Skip to main content
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.3.103
Capitolo 7, Problema 7.3.103

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

Guida verificata passo dopo passo
1
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.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
6m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
Percorso guidato
5:50
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.
Video consigliato:
Percorso guidato
5:50
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.
Video consigliato:
6:13
Exponential Functions
Pratica correlata
Domanda del libro di testo

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


∫ₑᵉ^³ dx / (x ln x ln²(ln x))

49
views
Domanda del libro di testo

7–28. Derivatives Evaluate the following derivatives.


d/dx (e^{-10x²})

59
views
Domanda del libro di testo

7–28. Derivatives Evaluate the following derivatives.


d/dy (y^{sin y})

36
views
Domanda del libro di testo

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.)

122
views
Domanda del libro di testo

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)

49
views
Domanda del libro di testo

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 → ∞?

82
views