교과서 질문
Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.
f(x) = sin x, a = π/2
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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 11.4.25
Power series for derivatives
a. Differentiate the Taylor series centered at 0 for the following functions.
b. Identify the function represented by the differentiated series.
c. Give the interval of convergence of the power series for the derivative.
f(x) = eˣ
검증된 단계별 안내1
Start with the Taylor series expansion of the function \(f(x) = e^x\) centered at 0, which is given by the infinite sum:
\[f(x) = \sum_{n=0}^{\infty} \frac{x^n}{n!}\]
Differentiate the series term-by-term with respect to \(x\). Using the power rule for differentiation, the derivative of each term \(\frac{x^n}{n!}\) is:
\[\frac{d}{dx} \left( \frac{x^n}{n!} \right) = \frac{n x^{n-1}}{n!} = \frac{x^{n-1}}{(n-1)!}\]
Note that for \(n=0\), the term is a constant and its derivative is zero, so the summation index will start from \(n=1\) after differentiation.
Rewrite the differentiated series by adjusting the index to start from \(n=0\) for convenience. Let \(m = n - 1\), then:
\[f'(x) = \sum_{m=0}^{\infty} \frac{x^m}{m!}\]
Recognize that the differentiated series is the same as the original Taylor series for \(e^x\). Therefore, the function represented by the differentiated series is also \(e^x\).
Determine the interval of convergence. Since the original series for \(e^x\) converges for all real \(x\), the differentiated series will have the same interval of convergence, which is \((-\infty, \infty)\).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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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, often zero (Maclaurin series). For eˣ, the series is ∑(xⁿ/n!), which converges for all real x. Understanding this expansion is essential to differentiate the series term-by-term.
추천 영상:
08:42
Taylor Series
Term-by-Term Differentiation of Power Series
Power series can be differentiated term-by-term within their interval of convergence. Differentiating each term of the Taylor series of eˣ involves applying the power rule to xⁿ/n!, resulting in a new series representing the derivative function. This process helps identify the derivative's power series.
추천 영상:
05:58Intro to Power Series
Interval of Convergence
The interval of convergence is the set of x-values for which a power series converges. Differentiating a power series does not change its radius of convergence, so the interval for the derivative's series is the same as the original. For eˣ, the series converges for all real numbers, meaning the interval is (-∞, ∞).
추천 영상:
08:44Interval of Convergence
관련 실천
교과서 질문
Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.
f(x) = ∫₀ˣ sin t² dt
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교과서 질문
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ k(x−1)ᵏ
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교과서 질문
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ 2ᵏ x²ᵏ⁺¹
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교과서 질문
Differentiating and integrating power series Find the power series representation for g centered at 0 by differentiating or integrating the power series for f (perhaps more than once). Give the interval of convergence for the resulting series.
g(x) = x/(1 + x²)² using f(x) = 1/(1 + x²)
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교과서 질문
Does the accuracy of an approximation given by a Taylor polynomial generally increase or decrease with the order of the approximation? Explain.
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