Textbook Question
Show that increasing functions and decreasing functions are one-to-one. That is, show that for any x₁ and x₂ in I, x₂ ≠ x₁ implies f(x₂) ≠ f(x₁).
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 7.1.53
Suppose that the differentiable function y = f(x) has an inverse and that the graph of f passes through the point (2, 4) and has a slope of 1/3 there. Find the value of df⁻¹/dx at x = 4.
Verified step by step guidance1
Recall that if \( y = f(x) \) has an inverse function \( f^{-1}(x) \), then the derivative of the inverse at a point \( x \) is given by the formula:
\[ \frac{d}{dx} f^{-1}(x) = \frac{1}{f'(f^{-1}(x))} \]
Identify the point on the original function \( f \) given in the problem: the function passes through \( (2, 4) \) and has a slope \( f'(2) = \frac{1}{3} \).
Since \( f(2) = 4 \), the inverse function satisfies \( f^{-1}(4) = 2 \). This means when \( x = 4 \) in the inverse function, the corresponding \( y \) value is 2.
Apply the formula for the derivative of the inverse at \( x = 4 \):
\[ \frac{d}{dx} f^{-1}(4) = \frac{1}{f'(f^{-1}(4))} = \frac{1}{f'(2)} \]
Substitute the known slope \( f'(2) = \frac{1}{3} \) into the expression to find \( \frac{d}{dx} f^{-1}(4) \).
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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) is invertible, then f⁻¹(y) returns the original x. Understanding this relationship is crucial for connecting values of f and f⁻¹ at corresponding points.
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Derivative of an Inverse Function
The derivative of the inverse function at a point x is the reciprocal of the derivative of the original function at the corresponding point y = f⁻¹(x). Formally, (df⁻¹/dx)(x) = 1 / (df/dx)(f⁻¹(x)). This formula allows us to find the slope of the inverse function using the slope of the original function.
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Evaluating Derivatives at Specific Points
To find the derivative of the inverse at x = 4, identify the corresponding point on f where f(x) = 4. Using the given point (2, 4), we know f(2) = 4, so f⁻¹(4) = 2. Then, use the slope of f at x = 2 to compute the derivative of the inverse at x = 4.
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Related Practice
Textbook Question
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Textbook Question
47. Carbon-14 The oldest known frozen human mummy, discovered in the Schnalstal glacier of the Italian Alps in 1991 and called Otzi, was found wearing straw shoes and a leather coat with goat fur, and holding a copper ax and stone dagger. It was estimated that Otzi died 5000 years before he was discovered in the melting glacier. How much of the original carbon-14 remained in Otzi at the time of his discovery?
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