Textbook Question
Find a Taylor series for f centered at 2 given that f⁽ᵏ⁾(2)=1, for all nonnegative integers k.
75
views
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 11.4.38
{Use of Tech} Approximating definite integrals Use a Taylor series to approximate the following definite integrals. Retain as many terms as needed to ensure the error is less than 10⁻⁴.
∫₀⁰ᐧ² sin x² dx
Verified step by step guidance1
Recognize that the integral involves the function \( \sin(x^2) \), which does not have an elementary antiderivative, making a Taylor series approximation a suitable approach.
Write the Taylor series expansion of \( \sin z \) around \( z=0 \):
\[
\sin z = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2n+1}}{(2n+1)!} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \cdots
\]
Substitute \( z = x^2 \) to get the series for \( \sin(x^2) \):
\[
\sin(x^2) = x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!} - \cdots
\]
Set up the integral of the series term-by-term from 0 to 0.2:
\[
\int_0^{0.2} \sin(x^2) \, dx = \int_0^{0.2} \left( x^2 - \frac{x^6}{3!} + \frac{x^{10}}{5!} - \cdots \right) dx = \sum_{n=0}^{\infty} (-1)^n \frac{1}{(2n+1)!} \int_0^{0.2} x^{4n+2} \, dx
\]
Integrate each term using the power rule:
\[
\int_0^{0.2} x^{m} \, dx = \frac{(0.2)^{m+1}}{m+1}
\]
where \( m = 4n + 2 \). So each term becomes:
\[
(-1)^n \frac{(0.2)^{4n+3}}{(2n+1)! (4n+3)}
\]
Add terms of the series until the absolute value of the next term is less than \( 10^{-4} \) to ensure the error is within the required tolerance. Sum all included terms to approximate the value of the integral.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from the function's derivatives at a single point. For sin(x²), expanding around x=0 allows approximation by polynomials, making integration manageable. Truncating the series after enough terms controls the approximation error.
Recommended video:
08:42
Taylor Series
Definite Integral Approximation Using Series
Integrating a function approximated by a Taylor series involves integrating each polynomial term individually over the given interval. This method simplifies complex integrals into sums of integrals of powers of x, which are straightforward to compute, enabling approximate evaluation of the definite integral.
Recommended video:
05:43Definition of the Definite Integral
Error Estimation and Control
When truncating a Taylor series, the remainder term estimates the error between the true function and its polynomial approximation. Ensuring the error is less than 10⁻⁴ requires calculating or bounding this remainder, guiding how many terms to retain for a sufficiently accurate integral approximation.
Recommended video:
04:57Determining Error and Relative Error
Related Practice
Textbook Question
Use of Tech Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at a.
c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
f(x) = 8x^(3/2), a=1; approximate 8 ⋅ 1.1^(3/2)
78
views
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) = tan⁻¹(4x), a = 0
70
views
Textbook Question
Use of Tech Linear and quadratic approximation
a. Find the linear approximating polynomial for the following functions centered at the given point a.
b. Find the quadratic approximating polynomial for the following functions centered at a.
c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.
Find the Taylor polynomials p₁, …, p₅ centered at a=0 for f(x)=e⁻ˣ
37
views
Textbook Question
Derivative trick Here is an alternative way to evaluate higher derivatives of a function f that may save time. Suppose you can find the Taylor series for f centered at the point a without evaluating derivatives (for example, from a known series). Then f⁽ᵏ⁾(a)=k! multiplied by the coefficient of (x−a)ᵏ. Use this idea to evaluate f⁽³⁾(0) and f⁽⁴⁾(0) for the following functions. Use known series and do not evaluate derivatives.
f(x) = eᶜᵒˢ ˣ
107
views
Textbook Question
{Use of Tech} Approximating sin x Let f(x)=sin x, and let pₙ and qₙ be nth−order Taylor polynomials for f centered at 0 and π, respectively.
a. Find p₅ and q₅
b. Graph f, p₅, and q₅ on the interval [−π, 2π]. On what interval is p₅ a better approximation to f than q₅? On what interval is q₅ a better approximation to f than p₅?
c. Complete the following table showing the errors in the approximations given by p₅ and q₅ at selected points.
d. At which points in the table is p₅ a better approximation to f than q₅? At which points do p₅ and q₅ give equal approximations to f? Explain your observations.
41
views