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

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

Briggs 3rd Edition

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

문제 7.3.5

7장, 문제 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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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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