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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 7.3.79f
Capitolo 7, Problema 7.3.79f

Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.
f. sinh (2 ln 3)

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1
Recall the definition of the hyperbolic sine function: \(\sinh(x) = \frac{e^{x} - e^{-x}}{2}\).
Substitute \(x = 2 \ln 3\) into the definition: \(\sinh(2 \ln 3) = \frac{e^{2 \ln 3} - e^{-2 \ln 3}}{2}\).
Use the property of exponents and logarithms: \(e^{a \ln b} = b^{a}\). So, \(e^{2 \ln 3} = 3^{2}\) and \(e^{-2 \ln 3} = 3^{-2}\).
Rewrite the expression using these simplifications: \(\sinh(2 \ln 3) = \frac{3^{2} - 3^{-2}}{2}\).
Simplify the powers: \(3^{2} = 9\) and \(3^{-2} = \frac{1}{9}\). So, \(\sinh(2 \ln 3) = \frac{9 - \frac{1}{9}}{2}\).

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Definition of Hyperbolic Sine Function (sinh)

The hyperbolic sine function, sinh(x), is defined as (e^x - e^(-x)) / 2. It is analogous to the sine function but based on exponential functions, which allows simplification when the input is expressed in terms of logarithms or exponentials.
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Graph of Sine and Cosine Function

Properties of Logarithms and Exponentials

The natural logarithm ln(x) and the exponential function e^x are inverse functions. Using properties like e^(ln a) = a helps simplify expressions involving compositions of exponentials and logarithms, which is essential for evaluating sinh(2 ln 3).
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Change of Base Property

Double Angle Formula for Hyperbolic Sine

The double angle formula for sinh states that sinh(2x) = 2 sinh(x) cosh(x). This identity allows breaking down sinh(2 ln 3) into simpler parts involving sinh(ln 3) and cosh(ln 3), which can then be evaluated using exponential definitions.
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Slope of Polar Curves Example 1
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