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 7.3.5

Chapter 7, Problem 7.3.5

Express sinh⁻¹ x in terms of logarithms.

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Recall the definition of the inverse hyperbolic sine function: sinh⁻¹(x) is the value y such that sinh(y) = x.

Use the definition of the hyperbolic sine function: sinh(y) = (e^y - e^(-y)) / 2.

Set sinh(y) equal to x: (e^y - e^(-y)) / 2 = x.

Multiply through by 2 to eliminate the fraction: e^y - e^(-y) = 2x.

Rewrite the equation in terms of e^y: e^y = x + √(x² + 1). Then take the natural logarithm of both sides to express y in terms of logarithms: sinh⁻¹(x) = ln(x + √(x² + 1)).

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Key Concepts

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

Inverse Hyperbolic Functions

Inverse hyperbolic functions, such as sinh⁻¹ x, are the inverses of hyperbolic functions. They allow us to find the value of the original variable when given the output of the hyperbolic function. For example, sinh(x) = y implies that x = sinh⁻¹(y). Understanding these functions is crucial for expressing them in alternative forms, such as logarithmic expressions.

Logarithmic Identities

Logarithmic identities are mathematical properties that relate logarithms to one another and to exponential functions. For instance, the identity for the inverse hyperbolic sine function is sinh⁻¹(x) = ln(x + √(x² + 1)). This identity is essential for converting hyperbolic functions into logarithmic form, which is often more useful in calculus and analysis.

Domain and Range of Functions

The domain and range of functions describe the set of possible input values (domain) and the resulting output values (range). For sinh⁻¹ x, the domain is all real numbers, while the range is also all real numbers. Understanding these properties is important when working with inverse functions, as they help ensure that the transformations maintain valid outputs.

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Textbook Question

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

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.

Textbook Question

88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.

lim x → 0⁺ (tanh x)ˣ

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

Textbook Question

13–14. Absolute and relative growth rates Two functions f and g are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant.

f(t) = 100 + 10.5t, g(t) = 100e^(t/10)