Textbook Question
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
b. Give the approximate coordinates of the absolute maximum and minimum values of ƒ (if they exist).
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.R.81
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→1 ( x- 1)^sinπx
Verified step by step guidance1
First, recognize that the limit is in an indeterminate form of 0^0 as x approaches 1. This suggests that we might need to use a logarithmic transformation to simplify the expression.
Take the natural logarithm of the expression: ln((x - 1)^sin(πx)) = sin(πx) * ln(x - 1). This allows us to transform the power into a product, which is easier to handle.
Now, evaluate the limit of the transformed expression: lim_(x→1) sin(πx) * ln(x - 1). Notice that as x approaches 1, sin(πx) approaches 0 and ln(x - 1) approaches -∞, creating an indeterminate form of 0 * -∞.
To resolve this, rewrite the expression as a quotient: lim_(x→1) (sin(πx)) / (1/ln(x - 1)). This is now in the form 0/0, which is suitable for l'Hôpital's Rule.
Apply l'Hôpital's Rule by differentiating the numerator and the denominator: differentiate sin(πx) with respect to x to get πcos(πx), and differentiate 1/ln(x - 1) with respect to x using the chain rule. Evaluate the new limit as x approaches 1.
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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 concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or where the function is not explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
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One-Sided Limits
L'Hôpital's Rule
L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, and then re-evaluating the limit. This technique simplifies the process of finding limits in complex scenarios.
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5:50Power Rules
Exponential Functions and Continuity
Exponential functions, such as those involving sin or cos, can exhibit unique behaviors near certain points. In the given limit, (x - 1) approaches 0 as x approaches 1, while sin(πx) oscillates between -1 and 1. Understanding the continuity of these functions and their limits is essential for accurately evaluating the limit, especially when combined with the properties of exponential growth or decay.
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Related Practice
Textbook Question
45–46. Linear approximation
a. Find the linear approximation to f at the given point a.
b. Use your answer from part (a) to estimate the given function value. Does your approximation underestimate or overestimate the exact function value?
ƒ(x) = x²⸍³ ; a =27; ƒ(29)
Textbook Question
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x sin⁻¹ x on [-1, 1]
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Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ ln ((x +1) / (x-1))
Textbook Question
90–103. Indefinite integrals Determine the following indefinite integrals.
∫ (x² / (x⁴ + x²)) dx
Textbook Question
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x⁴ - 50x² on [-1, 5]
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