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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.78a
Chapter 7, Problem 7.3.78a

Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.
a. coth 4

Verified step by step guidance
1
Recall the definition of the hyperbolic cotangent function: \(\text{coth}(x) = \frac{\cosh(x)}{\sinh(x)}\).
Express \(\cosh(x)\) and \(\sinh(x)\) in terms of exponential functions: \(\cosh(x) = \frac{e^{x} + e^{-x}}{2}\) and \(\sinh(x) = \frac{e^{x} - e^{-x}}{2}\).
Substitute \(x = 4\) into these expressions to find \(\cosh(4)\) and \(\sinh(4)\).
Calculate the values of \(\cosh(4)\) and \(\sinh(4)\) using a calculator or computational tool.
Divide \(\cosh(4)\) by \(\sinh(4)\) to find \(\text{coth}(4)\) and round the result to four decimal places.

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Key Concepts

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

Hyperbolic Functions

Hyperbolic functions, such as sinh, cosh, and coth, are analogs of trigonometric functions but based on hyperbolas. The function coth(x) is defined as cosh(x) divided by sinh(x), and it is important to understand their definitions to evaluate expressions correctly.
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Domain and Existence of Hyperbolic Functions

The domain of hyperbolic functions like coth(x) excludes values where the denominator is zero. Since sinh(x) = 0 at x = 0, coth(x) is undefined there. For x = 4, sinh(4) ≠ 0, so coth(4) exists and can be evaluated.
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Using Calculators for Hyperbolic Functions

Modern calculators often have built-in hyperbolic function keys (sinh, cosh, tanh). To find coth(x), compute cosh(x) and sinh(x) separately and divide. Reporting answers to four decimal places requires rounding the calculator output accordingly.
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Related Practice
Textbook Question

61–62. Points of intersection and area

a. Sketch the graphs of the functions f and g and find the x-coordinate of the points at which they intersect.


f(x) = sech x, g(x) = tanh x; the region bounded by the graphs of f, g, and the y-axis

Textbook Question

Visual approximation


a. Use a graphing utility to sketch the graph of y = coth x and then explain why ∫₅¹⁰ coth x dx ≈ 5.

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Textbook Question

Zero net area Consider the function f(x) = (1 − x)/x

a. Are there numbers 0 < a < 1 such that ∫₁₋ₐ¹⁺ᵃ f(x) dx = 0?

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Textbook Question

ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.

a. Make a sketch of the function f(x) = 1/x on the interval [1, 2]. Explain why the area of the region bounded by y = f(x) and the x-axis on [1, 2] is ln 2.

Textbook Question

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


a. Based on these figures, find an exponential growth function for the power (rate of electricity use) for the city.

Textbook Question

Projection sensitivity

According to the 2014 national population projections published by the U.S. Census Bureau, the U.S. population is projected to be 334.4 million in 2020 with an estimated growth rate of 0.79%/yr.

a. Based on these figures, find the doubling time and the projected population in 2050. Assume the growth rate remains constant.