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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.3.139
Chapter 7, Problem 7.3.139

In Exercises 139–142, find the length of each curve.
139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.

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Recall the formula for the length of a curve defined by a function \(y = f(x)\) from \(x = a\) to \(x = b\): \[L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx\]
Identify the function given: \[y = \frac{1}{2} \left(e^x + e^{-x}\right)\] and the interval: \[x \in [0, 1]\]
Compute the derivative \(\frac{dy}{dx}\): \[\frac{dy}{dx} = \frac{1}{2} \left(e^x - e^{-x}\right)\]
Substitute \(\frac{dy}{dx}\) into the arc length formula under the square root: \[\sqrt{1 + \left(\frac{1}{2} (e^x - e^{-x})\right)^2} = \sqrt{1 + \frac{(e^x - e^{-x})^2}{4}}\]
Simplify the expression inside the square root and set up the integral for the length: \[L = \int_0^1 \sqrt{1 + \frac{(e^x - e^{-x})^2}{4}} \, dx\] This integral can then be evaluated to find the length of the curve.

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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 found using the integral formula L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the length by summing infinitesimal line segments along the curve.
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Arc Length of Parametric Curves

Derivative of Exponential Functions

To apply the arc length formula, you need the derivative dy/dx. For functions involving e^x and e^(-x), the derivative rules state that d/dx(e^x) = e^x and d/dx(e^(-x)) = -e^(-x). Understanding these helps compute dy/dx accurately.
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Derivatives of General Exponential Functions

Simplification of Expressions Involving Hyperbolic Functions

The given function y = (1/2)(e^x + e^(-x)) is the hyperbolic cosine function, cosh(x). Recognizing this allows simplification of derivatives and integrals, as cosh^2(x) - sinh^2(x) = 1, which can simplify the arc length integral.
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Integrals Resulting in Inverse Trig Functions: Substitution Example 5
Related Practice
Textbook Question

7. Order the following functions from slowest growing to fastest growing as x→∞.

a. e^x

b. x^x

c. (ln x)^x

d. e^(x/2)

Textbook Question

Theory and Applications

L’Hôpital’s Rule does not help with the limits in Exercises 69–76. 

Try it—you just keep on cycling. Find the limits some other way.

69. lim (x → ∞) (√(9x + 1)) / (√(x + 1))

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Textbook Question

17. Show that √(10x+1) and √(x+1) grow at the same rate as x→∞ by showing that they both grow at the same rate as √x as x→∞.

Textbook Question

Since the hyperbolic functions can be expressed in terms of exponential functions, it is possible to express the inverse hyperbolic functions in terms of logarithms, as shown in the following table.

sinh⁻¹x = ln(x + √(x² + 1)), -∞ < x < ∞

cosh⁻¹x = ln(x + √(x² - 1)), x ≥ 1

tanh⁻¹x = (1/2)ln((1+x)/(1-x)), |x| < 1

sech⁻¹x = ln((1+√(1-x²))/x), 0 < x ≤ 1

csch⁻¹x = ln(1/x + √(1+x²)/|x|), x ≠ 1

coth⁻¹x = (1/2)ln((x+1)/(x-1)), |x| > 1

Use these formulas to express the numbers in Exercises 61–66 in terms of natural logarithms.

63. tanh⁻¹(-1/2)

Textbook Question

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

13. lim(x → 1⁻)arcsin(x)

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Textbook Question

Evaluate the integrals in Exercises 39–56.

47. ∫(from 2 to 4)dx/(x(ln x)²)