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).
ƒ(x) = sin 2x + 3 on [-π , π]
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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.73
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ ln ((x +1) / (x-1))
Verified step by step guidance1
Identify the form of the limit as x approaches infinity: lim_(x→∞) ln((x + 1) / (x - 1)). This is an indeterminate form of type ∞/∞.
Apply l'Hôpital's Rule, which is used to evaluate limits of indeterminate forms. First, differentiate the numerator and the denominator of the argument of the logarithm separately.
Differentiate the numerator (x + 1) to get 1, and the denominator (x - 1) to get 1. The limit now becomes lim_(x→∞) ln(1).
Since ln(1) is a constant, the limit simplifies to 0.
Conclude that the limit of the original expression as x approaches infinity is 0.
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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 at points where it may not be explicitly defined, such as at infinity or discontinuities. Evaluating limits is crucial for determining the continuity and differentiability of functions.
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One-Sided Limits
Natural Logarithm
The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a key function in calculus, particularly in growth and decay problems, and is often used in limits involving exponential functions. Understanding its properties, such as its domain and range, is essential for evaluating limits that involve logarithmic expressions.
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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 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. This rule simplifies the process of finding limits, especially when dealing with complex functions.
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Guided course
5:50Power Rules
Related Practice
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).
ƒ(x) = x³ - 6x² on [-1, 5]
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Textbook Question
82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.
2ˣ and 4ˣ⸍²
275
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Textbook Question
24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ
267
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Textbook Question
Maximum printable area A rectangular page in a text (with width x and length y) has an area of 98 in² , top and bottom margins set at 1 in, and left and right margins set at 1/2 in. The printable area of the page is the rectangle that lies within the margins. What are the dimensions of the page that maximize the printable area?
227
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Textbook Question
24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.
ƒ(x) = ln( x² + 3) / (x -1)
172
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