Textbook Question
In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
5. lim (x → 0) (1 - cos x) / x²
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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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.1.21
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=x³-1
Verified step by step guidance1
Start with the given function: \(y = f(x) = x^3 - 1\).
To find the inverse function \(f^{-1}(x)\), swap the roles of \(x\) and \(y\): write \(x = y^3 - 1\).
Solve this equation for \(y\) to express \(y\) in terms of \(x\): add 1 to both sides to get \(x + 1 = y^3\).
Take the cube root of both sides to isolate \(y\): \(y = \sqrt[3]{x + 1}\).
Therefore, the formula for the inverse function is \(f^{-1}(x) = \sqrt[3]{x + 1}\).
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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. If y = f(x), then x = f⁻¹(y). The graph of an inverse function is a reflection of the original function's graph across the line y = x.
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Inverse Cosine
Finding the Inverse Function Algebraically
To find the inverse function f⁻¹(x), start with y = f(x), then solve the equation for x in terms of y. Finally, interchange x and y to express y = f⁻¹(x). This process reverses the roles of dependent and independent variables.
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4:49Inverse Cosine
Graphical Relationship Between f and f⁻¹
The graphs of a function and its inverse are symmetric about the line y = x. Points (a, b) on f correspond to points (b, a) on f⁻¹. This symmetry helps visualize and verify the correctness of the inverse function.
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Related Practice
Textbook Question
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
35. y = sinh⁻¹(tan x)
Textbook Question
Evaluate the integrals in Exercises 87–96.
93. ∫₀^(π/2) 7^(cos t) sin t dt
Textbook Question
Evaluate the integrals in Exercises 53–76.
57. ∫dx/(x√(25x²-2))
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = (x^2 - 2x + 2)e^(x)
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.
126. eʸ = y^(ln x)