Textbook Question
Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.
f(x) = x²/₃ (x²-4)
Ch. 4 - Applications of the Derivative
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 4, Problem 4.7.83
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→0 (x + cos x)¹/ˣ
Verified step by step guidance1
First, recognize that the limit \( \lim_{x \to 0} (x + \cos x)^{1/x} \) is of the indeterminate form \( 1^{\infty} \). This suggests that we can use the natural logarithm to simplify the expression.
Take the natural logarithm of the expression: \( \ln L = \lim_{x \to 0} \frac{\ln(x + \cos x)}{x} \). This transforms the problem into a form where l'Hôpital's Rule can be applied.
Check the new limit \( \lim_{x \to 0} \frac{\ln(x + \cos x)}{x} \). As \( x \to 0 \), both the numerator and the denominator approach 0, creating a \( \frac{0}{0} \) indeterminate form, which is suitable for l'Hôpital's Rule.
Apply l'Hôpital's Rule by differentiating the numerator and the denominator: Differentiate \( \ln(x + \cos x) \) with respect to \( x \) to get \( \frac{1}{x + \cos x} (1 - \sin x) \), and differentiate \( x \) to get 1.
Evaluate the new limit: \( \lim_{x \to 0} \frac{1 - \sin x}{x + \cos x} \). After finding this limit, exponentiate the result to find the original limit \( L \), since \( L = e^{\ln L} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this problem, evaluating the limit as x approaches 0 is crucial for determining the behavior of the function (x + cos x)^(1/x).
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One-Sided Limits
L'Hôpital's Rule
L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in simplifying complex limits, like the one presented in the question.
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5:50Power Rules
Exponential Functions
Exponential functions, such as e^x, are functions where a constant base is raised to a variable exponent. In the context of limits, they often arise when evaluating expressions of the form (1 + f(x))^(g(x)) as x approaches a limit. Understanding how to manipulate these functions is key to solving the limit problem, especially when applying logarithmic transformations to simplify the expression.
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Related Practice
Textbook Question
{Use of Tech} Critical points and extreme values
a. Find the critical points of the following functions on the given interval. Use a root finder, if necessary.
b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.
c. Find the absolute maximum and minimum values on the given interval, if they exist
h(x) (5-x)/(x² + 2x - 3) on [-10,10]
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_v→3 (v-1-√(v²-5)) / (v-3)
Textbook Question
Finding antiderivatives. Find all the antiderivatives of the following functions. Check your work by taking derivatives.
p(x) = 3 sec² x
Textbook Question
Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.
f(x) = 8x³ + sin x; F(0) = 2
Textbook Question
{Use of Tech} Absolute maxima and minima
a. Find the critical points of f on the given interval.
b. Determine the absolute extreme values of f on the given interval.
c. Use a graphing utility to confirm your conclusions.
f(x) = 2ᶻ sin x on [-2,6]