Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
a. inverse hyperbolic functions.
67. ∫(from 0 to 2√3)dx/√(4+x²)
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 7.1.41a
In Exercises 41–44:
a. Find f⁻¹(x).
41. f(x) = 2x + 3, a = −1
Verified step by step guidance1
Start with the given function: \(f(x) = 2x + 3\).
To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\): \(y = 2x + 3\).
Swap the variables \(x\) and \(y\) to get \(x = 2y + 3\); this step reflects the idea that the inverse function reverses the roles of inputs and outputs.
Solve the equation \(x = 2y + 3\) for \(y\): subtract 3 from both sides to get \(x - 3 = 2y\), then divide both sides by 2 to isolate \(y\): \(y = \frac{x - 3}{2}\).
Rewrite \(y\) as \(f^{-1}(x)\) to express the inverse function: \(f^{-1}(x) = \frac{x - 3}{2}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Functions
An inverse function reverses the effect of the original function, swapping inputs and outputs. For a function f(x), its inverse f⁻¹(x) satisfies f(f⁻¹(x)) = x. Finding the inverse involves solving the equation y = f(x) for x in terms of y.
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Solving Linear Equations
To find the inverse of a linear function like f(x) = 2x + 3, you solve for x by isolating it on one side of the equation. This involves algebraic manipulation such as subtracting constants and dividing by coefficients to express x as a function of y.
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07:17Linearization
Domain and Range Considerations
When finding an inverse, it's important to consider the domain and range of the original function and its inverse. For linear functions, the domain and range are typically all real numbers, but specifying points like a = -1 helps evaluate the inverse at particular values.
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Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
7. a. sec^(-1)(-√2)
Textbook Question
In Exercises 41–44:
a. Find f⁻¹(x).
42. f(x) = (x + 2) / (1 − x), a = 1/2
Textbook Question
78. Which one is correct, and which one is wrong? Give reasons for your answers.
a. lim (x → 0) (x² - 2x) / (x² - sin x) = lim (x → 0) (2x - 2) / (2x - cos x) = lim (x → 0) 2 / (2 + sin x) = 2 / (2 + 0) = 1
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Textbook Question
Find the volumes of the solids in Exercises 135 and 136.
135. The solid lies between planes perpendicular to the x-axis at x=-1 and x=1. The cross-sections perpendicular to the x-axis are
a. circles whose diameters stretch from the curve y=-1/√(1+x²) to the curve y=1/√(1+x²).
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Textbook Question
21. a. Show that ln(x) grows slower as x→∞ than x^(1/n) for any positive integer n, even x^(1/1,000,000).