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 7.3.105

7장, 문제 7.3.105

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.

검증된 단계별 안내

Recall the definition of the inverse hyperbolic cosine function: \(\cosh^{-1} t = \ln \left(t + \sqrt{t^2 - 1} \right)\) for \(t \geq 1\).

Substitute \(t = \cosh x\) into the formula to get \(\cosh^{-1}(\cosh x) = \ln \left( \cosh x + \sqrt{\cosh^2 x - 1} \right)\).

Use the identity \(\cosh^2 x - 1 = \sinh^2 x\) to rewrite the expression as \(\ln \left( \cosh x + |\sinh x| \right)\).

Consider the two cases separately: - For \(x \geq 0\), \(\sinh x \geq 0\), so \(|\sinh x| = \sinh x\). - For \(x < 0\), \(\sinh x < 0\), so \(|\sinh x| = -\sinh x\).

Evaluate the expression in each case: - When \(x \geq 0\), \(\ln (\cosh x + \sinh x) = \ln (e^x) = x\). - When \(x < 0\), \(\ln (\cosh x - \sinh x) = \ln (e^{-x}) = -x = |x|\). Thus, \(\cosh^{-1}(\cosh x) = |x|\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Inverse Hyperbolic Cosine Function

The inverse hyperbolic cosine, denoted as cosh⁻¹(t), is the function that returns the value x such that cosh(x) = t for t ≥ 1. It is defined by the formula cosh⁻¹(t) = ln(t + √(t² – 1)), which expresses the inverse in terms of natural logarithms and square roots.

Properties of the Hyperbolic Cosine Function

The hyperbolic cosine function, cosh(x), is an even function, meaning cosh(x) = cosh(–x). It is always greater than or equal to 1 and increases exponentially for large |x|. This symmetry is key to understanding why cosh⁻¹(cosh x) equals the absolute value of x.

Piecewise Analysis Based on Domain

To prove cosh⁻¹(cosh x) = |x|, it is essential to consider two cases: x ≥ 0 and x < 0. Since cosh is even, analyzing these cases separately helps show that the inverse function returns x when x is nonnegative and –x when x is negative, resulting in the absolute value.

관련 실천

교과서 질문

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.

views

교과서 질문

Express sinh⁻¹ x in terms of logarithms.

169

views

교과서 질문

How does the graph of the catenary y = a cosh x/a change as a > 0 increases?

151

views

교과서 질문

22–36. Derivatives Find the derivatives of the following functions.

f(x) = cosh²x

views

교과서 질문

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.

views

교과서 질문

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫ 3^{-2x} dx

views