Textbook Question
Inverse identity Show that cosh⁻¹(cosh x) = |x| by using the formula cosh⁻¹ t = ln (t + √(t² – 1)) and considering the cases x ≥ 0 and x < 0.
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 7.3.23
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.23Chapter 7, Problem 7.3.23
22–36. Derivatives Find the derivatives of the following functions.
f(x) = cosh²x
Verified step by step guidance1
Step 1: Recognize that the function f(x) = cosh²(x) is a composite function. It involves the square of the hyperbolic cosine function, cosh(x). To differentiate it, we will use the chain rule.
Step 2: Recall the chain rule formula: If a function is composed as g(h(x)), then its derivative is g'(h(x)) * h'(x). Here, g(x) = x² and h(x) = cosh(x).
Step 3: Differentiate the outer function g(x) = x². The derivative of x² is 2x. Applying this to g(h(x)), we get 2 * cosh(x).
Step 4: Differentiate the inner function h(x) = cosh(x). Recall that the derivative of cosh(x) is sinh(x).
Step 5: Combine the results from Step 3 and Step 4 using the chain rule. The derivative of f(x) = cosh²(x) is 2 * cosh(x) * sinh(x).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivatives
A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that allows us to determine how a function behaves at any given point. The derivative can be interpreted as the slope of the tangent line to the graph of the function at a specific point.
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Derivatives
Hyperbolic Functions
Hyperbolic functions, such as cosh(x), are analogs of trigonometric functions but are based on hyperbolas instead of circles. The function cosh(x) is defined as (e^x + e^(-x))/2 and is used frequently in calculus, particularly in problems involving exponential growth and decay. Understanding their properties is essential for differentiating functions that involve them.
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Chain Rule
The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative can be found by multiplying the derivative of the outer function f with the derivative of the inner function g. This rule is particularly useful when dealing with functions like cosh²(x), where one function is nested within another.
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Related Practice
Textbook Question
Evaluate the following derivatives.
d/dx ((1/x)ˣ)
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.
Population The population of Clark County, Nevada, was about 2.115 million in 2015. Assuming an annual growth rate of 1.5%/yr, what will the county population be in 2025?
Textbook Question
How does the graph of the catenary y = a cosh x/a change as a > 0 increases?
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Textbook Question
39–40. LED lighting LED (light-emitting diode) bulbs are rapidly decreasing in cost, and they are more energy-efficient than standard incandescent light bulbs and CFL (compact fluorescent light) bulbs. By some estimates, LED bulbs last more than 40 times longer than incandescent bulbs and more than 8 times longer than CFL bulbs. Haitz’s law, which is explored in the following two exercises, predicts that over time, LED bulbs will exponentially increase in efficiency and exponentially decrease in cost.
Haitz’s law predicts that the cost per lumen of an LED bulb decreases by a factor of 10 every 10 years. This means that 10 years from now, the cost of an LED bulb will be 1/10 of its current cost. Predict the cost of a particular LED bulb in 2021 if it costs 4 dollars in 2018.
Textbook Question
29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.
∫ 3^{-2x} dx