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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 7.3.13
Chapter 7, Problem 7.3.13

11–15. Identities Prove each identity using the definitions of the hyperbolic functions.


cosh 2x = cosh²x + sinh²x (Hint: Begin with the right side of the equation.)

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Start with the right-hand side of the equation: cosh²x + sinh²x.
Recall the definitions of the hyperbolic functions: cosh(x) = (e^x + e^(-x))/2 and sinh(x) = (e^x - e^(-x))/2.
Square both definitions: cosh²x = [(e^x + e^(-x))/2]² and sinh²x = [(e^x - e^(-x))/2]².
Expand the squares: cosh²x = (e^(2x) + 2 + e^(-2x))/4 and sinh²x = (e^(2x) - 2 + e^(-2x))/4.
Add the two results: cosh²x + sinh²x = [(e^(2x) + 2 + e^(-2x)) + (e^(2x) - 2 + e^(-2x))]/4 = (2e^(2x) + 2e^(-2x))/4 = (e^(2x) + e^(-2x))/2, which is the definition of cosh(2x).

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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(x) and cosh(x), are analogs of the trigonometric functions but are based on hyperbolas instead of circles. They are defined as sinh(x) = (e^x - e^(-x))/2 and cosh(x) = (e^x + e^(-x))/2. Understanding these definitions is crucial for manipulating and proving identities involving hyperbolic functions.
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Hyperbolic Identities

Hyperbolic identities are equations that hold true for hyperbolic functions, similar to trigonometric identities. One fundamental identity is cosh²(x) - sinh²(x) = 1. Recognizing and applying these identities is essential for proving new identities, such as the one in the question.
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Proof Techniques

Proof techniques in calculus often involve algebraic manipulation, substitution, and the application of known identities. In this case, starting with the right side of the equation and transforming it to match the left side using hyperbolic definitions and identities is a common strategy. Mastery of these techniques is vital for successfully proving mathematical statements.
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Related Practice
Textbook Question

7–28. Derivatives Evaluate the following derivatives.


d/dy (y^{sin y})

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.


d. 2ˣ = 2² ˡⁿ ˣ

Textbook Question

7–28. Derivatives Evaluate the following derivatives.


d/dt (t^{1/t})

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

Behavior at the origin Using calculus and accurate sketches, explain how the graphs of f(x) = xᵖ ln x differ as x → 0⁺ for p = 1/2, 1, and 2.