Textbook Question
Approximating ln 2 Consider the following three ways to approximate
ln 2.
e. Using four terms of the series, which of the three series derived in parts (a)–(d) gives the best approximation to ln 2? Can you explain why?
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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
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 11, Problem 11.R.6
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = e^(sin x), n = 2, a = 0
Verified step by step guidance1
Identify the function and the point of expansion: here, the function is \(f(x) = e^{\sin x}\) 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 \(x = a\).
Calculate the derivatives of \(f(x)\) up to order 2:
- First, find \(f(0) = e^{\sin 0}\).
- Then, find \(f'(x)\) using the chain rule and evaluate \(f'(0)\).
- Next, find \(f''(x)\) by differentiating \(f'(x)\) and evaluate \(f''(0)\).
Substitute the values of \(f(0)\), \(f'(0)\), and \(f''(0)\) into the Taylor polynomial formula:
\[T_2(x) = f(0) + f'(0)(x - 0) + \frac{f''(0)}{2!}(x - 0)^2.\]
Write the final expression for the 2nd-order Taylor polynomial centered at 0, which approximates \(f(x)\) near \(x=0\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Polynomials
A Taylor polynomial approximates a function near a point a by using the function's derivatives at that point. The nth-order Taylor polynomial includes terms up to the nth derivative, providing a polynomial that closely matches the function's behavior near a.
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Derivatives of Composite Functions
To find the Taylor polynomial of e^(sin x), you must compute derivatives of a composite function. This involves applying the chain rule repeatedly to differentiate e^(sin x), since it is a composition of the exponential and sine functions.
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3:48Evaluate Composite Functions - Special Cases
Evaluating Derivatives at the Center Point
After finding the derivatives, evaluate each at the center point a=0. These values are used as coefficients in the Taylor polynomial, ensuring the polynomial matches the function and its derivatives at that point.
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5:14Evaluate Logarithms
Related Practice
Textbook Question
Approximating ln 2 Consider the following three ways to approximate
ln 2.
b. Use the Taylor series for ln (1 - x) centered at 0 and the identity ln 2 = -ln 1/2. Write the resulting infinite series.
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Textbook Question
Find the remainder term Rₙ(x) for the Taylor series centered at 0 for the following functions. Find an upper bound for the magnitude of the remainder on the given interval for the given value of n. (The bound is not unique.)
ƒ(x) = eˣ; bound R₃(x), for |x| < 1
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Textbook Question
Power series from the geometric series Use the geometric series a Σₖ ₌ ₀ ∞ (x)ᵏ = 1/(1 - x), for |x| < 1, to determine the Maclaurin series and the interval of convergence for the following functions.
ƒ(x) = ln (1 - 4x)
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Textbook Question
ƒ(x) = eˣ, a = 0; e-0.08
b. Use the Taylor polynomials to approximate the given expression. Make a table showing the approximations and the absolute error in these approximations using a calculator for the exact function value.
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Textbook Question
Radius and interval of convergence Use the Ratio Test or the Root Test to determine the radius of convergence of the following power series. Test the endpoints to determine the interval of convergence, when appropriate.
x +x³/3 +x⁵/5 +x⁷/7 + ...
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