교과서 질문
Use the Taylor series for cos x centered at 0 to verify that lim ₓ→ₐ (1− cos x)/x = 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
11장, 문제 11.1.37
{Use of Tech} Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with n = 3.
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
√1.06
검증된 단계별 안내1
Identify the function to approximate: here, we want to approximate \( f(x) = \sqrt{x} \) near a point where the function and its derivatives are easy to compute. A good choice is \( a = 1 \) because \( \sqrt{1} = 1 \).
Write the Taylor polynomial of degree 3 for \( f(x) \) centered at \( a = 1 \). The general formula is:
\[
T_3(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3
\]
Calculate the first, second, and third derivatives of \( f(x) = \sqrt{x} = x^{1/2} \).
Evaluate each derivative at \( x = 1 \) to find \( f(1), f'(1), f''(1), \) and \( f'''(1) \). Substitute these values into the Taylor polynomial formula.
Substitute \( x = 1.06 \) into the Taylor polynomial \( T_3(x) \) to approximate \( \sqrt{1.06} \). This gives the approximate value using the third-degree Taylor polynomial.
To find the absolute error, calculate the exact value of \( \sqrt{1.06} \) using a calculator, then subtract the Taylor polynomial approximation from this exact value. The absolute error is:
\[
\text{Absolute Error} = |\sqrt{1.06} - T_3(1.06)|
\]
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
3m도움이 되었나요?
주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Taylor Polynomials
Taylor polynomials approximate functions near a point by using derivatives at that point. For n=3, the polynomial includes terms up to the cubic degree, providing a close estimate of the function's value near the expansion point.
추천 영상:
07:00
Taylor Polynomials
Error Estimation in Approximations
The absolute error measures the difference between the exact value and the approximation. Calculating this helps assess the accuracy of the Taylor polynomial approximation.
추천 영상:
04:57Determining Error and Relative Error
Function Expansion Point and Domain
Choosing the expansion point (often near the value to approximate) is crucial for accuracy. Understanding the domain and behavior of the function, like √x near x=1, ensures the Taylor polynomial converges well.
추천 영상:
5:10Finding the Domain and Range of a Graph
관련 실천
교과서 질문
{Use of Tech} Approximations with Taylor polynomials
a. Approximate the given quantities using Taylor polynomials with n = 3.
b. Compute the absolute error in the approximation, assuming the exact value is given by a calculator.
e⁰ᐧ¹²
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교과서 질문
Suppose a power series converges if |x−3|<4 and diverges if |x−3| ≥ 4. Determine the radius and interval of convergence.
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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) = − 1/(1 + x)² using f(x) = 1/(1 + x)
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교과서 질문
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (sin 2x)/x
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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∞ (x/3)ᵏ
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