Textbook Question
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) = e⁻ˣ, a = 0
Ch. 11 - Power Series
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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.4
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
Verified step by step guidance1
Identify the function and the point of expansion: here, the function is \(f(x) = \cos^{-1} x\) and the center is \(a = \frac{1}{2}\).
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 \(f^{(k)}(a)\) denotes the \(k\)th derivative of \(f\) evaluated at \(x = a\).
Calculate the function value at \(a\):
\[f(a) = \cos^{-1} \left( \frac{1}{2} \right).\]
Find the first and second derivatives of \(f(x)\):
- First derivative:
\[f'(x) = \frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}},\]
- Second derivative:
\[f''(x) = \frac{d}{dx} f'(x) = \frac{x}{(1 - x^2)^{3/2}}.\]
Evaluate the first and second derivatives at \(x = a = \frac{1}{2}\):
\[f'(a) = -\frac{1}{\sqrt{1 - (\frac{1}{2})^2}}, \quad f''(a) = \frac{\frac{1}{2}}{(1 - (\frac{1}{2})^2)^{3/2}}.\]
Then, substitute these values into the Taylor polynomial formula up to order 2:
\[T_2(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2.\]
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
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.
Recommended video:
07:00
Taylor Polynomials
Derivatives of Inverse Trigonometric Functions
To find the Taylor polynomial of ƒ(x) = cos⁻¹(x), it is essential to know the derivatives of the inverse cosine function. These derivatives involve expressions with square roots and powers, which are used to compute the coefficients of the polynomial.
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Centering the Polynomial at a Specific Point
Centering the Taylor polynomial at a = 1/2 means all derivatives are evaluated at x = 1/2. This shifts the polynomial's approximation to be most accurate near this point, affecting the polynomial's terms and their values.
Recommended video:
07:00Taylor Polynomials
Related Practice
Textbook Question
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
Textbook Question
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
Textbook Question
Binomial series Write out the first three terms of the Maclaurin series for the following functions.
ƒ(x) = (1 + 2x)^(-5)
Textbook Question
Representing functions by power series Identify the functions represented by the following power series.
∑ₖ₌₀∞ (xᵏ)/(2ᵏ)
Textbook Question
Approximating ln 2 Consider the following three ways to approximate
ln 2.
a. Use the Taylor series for ln (1 + x) centered at 0 and evaluate it at x = 1 (convergence was asserted in Table 11.5). Write the resulting infinite series.