SOLVED:Find The Numerical Value Of Each Expression. (a) \sinh 4 (b ...

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Find the numerical value of each expression. (a) $ \sinh 4 $ (b) $ \sinh (\ln 4) $ Find the numerical value of each expression.(a) $ \sinh 4 $ (b) $ \sinh (\ln 4) $ Calculus: Early Transcendentals James Stewart 8th Edition Chapter 3, Problem 4 View All Chapters

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Step 1: We know that the hyperbolic sine function is defined as $\sinh x = \frac{e^x - e^{-x}}{2}$. Show more…

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Key Concepts

Hyperbolic Sine Function The hyperbolic sine, denoted as sinh(x), is a fundamental hyperbolic function defined by the formula sinh(x) = (e^x - e^(-x))/2. It parallels the properties of the trigonometric sine function but is based on exponential functions. This function plays a central role in areas such as calculus, differential equations, and various physical models, often emerging in problems involving hyperbolic geometry and complex numbers. Exponential Functions Exponential functions involve expressions where a constant base (commonly e, Euler's number) is raised to a variable exponent. They have unique properties such as rapid growth and a strictly positive range. Understanding exponential functions is crucial when dealing with hyperbolic functions, as these functions are defined in terms of exponential expressions, which are fundamental in analyzing growth and decay processes and solving differential equations. Logarithmic Functions and Their Relationship with Exponentials Logarithms, especially the natural logarithm ln, are the inverse functions of the exponential functions. This relationship is key when simplifying expressions that involve both logarithms and exponentials, such as when a hyperbolic function is applied to a logarithmic value. Recognizing this inverse relationship enables the transformation and simplification of complex expressions in a straightforward manner. Algebraic Manipulation of Composite Functions Algebraic manipulation refers to the process of rewriting and simplifying expressions using known identities and properties. In the context of composite functions like sinh(ln(a)), this involves substituting the definitions of both the hyperbolic sine and the logarithmic function and applying the properties of exponents. This skill is essential for evaluating expressions efficiently and is broadly applicable across various mathematical disciplines. Key Concept Premium Feature Explore the core concept behind this problem. View Video View Video Key Concept Premium Feature Explore the core concept behind this problem. Your browser does not support the video tag. Close Try it Now

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Transcript

00:01 Okay, we know we have e to the fourth minus e to the negative fourth over two, which is e to the eighth minus one over two e to the fourth, which is 27 .29. 00:19 Moving on to part b, we have e to the natural log of four minus e to the negative natural log of four over two... 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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