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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.R.13
Capitolo 7, Problema 7.R.13

10–19. Derivatives Find the derivatives of the following functions.


f(t) = cosh t sinh t

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1
Recall the product rule for derivatives: if you have a function \(f(t) = u(t) v(t)\), then its derivative is \(f'(t) = u'(t) v(t) + u(t) v'(t)\).
Identify the two functions in the product: \(u(t) = \cosh t\) and \(v(t) = \sinh t\).
Find the derivatives of each function separately: \(\frac{d}{dt} \cosh t = \sinh t\) and \(\frac{d}{dt} \sinh t = \cosh t\).
Apply the product rule: \(f'(t) = (\sinh t)(\sinh t) + (\cosh t)(\cosh t)\).
Simplify the expression if possible, using hyperbolic identities such as \(\cosh^2 t - \sinh^2 t = 1\).

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Hyperbolic Functions

Hyperbolic functions, such as sinh(t) and cosh(t), are analogs of trigonometric functions but based on hyperbolas. They have unique properties and identities, like cosh²(t) - sinh²(t) = 1, which are useful in differentiation and integration.
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Asymptotes of Hyperbolas

Product Rule for Differentiation

The product rule states that the derivative of a product of two functions u(t) and v(t) is u'(t)v(t) + u(t)v'(t). This rule is essential when differentiating functions like f(t) = cosh(t) * sinh(t), where both factors depend on t.
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The Product Rule

Derivatives of Hyperbolic Functions

The derivatives of basic hyperbolic functions are: d/dt[sinh(t)] = cosh(t) and d/dt[cosh(t)] = sinh(t). Knowing these derivatives allows you to apply the product rule correctly to find the derivative of functions involving hyperbolic terms.
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Asymptotes of Hyperbolas
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Domanda del libro di testo

Surface area of a catenoid When the catenary y = a cosh x/a is revolved about the x-axis, it sweeps out a surface of revolution called a catenoid. Find the area of the surface generated when y = cosh x on [–ln 2, ln 2] is rotated about the x-axis.

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63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.


limₕ→₀ (1 + 3h)^{2/h}

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27–28. Curve sketching Use the graphing techniques of Section 4.4 to graph the following functions on their domains. Identify local extreme points, inflection points, concavity, and end behavior. Use a graphing utility only to check your work.

f(x) = ln x – ln² x

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10–19. Derivatives Find the derivatives of the following functions.

f(x) = tanh⁻¹(cos x)

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. ln xy = (ln x)(ln y)

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Domanda del libro di testo

10–19. Derivatives Find the derivatives of the following functions.

f(x) = (sinh x) / (1 + sinh x)

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