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

Briggs 3rd Edition

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

Problem 7.3.109

Chapter 7, Problem 7.3.109

Arc length Use the result of Exercise 108 to find the arc length of the curve: f(x) = ln |tanh(x / 2)| on [ln 2, ln 8].

Verified step by step guidance

Recall the formula for the arc length of a curve \(y = f(x)\) on the interval \([a, b]\): \[L = \int_a^b \sqrt{1 + \left( f'(x) \right)^2} \, dx\]

Identify the function given: \[f(x) = \ln \left| \tanh \left( \frac{x}{2} \right) \right|\] and the interval: \[[a, b] = [\ln 2, \ln 8]\]

Find the derivative \(f'(x)\) using the chain rule and the derivative of \(\ln |u|\) which is \(\frac{u'}{u}\): Let \[u = \tanh \left( \frac{x}{2} \right)\] then \[f'(x) = \frac{u'}{u}\] Calculate \(u'\) by differentiating \(\tanh \left( \frac{x}{2} \right)\) with respect to \(x\).

Substitute \(f'(x)\) into the arc length formula: \[L = \int_{\ln 2}^{\ln 8} \sqrt{1 + \left( f'(x) \right)^2} \, dx\] Simplify the expression under the square root as much as possible to make the integral easier to evaluate.

Evaluate the integral either by direct integration or by using a suitable substitution if needed, to find the arc length over the given interval.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

15m

Was this helpful?

Key Concepts

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

Arc Length Formula

The arc length of a curve y = f(x) from x = a to x = b is given by the integral \( L = \int_a^b \sqrt{1 + (f'(x))^2} \, dx \). This formula calculates the length of the curve by summing infinitesimal line segments along the curve.

Derivative of the Function

To apply the arc length formula, you need the derivative f'(x). For f(x) = ln|tanh(x/2)|, use the chain rule and derivatives of hyperbolic functions to find f'(x), which is essential for evaluating the integrand \( \sqrt{1 + (f'(x))^2} \).

Properties of Hyperbolic Functions

Understanding hyperbolic functions like tanh(x) and their derivatives is crucial. For example, \( \frac{d}{dx} \tanh(x) = \text{sech}^2(x) \). These properties help simplify the derivative and the integrand in the arc length integral.

Recommended video:

06:21

Properties of Functions

Related Practice

Textbook Question

88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.

lim x → 0⁺ (tanh x)ˣ

Textbook Question

Newton’s method Use Newton’s method to find all local extreme values of ƒ(x) = x sech x.

Textbook Question

22–36. Derivatives Find the derivatives of the following functions.

f(t) = 2 tanh⁻¹ √t

Textbook Question

Uranium dating Uranium-238 (U-238) has a half-life of 4.5 billion years. Geologists find a rock containing a mixture of U-238 and lead, and they determine that 85% of the original U-238 remains; the other 15% has decayed into lead. How old is the rock?

Textbook Question

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

∫₀ˡⁿ ² (e^{3x} − e^{−3x}) / (e^{3x} + e^{−3x}) dx

Textbook Question

What is the inverse function of ln x, and what are its domain and range?