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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.103
Chapter 7, Problem 7.3.103

101–104. Proving identities Prove the following identities.
cosh (x + y) = cosh x cosh y + sinh x sinh y

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Recall the definitions of hyperbolic cosine and hyperbolic sine: \(\cosh z = \frac{e^{z} + e^{-z}}{2}\) and \(\sinh z = \frac{e^{z} - e^{-z}}{2}\) for any variable \(z\).
Express \(\cosh(x + y)\) using its definition: \(\cosh(x + y) = \frac{e^{x+y} + e^{-(x+y)}}{2}\).
Rewrite the right-hand side of the identity using the definitions: \(\cosh x \cosh y + \sinh x \sinh y = \left(\frac{e^{x} + e^{-x}}{2}\right) \left(\frac{e^{y} + e^{-y}}{2}\right) + \left(\frac{e^{x} - e^{-x}}{2}\right) \left(\frac{e^{y} - e^{-y}}{2}\right)\).
Expand both products on the right-hand side carefully, combining like terms and simplifying the expression.
Show that after simplification, the right-hand side equals \(\frac{e^{x+y} + e^{-(x+y)}}{2}\), which matches the left-hand side, thus proving the identity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hyperbolic Functions

Hyperbolic functions, such as sinh and cosh, are analogs of trigonometric functions but based on hyperbolas instead of circles. They are defined using exponential functions: cosh x = (e^x + e^{-x})/2 and sinh x = (e^x - e^{-x})/2. Understanding their definitions is essential for manipulating and proving identities involving them.
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Asymptotes of Hyperbolas

Addition Formulas for Hyperbolic Functions

Addition formulas express the value of hyperbolic functions at sums of arguments in terms of functions at individual arguments. For example, cosh(x + y) can be expanded using the definitions of cosh and sinh. These formulas are key tools for proving identities and simplifying expressions involving sums.
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Asymptotes of Hyperbolas

Algebraic Manipulation of Exponentials

Proving hyperbolic identities often requires rewriting functions in terms of exponentials and then applying algebraic operations like expansion, grouping, and factoring. Mastery of these algebraic techniques allows one to transform complex expressions into simpler, recognizable forms to verify identities.
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Exponential Functions
Related Practice
Textbook Question

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


∫ₑᵉ^³ dx / (x ln x ln²(ln x))

Textbook Question

7–28. Derivatives Evaluate the following derivatives.


d/dx (e^{-10x²})

Textbook Question

7–28. Derivatives Evaluate the following derivatives.


d/dy (y^{sin y})

Textbook Question

Harmonic sum In Chapter 10, we will encounter the harmonic sum 1 + 1/2 + 1/3 + ⋯ + 1/n. Use a left Riemann sum to approximate ∫[1 to n+1] (dx/x) (with unit spacing between the grid points) to show that 1 + 1/2 + 1/3 + ⋯ + 1/n > ln(n + 1). Use this fact to conclude that lim (n → ∞) (1 + 1/2 + 1/3 + ⋯ + 1/n) does not exist.

Textbook Question

63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.

∫₁ᵉ^² dx/x√(ln²x + 1)

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Textbook Question

Average value What is the average value of f(x) = 1/x on the interval [1, p] for p > 1? What is the average value of f as p → ∞?