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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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.43
Write out the first three terms of the Maclaurin series for the following functions.
ƒ(x) = (1 + x)^(1/3)"
검증된 단계별 안내1
Recall that the Maclaurin series is the Taylor series expansion of a function about \(x = 0\). It is given by the formula:
\[f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots\]
Identify the function: \(f(x) = (1 + x)^{\frac{1}{3}}\). We will need to find the first and second derivatives of \(f(x)\) to get the first three terms.
Calculate the first derivative using the power rule for fractional exponents:
\[f'(x) = \frac{1}{3}(1 + x)^{-\frac{2}{3}}\]
Calculate the second derivative by differentiating \(f'(x)\) again:
\[f''(x) = \frac{1}{3} \times \left(-\frac{2}{3}\right)(1 + x)^{-\frac{5}{3}} = -\frac{2}{9}(1 + x)^{-\frac{5}{3}}\]
Evaluate \(f(0)\), \(f'(0)\), and \(f''(0)\) by substituting \(x = 0\) into each expression, then write the first three terms of the Maclaurin series as:
\[f(0) + f'(0)x + \frac{f''(0)}{2!}x^2\]
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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4m주요 개념
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Maclaurin Series
The Maclaurin series is a special case of the Taylor series expanded at x = 0. It represents a function as an infinite sum of its derivatives evaluated at zero, multiplied by powers of x. This series helps approximate functions near zero using polynomials.
추천 영상:
08:26
Convergence of Taylor & Maclaurin Series
Binomial Series Expansion
The binomial series generalizes the binomial theorem to any real exponent, allowing expansion of expressions like (1 + x)^r. It uses coefficients derived from generalized binomial coefficients, which involve factorials or the Gamma function for non-integer powers.
추천 영상:
가이드 코스
06:00Geometric Series
Derivatives of Power Functions
To find terms in the Maclaurin series, you compute derivatives of the function at zero. For functions like (1 + x)^(1/3), derivatives involve applying the power rule repeatedly, which helps determine coefficients for each term in the series.
추천 영상:
07:32Representing Functions as Power Series
관련 실천
교과서 질문
Binomial series Write out the first three terms of the Maclaurin series for the following functions.
ƒ(x) = (1 + 2x)^(-5)
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교과서 질문
Radius of convergence Find the radius of convergence for the following power series.
∑ₖ₌₁∞ (k!xᵏ)/(kᵏ)
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교과서 질문
Combining power series Use the power series representation
f(x ) =ln (1 − x) = −∑ₖ₌₁∞ xᵏ/k, for −1 ≤ x < 1,
to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.
p(x) = 2x⁶ ln(1 − 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
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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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