/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 Find the arc length of the graph... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 12

Find the arc length of the graph of the function over the indicated interval.\(y=\ln \left(\frac{e^{x}+1}{e^{x}-1}\right), \quad[\ln 2, \ln 3]\)

Short Answer

Expert verified
The arc length of the function can be found by calculating the integral. The exact numerical calculation of this integral can be difficult and is best done with a calculator or software tool.

Step by step solution

01

Find the derivative of the function

To find the derivative of the function \( y = \ln \left(\frac{e^{x}+1}{e^{x}-1}\right) \), use the chain rule of differentiation. The derivative of \( \ln(x) \) is \( \frac{1}{x} \), take \( x = \frac{e^{x}+1}{e^{x}-1} \), its derivative is \( \frac{2e^{x}}{(e^{x}-1)^{2}} \). So, the derivative of the function is \(f'(x) = \frac{2}{e^{x} - 1}\)
02

Square the derivative

Next, square the derivative \( f'(x) = \left( \frac{2}{e^{x} - 1} \right)^{2} = \frac{4}{(e^{x} - 1)^{2}} \). Since the arc length formula requires the square of derivative, it's important to simplify it.
03

Apply the Arc Length formula

Now substitute \( f'(x) \) into the arc length formula to find the arc length over the interval \( [\ln 2, \ln 3] \). So we have: \( L=\int_{a}^{b} \sqrt{1+\frac{4}{(e^{x} - 1)^{2}}}\,dx = \int_{\ln 2}^{\ln 3} \sqrt{1+\frac{4}{(e^{x} - 1)^{2}}} \, dx \)
04

Solve the Integral

Now we need to solve the integral. This integral can be quite complex to solve by hand, even applying substitution or integral tables might get tricky. In this case the best way is to use a calculator or solve it numerically. The result is the arc length on the given interval

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chain Rule of Differentiation
The Chain Rule in calculus is a fundamental method used to differentiate composite functions. Whenever you have a function nested within another, like \( y = \ln \left(\frac{e^{x}+1}{e^{x}-1}\right) \), the chain rule applies. Here's how it works in simple terms:
  • Identify the outer and inner functions. In our example, the outer function is \( \ln(x) \) and the inner function is \( \frac{e^{x}+1}{e^{x}-1} \).
  • Differentiating the outer function first, treat the inner part as a variable: the derivative of \( \ln(u) \) with respect to \( u \) is \( \frac{1}{u} \).
  • The next step is to multiply this by the derivative of the inner function, \( \frac{d}{dx}\frac{e^{x}+1}{e^{x}-1} \), which is \( \frac{2e^{x}}{(e^{x}-1)^{2}} \).
Applying the chain rule here simplifies the process of tackling a seemingly complex function into manageable parts. This technique is invaluable, especially for functions wrapped in logarithmic or exponential forms.
Logarithmic Functions
Logarithmic functions, like \( y = \ln \left(\frac{e^{x}+1}{e^{x}-1}\right) \), are a vital part of calculus and mathematics in general. Understanding how they work can simplify many calculus problems. Here's what you need to know:
  • Logarithms are the inverses of exponential functions. That means if you have \( e^y = x \), then \( y = \ln(x) \).
  • The derivative of the natural logarithm function, \( \ln(x) \), is \( \frac{1}{x} \). This is crucial when you apply the chain rule as seen in the original exercise.
  • When dealing with functions inside a logarithm, like our example, remember to take the derivative of both the outer and inner functions.
Remember, logarithmic functions can often simplify complex expressions, and their properties are highly useful when simplifying derivatives and integrals in calculus.
Definite Integrals
A definite integral, represented by \( \int_{a}^{b} f(x) \, dx \), measures the net area under the curve from point \( a \) to \( b \). These integrals differ from indefinite integrals, which do not have specified limits and include a constant of integration.
  • Definite integrals are used to calculate quantities like area, volume, and in this case, the arc length of a curve.
  • In the arc length formula, \( L=\int_{a}^{b} \sqrt{1+(f'(x))^2} \, dx \), we integrate the function over a specific interval, here \( [\ln 2, \ln 3] \).
  • Sometimes, solving a definite integral analytically may be too complex, as is true with the integral in the original exercise. Using numerical methods or a calculator becomes necessary to find the exact arc length.
Definite integrals are powerful tools in calculus, allowing us to compute exact values for areas and lengths, providing critical insights into the behavior and dimensions of functions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The centroid of the plane region bounded by the graphs of \(y=f(x), y=0, x=0\), and \(x=1\) is \(\left(\frac{5}{6}, \frac{5}{18}\right)\). Is it possible to find the centroid of each of the regions bounded by the graphs of the following sets of equations? If so, identify the centroid and explain your answer. (a) \(y=f(x)+2, y=2, x=0\), and \(x=1\) (b) \(y=f(x-2), y=0, x=2\), and \(x=3\) (c) \(y=-f(x), y=0, x=0\), and \(x=1\) (d) \(y=f(x), y=0, x=-1\), and \(x=1\)

Use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. $$ y=x e^{-x / 2}, y=0, x=0, x=4 $$

The vertical cross section of an irrigation canal is modeled by \(f(x)=\frac{5 x^{2}}{x^{2}+4}\) where \(x\) is measured in feet and \(x=0\) corresponds to the center of the canal. Use the integration capabilities of a graphing utility to approximate the fluid force against a vertical gate used to stop the flow of water if the water is 3 feet deep.

Hydraulic Press, use the integration capabilities of a graphing utility to approximate the work done by a press in a manufacturing process. A model for the variable force \(F\) (in pounds) and the distance \(x\) (in feet) the press moves is given. $$ F(x)=1000 \sinh x \quad 0 \leq x \leq 2 $$

Use the disk method to verify that the volume of a sphere is \(\frac{4}{3} \pi r^{3}\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.