Textbook Question
{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.
∫₀⁰ᐧ³⁵ tan ⁻¹x dx
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 11.4.7
Limits Evaluate the following limits using Taylor series.
lim ₓ→₀ (eˣ − 1)/x
Verified step by step guidance1
Recall that the Taylor series expansion of the exponential function \(e^x\) around \(x=0\) is given by: \[e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\]
Substitute the Taylor series expansion of \(e^x\) into the given limit expression: \[\frac{e^x - 1}{x} = \frac{\left(1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\right) - 1}{x}\]
Simplify the numerator by canceling the 1's: \[\frac{x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots}{x}\]
Factor out \(x\) from the numerator: \[\frac{x \left(1 + \frac{x}{2!} + \frac{x^2}{3!} + \cdots \right)}{x}\]
Cancel the \(x\) terms and then evaluate the limit as \(x\) approaches 0 by substituting \(x=0\) into the remaining series inside the parentheses: \[\lim_{x \to 0} \left(1 + \frac{x}{2!} + \frac{x^2}{3!} + \cdots \right)\]
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits and Continuity
Limits describe the behavior of a function as the input approaches a particular value. Understanding limits is essential for analyzing functions near points where direct substitution may be undefined or indeterminate, such as 0/0 forms.
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Intro to Continuity
Taylor Series Expansion
A Taylor series represents a function as an infinite sum of terms calculated from its derivatives at a single point. Using Taylor series near zero (Maclaurin series) helps approximate functions and evaluate limits by simplifying complex expressions.
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Indeterminate Forms and L'Hôpital's Rule
When evaluating limits, expressions like 0/0 are indeterminate forms that require special techniques. Taylor series or L'Hôpital's Rule can resolve these by transforming the expression into a determinate form to find the limit.
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Related Practice
Textbook Question
A limit by Taylor series Use Taylor series to evaluate lim ₓ→₀ ((sin x)/x)¹/ˣ²
1
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Textbook Question
Taylor polynomials Find the nth-order Taylor polynomial for the following functions centered at the given point a.
ƒ(x) = cosh x, n = 3, a = ln 2
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Textbook Question
{Use of Tech} Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10⁻³ ? (The answer depends on your choice of a center.)
ln 0.85
Textbook Question
{Use of Tech} Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.
f(x) = ∜x with a=16; approximate ∜13.
Textbook Question
Radius and interval of convergence Determine the radius and interval of convergence of the following power series.
∑ₖ₌₀∞ (-x/10)²ᵏ