Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.3.105

Chapter 7, Problem 7.3.105

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.

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Recall the definition of the inverse hyperbolic cosine function: \(\cosh^{-1} t = \ln \left(t + \sqrt{t^2 - 1} \right)\) for \(t \geq 1\).

Substitute \(t = \cosh x\) into the formula to get \(\cosh^{-1}(\cosh x) = \ln \left( \cosh x + \sqrt{\cosh^2 x - 1} \right)\).

Use the identity \(\cosh^2 x - 1 = \sinh^2 x\) to rewrite the expression as \(\ln \left( \cosh x + |\sinh x| \right)\).

Consider the two cases separately: - For \(x \geq 0\), \(\sinh x \geq 0\), so \(|\sinh x| = \sinh x\). - For \(x < 0\), \(\sinh x < 0\), so \(|\sinh x| = -\sinh x\).

Evaluate the expression in each case: - When \(x \geq 0\), \(\ln (\cosh x + \sinh x) = \ln (e^x) = x\). - When \(x < 0\), \(\ln (\cosh x - \sinh x) = \ln (e^{-x}) = -x = |x|\). Thus, \(\cosh^{-1}(\cosh x) = |x|\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Hyperbolic Cosine Function

The inverse hyperbolic cosine, denoted as cosh⁻¹(t), is the function that returns the value x such that cosh(x) = t for t ≥ 1. It is defined by the formula cosh⁻¹(t) = ln(t + √(t² – 1)), which expresses the inverse in terms of natural logarithms and square roots.

Properties of the Hyperbolic Cosine Function

The hyperbolic cosine function, cosh(x), is an even function, meaning cosh(x) = cosh(–x). It is always greater than or equal to 1 and increases exponentially for large |x|. This symmetry is key to understanding why cosh⁻¹(cosh x) equals the absolute value of x.

Piecewise Analysis Based on Domain

To prove cosh⁻¹(cosh x) = |x|, it is essential to consider two cases: x ≥ 0 and x < 0. Since cosh is even, analyzing these cases separately helps show that the inverse function returns x when x is nonnegative and –x when x is negative, resulting in the absolute value.

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Piecewise Functions

Related Practice

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Population of Texas Texas was the third fastest growing state in the United States in 2016. Texas grew from 25.1 million in 2010 to 26.47 million in 2016. Use an exponential growth model to predict the population of Texas in 2025.

Textbook Question

Express sinh⁻¹ x in terms of logarithms.

Textbook Question

Evaluate the following derivatives.

d/dx ((1/x)ˣ)

Textbook Question

22–36. Derivatives Find the derivatives of the following functions.

f(x) = cosh²x

Textbook Question

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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