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 7.7.4
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
Verified step by step guidance1
Recall the fundamental identity for hyperbolic functions: \(\cosh^{2}x - \sinh^{2}x = 1\).
Given \(\cosh x = \frac{13}{5}\) and \(x > 0\), substitute into the identity to find \(\sinh x\):
\(\left(\frac{13}{5}\right)^{2} - \sinh^{2}x = 1\).
Solve for \(\sinh^{2}x\):
\(\sinh^{2}x = \left(\frac{13}{5}\right)^{2} - 1\).
Since \(x > 0\), take the positive square root to find \(\sinh x\):
\(\sinh x = \sqrt{\left(\frac{13}{5}\right)^{2} - 1}\).
Use the definitions of the other hyperbolic functions in terms of \(\sinh x\) and \(\cosh x\) to find their values:
- \(\tanh x = \frac{\sinh x}{\cosh x}\)
- \(\coth x = \frac{\cosh x}{\sinh x}\)
- \(\sech x = \frac{1}{\cosh x}\)
- \(\csch x = \frac{1}{\sinh x}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Functions and Their Definitions
Hyperbolic functions such as sinh x and cosh x are analogs of trigonometric functions but based on exponential functions. Specifically, sinh x = (e^x - e^{-x})/2 and cosh x = (e^x + e^{-x})/2. Understanding these definitions helps in expressing and manipulating hyperbolic functions.
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Fundamental Hyperbolic Identity
The identity cosh²x - sinh²x = 1 is a key relationship between hyperbolic sine and cosine, similar to the Pythagorean identity in trigonometry. It allows solving for one function when the other is known, which is essential for finding the remaining hyperbolic functions.
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Definition of Remaining Hyperbolic Functions
Besides sinh and cosh, the other hyperbolic functions include tanh x = sinh x / cosh x, coth x = cosh x / sinh x, sech x = 1 / cosh x, and csch x = 1 / sinh x. Knowing these definitions enables calculation of all hyperbolic functions once sinh x and cosh x are determined.
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Related Practice
Textbook Question
Find the values in Exercises 9–12.
11. tan(arcsin(-1/2))
Textbook Question
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
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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→∞.
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In Exercises 139–142, find the length of each curve.
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Textbook Question
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
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