SOLVED:Find The Numerical Value Of Each Expression ... - Numerade

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Question

Find the numerical value of each expression. $${ (a) }\sinh (\ln 2) \quad \text { (b) } \sinh 2$$ Find the numerical value of each expression.$${ (a) }\sinh (\ln 2) \quad \text { (b) } \sinh 2$$ Calculus Early Transcendentals James Stewart 6th Edition Chapter 3, Problem 3 View All Chapters

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Step 1: We know that the definition of hyperbolic sine (sinh) is $\sinh(x) = \frac{e^x - e^{-x}}{2}$. Show more…

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Key Concepts

Hyperbolic Sine Function The hyperbolic sine function, sinh(x), is defined using exponential functions as (e^x - e^(-x)) / 2. It behaves similarly to the regular sine function in many ways, although it is not periodic and arises in contexts such as the description of hyperbolic geometry, the solution of certain differential equations, and various physical applications. Exponential Function The exponential function, expressed as e^x where e is a mathematical constant approximately equal to 2.71828, is a fundamental function in mathematics due to its unique growth properties. It forms the basis of defining hyperbolic functions and appears widely in calculus, differential equations, and complex analysis. Natural Logarithm The natural logarithm, denoted ln(x), is the inverse function of the exponential function. It is used to solve equations involving exponentials and to transform multiplicative relationships into additive ones. Understanding ln(x) is key to evaluating expressions where logarithms are inputs to other functions, such as in compositions with hyperbolic functions.

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Transcript

00:01 In this problem, we need to evaluate the numerical values of cinch of the natural log of 2 and the cinch of 2. 00:19 Now, the definition of cinch is that cinch of any number x equals e raise to the power x minus e raise to the power negative x divided by 2. 00:30 And this gives us cinch of ln2... Need help? Use Ace Ace is your personal tutor. It breaks down any question with clear steps so you can learn. Start Using Ace Ace is your personal tutor for learning Step-by-step explanations Instant summaries Summarize YouTube videos Understand textbook images or PDFs Study tools like quizzes and flashcards Listen to your notes as a podcast

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