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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.79g
Capitolo 7, Problema 7.3.79g

Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.
g. cosh² 1

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1
Recall the definition of the hyperbolic cosine function: \(\cosh x = \frac{e^{x} + e^{-x}}{2}\).
Express \(\cosh^2 1\) as \(\left( \cosh 1 \right)^2 = \left( \frac{e^{1} + e^{-1}}{2} \right)^2\).
Square the expression inside the parentheses: \(\left( \frac{e^{1} + e^{-1}}{2} \right)^2 = \frac{(e^{1} + e^{-1})^2}{4}\).
Expand the numerator using the formula \((a + b)^2 = a^2 + 2ab + b^2\): \((e^{1})^2 + 2 e^{1} e^{-1} + (e^{-1})^2 = e^{2} + 2 + e^{-2}\).
Combine the results to write \(\cosh^2 1 = \frac{e^{2} + 2 + e^{-2}}{4}\), which is the simplified exact expression.

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Definition of Hyperbolic Cosine (cosh)

The hyperbolic cosine function, cosh(x), is defined as (e^x + e^(-x)) / 2. It is an even function and is analogous to the cosine function in trigonometry but based on exponential functions. Understanding this definition allows direct evaluation of cosh at any real number.
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Definition of the Definite Integral

Squaring Hyperbolic Functions

Squaring cosh(x) means computing (cosh(x))^2, which can be expressed using exponential terms or simplified using hyperbolic identities. Recognizing how to handle powers of hyperbolic functions is essential for simplification and evaluation.
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Asymptotes of Hyperbolas

Hyperbolic Identity: cosh²(x) - sinh²(x) = 1

This fundamental identity relates cosh and sinh functions, similar to the Pythagorean identity in trigonometry. It can be used to rewrite or simplify expressions involving cosh²(x), especially when combined with expressions involving sinh(x).
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Integrals of Natural Exponential Functions (e^x)
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