Textbook Question
Critical points Find the critical points of the function ƒ(x) = sinh² x cosh x.
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
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
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.1.40
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.1.40Chapter 7, Problem 7.1.40
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₋₂² (e^{z/2}) / (e^{z/2} + 1) dz
Verified step by step guidance1
First, observe the integral \( \int_{-2}^{2} \frac{e^{z/2}}{e^{z/2} + 1} \, dz \). Notice the integrand involves an exponential function in the numerator and denominator.
To simplify the integral, consider the substitution \( u = e^{z/2} \). Then, \( du = \frac{1}{2} e^{z/2} dz = \frac{1}{2} u \, dz \), which implies \( dz = \frac{2}{u} du \). However, since the limits are in terms of \( z \), it might be easier to explore symmetry or another approach first.
Check if the integrand has any symmetry properties by evaluating \( f(-z) \) and comparing it to \( f(z) \). This can help determine if the integral can be simplified by splitting or combining parts.
Rewrite the integrand as \( \frac{e^{z/2}}{e^{z/2} + 1} = 1 - \frac{1}{e^{z/2} + 1} \). This decomposition can make the integral easier to handle by splitting it into two integrals.
Express the original integral as \( \int_{-2}^{2} 1 \, dz - \int_{-2}^{2} \frac{1}{e^{z/2} + 1} \, dz \). Then, evaluate each integral separately, using substitution or symmetry as appropriate.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals and Symmetry
Definite integrals calculate the net area under a curve between two limits. When the integrand exhibits symmetry (even, odd, or other), this property can simplify evaluation by reducing the interval or transforming the integral.
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Definition of the Definite Integral
Substitution Method
The substitution method involves changing variables to simplify an integral. By letting a part of the integrand equal a new variable, the integral can often be rewritten in a more manageable form, facilitating easier integration.
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Exponential Functions and Their Properties
Exponential functions like e^{x} have unique properties, such as their derivatives and integrals being proportional to themselves. Understanding how to manipulate expressions involving exponentials is crucial for integrating functions with exponential terms.
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Related Practice
Textbook Question
Area of region Find the area of the region bounded by y = sech x, x = 1, and the unit circle (see figure).
Textbook Question
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫₀³ (2x - 1) / (x + 1) dx
Textbook Question
Solid of revolution Compute the volume of the solid of revolution that results when the region in Exercise 85 is revolved about the x-axis.
Textbook Question
After the introduction of foxes on an island, the number of rabbits on the island decreases by 4.5% per month. If y(t) equals the number of rabbits on the island t months after foxes were introduced, find the rate constant k for the exponential decay function y(t) = y₀eᵏᵗ.
Textbook Question
15–20. Designing exponential growth functions Complete the following steps for the given situation.
a. Find the rate constant k and use it to devise an exponential growth function that fits the given data.
b. Answer the accompanying question.
Rising costs Between 2010 and 2016, the average rate of inflation was about 1.6%/yr. If a cart of groceries cost \$100 in 2010, what will it cost in 2025, assuming the rate of inflation remains constant at 1.6%?