교과서 질문
Approximating real numbers Use an appropriate Taylor series to find the first four nonzero terms of an infinite series that is equal to the following numbers. There is more than one way to choose the center of the series.
sin 20°
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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.2
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
검증된 단계별 안내1
Identify the function and the point of expansion: here, the function is \(f(x) = \sin(2x)\) and the center is \(a = 0\).
Recall the formula for the nth-order Taylor polynomial centered at \(a\):
\[T_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k,\]
where \(f^{(k)}(a)\) is the \(k\)th derivative of \(f\) evaluated at \(x = a\).
Compute the derivatives of \(f(x) = \sin(2x)\) up to the 3rd order:
- \(f(x) = \sin(2x)\)
- \(f'(x) = 2 \cos(2x)\)
- \(f''(x) = -4 \sin(2x)\)
- \(f^{(3)}(x) = -8 \cos(2x)\)
Evaluate each derivative at \(x = 0\):
- \(f(0) = \sin(0) = 0\)
- \(f'(0) = 2 \cos(0) = 2\)
- \(f''(0) = -4 \sin(0) = 0\)
- \(f^{(3)}(0) = -8 \cos(0) = -8\)
Construct the 3rd-order Taylor polynomial using the formula:
\[T_3(x) = f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3,\]
and substitute the values found in the previous step.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Taylor Polynomial
A Taylor polynomial approximates a function near a point a by using a finite sum of derivatives of the function at a. The nth-order Taylor polynomial includes terms up to the nth derivative, providing a polynomial that closely matches the function's behavior near a.
추천 영상:
07:00
Taylor Polynomials
Derivatives of Trigonometric Functions
To construct the Taylor polynomial for ƒ(x) = sin(2x), you need to compute derivatives of sin(2x) at the point a. Knowing the derivatives of sine and cosine functions, and applying the chain rule for the inner function 2x, is essential for finding these values.
추천 영상:
06:35Derivatives of Other Inverse Trigonometric Functions
Centering at a Point (a = 0)
Centering the Taylor polynomial at a = 0 means the polynomial is expanded around x = 0, also called a Maclaurin polynomial. This simplifies calculations since derivatives are evaluated at zero, and powers of (x - 0) become powers of x.
추천 영상:
04:50Critical Points
관련 실천
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.
∫₀1/2 tan⁻¹ x dx
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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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