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.R.31
Capitolo 7, Problema 7.R.31

Linear approximation Find the linear approximation to ƒ(x) = cosh x at a = ln 3 and then use it to approximate the value of cosh 1.

Guida verificata passo dopo passo
1
Recall that the linear approximation of a function ƒ(x) at a point a is given by the formula: \[L(x) = ƒ(a) + ƒ'(a)(x - a)\] where ƒ'(a) is the derivative of ƒ evaluated at x = a.
Identify the function and the point of approximation: here, ƒ(x) = \(\cosh\) x and a = \(\ln\) 3.
Calculate ƒ(a) by evaluating \(\cosh\)(\(\ln\) 3). Remember that \(\cosh\) x = \(\frac{e^x + e^{-x}\)}{2}, so substitute x = \(\ln\) 3 to find ƒ(a).
Find the derivative of ƒ(x). Since ƒ(x) = \(\cosh\) x, its derivative is ƒ'(x) = \(\sinh\) x. Then evaluate ƒ'(a) = \(\sinh\)(\(\ln\) 3). Use the definition \(\sinh\) x = \(\frac{e^x - e^{-x}\)}{2} to compute this.
Write the linear approximation formula using the values found: \[L(x) = \cosh(\ln 3) + \sinh(\ln 3)(x - \ln 3)\] Finally, substitute x = 1 into this linear approximation to estimate \(\cosh\) 1.

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:
4m

Concetti chiave

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

Linear Approximation

Linear approximation uses the tangent line at a point a to estimate the value of a function near a. It is given by L(x) = f(a) + f'(a)(x - a), providing a simple way to approximate function values close to a.
Video consigliato:
Percorso guidato
07:17
Linearization

Derivative of cosh x

The derivative of cosh x is sinh x, which represents the slope of the tangent line to the curve at any point x. Knowing this derivative is essential to find the slope f'(a) needed for the linear approximation.
Video consigliato:
04:56
Derivative of the Natural Exponential Function (e^x)

Evaluating at a specific point

After finding the linear approximation at a = ln 3, you substitute x = 1 into the linear function to approximate cosh 1. This step applies the approximation formula to estimate values near the chosen point.
Video consigliato:
5:14
Evaluate Logarithms
Pratica correlata
Domanda del libro di testo

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

f(x) = ln(3 sin² 4x)

72
views
Domanda del libro di testo

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


g(t) = sinh⁻¹(√t)

55
views
Domanda del libro di testo

Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is

f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0

where ln x has zero mean and standard deviation σ > 0.

e. For what value of σ > 0 in part (d) does ƒ(x*) have a minimum?

80
views
Domanda del libro di testo

Derivatives of hyperbolic functions Compute the following derivatives.

b. d/dx (x sech x)

105
views
Domanda del libro di testo

Limit Evaluate lim x → ∞ (tanh x)ˣ.

104
views
Domanda del libro di testo

Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is

f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0

where ln x has zero mean and standard deviation σ > 0.

b. Evaluate lim x → 0 ƒ(x). (Hint: Let x = eʸ.)

92
views