Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
120. y = x^(sin x)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.141
In Exercises 139–142, find the length of each curve.
141. y = ln(cos(x)) from x = 0 to x = π/4.
Verified step by step guidance1
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:
\[y = \ln(\cos(x))\]
with the interval from \(x = 0\) to \(x = \frac{\pi}{4}\).
Find the derivative \(\frac{dy}{dx}\) using the chain rule:
Since \(y = \ln(u)\) where \(u = \cos(x)\), then
\[\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} = \frac{1}{\cos(x)} \cdot (-\sin(x)) = -\tan(x)\]
Substitute \(\frac{dy}{dx} = -\tan(x)\) into the arc length formula:
\[L = \int_0^{\frac{\pi}{4}} \sqrt{1 + (-\tan(x))^2} \, dx = \int_0^{\frac{\pi}{4}} \sqrt{1 + \tan^2(x)} \, dx\]
Use the trigonometric identity \(1 + \tan^2(x) = \sec^2(x)\) to simplify the integrand:
\[L = \int_0^{\frac{\pi}{4}} \sqrt{\sec^2(x)} \, dx = \int_0^{\frac{\pi}{4}} |\sec(x)| \, dx\]
Since \(\sec(x)\) is positive on \([0, \frac{\pi}{4}]\), this becomes
\[L = \int_0^{\frac{\pi}{4}} \sec(x) \, dx\]
The next step would be to evaluate this integral.
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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 given by the integral L = ∫_a^b √(1 + (dy/dx)^2) dx. This formula calculates the length by summing infinitesimal line segments along the curve.
Recommended video:
06:29
Arc Length of Parametric Curves
Derivative of y = ln(cos(x))
To find the arc length, you need the derivative dy/dx. For y = ln(cos(x)), use the chain rule: dy/dx = -tan(x), since the derivative of ln(u) is 1/u * du/dx and d/dx[cos(x)] = -sin(x).
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04:56Derivative of the Natural Exponential Function (e^x)
Integration Techniques for Arc Length
Evaluating the arc length integral often requires simplifying the integrand and applying appropriate integration methods, such as substitution or recognizing standard integral forms, to compute the exact length over the given interval.
Recommended video:
06:29Arc Length of Parametric Curves
Related Practice
Textbook Question
13. When is a polynomial f(x) of at most the order of a polynomial g(x) as x→∞? Give reasons for your answer.
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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.
65. sech⁻¹(3/5)
Textbook Question
Evaluate the integrals in Exercises 31–78.
69. ∫dy/(y√(4y²-1))
Textbook Question
Solve the differential equation in Exercises 9–22.
13. (dy/dx) = √y cos²√y
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Textbook Question
In Exercises 27–32, find dy/dx.
ln y = e^y sinx