Textbook Question
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 1 - x/2, x ≤ 0
x/(x + 2), x > 0
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 7.3.139
In Exercises 139–142, find the length of each curve.
139. y = (1/2)(e^x + e^(−x)) from x = 0 to x = 1.
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:
\[y = \frac{1}{2} \left(e^x + e^{-x}\right)\]
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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04:50Derivatives 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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03:18 Integrals Resulting in Inverse Trig Functions: Substitution Example 5
Related Practice
Textbook Question
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
17. lim(x→∞)arcsec(x)
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Textbook Question
Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.
4. cosh x = 13/5, x>0
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
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
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Textbook Question
Evaluate the integrals in Exercises 39–56.
47. ∫(from 2 to 4)dx/(x(ln x)²)