^{2n+1} \, dx \).
Integrate each term inside the sum: \( \int_0^{1/2} x^{2n+1} \, dx = \left[ \frac{x^{2n+2}}{2n+2} \right]_0^{1/2} = \frac{(1/2)^{2n+2}}{2n+2} \). Substitute this back into the sum to get \( \sum_{n=0}^{\infty} (-1)^n \frac{(1/2)^{2n+2}}{(2n+1)(2n+2)} \).
Determine how many terms to keep by estimating the error. Since the series is alternating with decreasing terms in absolute value, the error after \( N \) terms is less than or equal to the absolute value of the \( (N+1)^{th} \) term. Calculate the magnitude of terms until the next term is less than \( 10^{-3} \).
Sum the terms up to the required \( N \) to approximate the integral \( \int_0^{1/2} \tan^{-1} x \, dx \) with an error less than \( 10^{-3} \).
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. For functions like arctan(x), the series allows approximation by polynomials, which simplifies integration and error estimation.
추천 영상:
08:42
Taylor Series
Definite Integration of Power Series
Integrating a power series term-by-term within its interval of convergence is valid and often easier than integrating the original function. This approach transforms the integral into a sum of integrals of polynomial terms, facilitating approximation.
추천 영상:
05:58Intro to Power Series
Error Estimation in Series Approximations
To ensure the approximation error is below a threshold (e.g., 10⁻³), one uses remainder terms or bounds on the next omitted term in the Taylor series. This guarantees the truncated series provides a sufficiently accurate estimate of the integral.
추천 영상:
가이드 코스
04:57Determining Error and Relative Error
관련 실천
교과서 질문
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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교과서 질문
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 x cos x dx
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교과서 질문
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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교과서 질문
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = cos⁻¹ x, n = 2, a = 1/2
80
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교과서 질문
Approximating ln 2 Consider the following three ways to approximate
ln 2.
a. Use the Taylor series for ln (1 + x) centered at 0 and evaluate it at x = 1 (convergence was asserted in Table 11.5). Write the resulting infinite series.
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