Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problem 7.R.31

Chapter 7, Problem 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.

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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.

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Key Concepts

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

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.

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.

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.

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

g(t) = sinh⁻¹(√t)

Textbook Question

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?

Textbook Question

Derivatives of hyperbolic functions Compute the following derivatives.

b. d/dx (x sech x)

Textbook Question

Limit Evaluate lim x → ∞ (tanh x)ˣ.

Textbook Question

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

f(t) = cosh t sinh t

Textbook Question

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)