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.
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.8
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = sinh (-3x), n = 3, a = 0
Verified step by step guidance1
Recall the definition of the nth-order Taylor polynomial of a function \( f(x) \) 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 \( a \).
Identify the function and the center: here, \( f(x) = \sinh(-3x) \), \( n = 3 \), and \( a = 0 \). We will need to find \( f(0) \), \( f'(0) \), \( f''(0) \), and \( f^{(3)}(0) \).
Compute the derivatives of \( f(x) = \sinh(-3x) \) step-by-step:
- First derivative: \( f'(x) = \frac{d}{dx} \sinh(-3x) \)
- Second derivative: \( f''(x) = \frac{d}{dx} f'(x) \)
- Third derivative: \( f^{(3)}(x) = \frac{d}{dx} f''(x) \)
Remember to apply the chain rule carefully since the argument is \( -3x \).
Evaluate each derivative at \( x = 0 \):
Calculate \( f(0) \), \( f'(0) \), \( f''(0) \), and \( f^{(3)}(0) \) by substituting \( x = 0 \) into each derivative expression.
Construct the Taylor polynomial of order 3 using the formula:
\[ T_3(x) = f(0) + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f^{(3)}(0)}{3!} x^3 \]
This polynomial 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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Taylor Polynomials
Hyperbolic Sine Function (sinh)
The hyperbolic sine function, sinh(x), is defined as (e^x - e^(-x))/2. It is an odd function with derivatives that cycle between sinh and cosh, which is important when computing derivatives for the Taylor polynomial.
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Evaluating Derivatives at the Center Point
To construct the Taylor polynomial centered at a, you must compute the function's derivatives at x = a. These values determine the coefficients of the polynomial terms, making accurate evaluation at the center essential.
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Related Practice
Textbook Question
Limits by power series Use Taylor series to evaluate the following limits.
lim ₙ → ₄ ln (x - 3)/(x² - 16)
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) = 1/(1 + 5x)
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)
Textbook Question
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°
Textbook Question
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) = cos x, a = π/2
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