Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Series Expansions and Approximations
In calculus, finding series expansions and approximations of functions is vital for simplifying complex expressions and solving integrals. Let's consider some interesting functions and tasks related to these expansions.
#### Functions
**47.** Given \( f(x) = \frac{x^2}{1 + x} \)
**48.** Given \( f(x) = \tan^{-1}(x^2) \)
**49.** Given \( f(x) = \ln(4 - x) \)
**50.** Given \( f(x) = xe^{2x} \)
**51.** Given \( f(x) = \sin(x^4) \)
**52.** Given \( f(x) = 10^x \)
**53.** Given \( f(x) = \frac{1}{\sqrt[4]{16 - x}} \)
**54.** Given \( f(x) = (1 - 3x)^{-5} \)
#### Tasks
**55.** Evaluate the integral \( \int \frac{e^x}{x} dx \) as an infinite series.
**56.** Use series to approximate \( \int_{0}^{1} \sqrt{1 + x^4} \, dx \) correct to two decimal places.
### Explanation of Functions and Tasks
1. **Functions 47 to 54** show various forms in which functions can appear, particularly those involving logarithmic, trigonometric, exponential, and algebraic terms. The goal is to express these functions in terms of series expansions.
2. **Task 55** involves evaluating an integral that may not have a straightforward antiderivative. The function \( \frac{e^x}{x} \) can be expanded into its series form, which can then be integrated term by term.
3. **Task 56** requires approximating a definite integral using series expansion. By expanding \( \sqrt{1 + x^4} \) into a series, we can integrate term-by-term and obtain an approximate value accurate to two decimal places.
These exercises enhance skills in manipulating series and understanding their applications in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18574973-f25e-4ab8-b7d6-6007b5b87fc4%2Fcf794f2a-f6fe-4884-a964-6d0d14ecc576%2Fhfqsq3x_processed.png&w=3840&q=75)
Transcribed Image Text:### Series Expansions and Approximations
In calculus, finding series expansions and approximations of functions is vital for simplifying complex expressions and solving integrals. Let's consider some interesting functions and tasks related to these expansions.
#### Functions
**47.** Given \( f(x) = \frac{x^2}{1 + x} \)
**48.** Given \( f(x) = \tan^{-1}(x^2) \)
**49.** Given \( f(x) = \ln(4 - x) \)
**50.** Given \( f(x) = xe^{2x} \)
**51.** Given \( f(x) = \sin(x^4) \)
**52.** Given \( f(x) = 10^x \)
**53.** Given \( f(x) = \frac{1}{\sqrt[4]{16 - x}} \)
**54.** Given \( f(x) = (1 - 3x)^{-5} \)
#### Tasks
**55.** Evaluate the integral \( \int \frac{e^x}{x} dx \) as an infinite series.
**56.** Use series to approximate \( \int_{0}^{1} \sqrt{1 + x^4} \, dx \) correct to two decimal places.
### Explanation of Functions and Tasks
1. **Functions 47 to 54** show various forms in which functions can appear, particularly those involving logarithmic, trigonometric, exponential, and algebraic terms. The goal is to express these functions in terms of series expansions.
2. **Task 55** involves evaluating an integral that may not have a straightforward antiderivative. The function \( \frac{e^x}{x} \) can be expanded into its series form, which can then be integrated term by term.
3. **Task 56** requires approximating a definite integral using series expansion. By expanding \( \sqrt{1 + x^4} \) into a series, we can integrate term-by-term and obtain an approximate value accurate to two decimal places.
These exercises enhance skills in manipulating series and understanding their applications in calculus.
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