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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.1.47c
Chapter 7, Problem 7.1.47c

Suppose that the function f and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.
Assuming the inverse function f^(-1) is differentiable, find the slope of f^(-1)(x) at
c. x=3
Table showing values of x, f(x), and f prime of x at points 0 to 4, used to find the slope of the inverse function at x=3.

Verified step by step guidance
1
Identify the value of \(c\) for which you need to find the slope of the inverse function \(f^{-1}(x)\). Here, \(c = 3\).
Recall the formula for the derivative of the inverse function: \(\left(f^{-1}\right)'(c) = \frac{1}{f'(f^{-1}(c))}\).
Find \(f^{-1}(3)\) by looking for the \(x\) value such that \(f(x) = 3\). From the table, \(f(0) = 3\), so \(f^{-1}(3) = 0\).
Use the value of \(f^{-1}(3)\) to find \(f'(f^{-1}(3)) = f'(0)\). From the table, \(f'(0) = \frac{4}{3}\).
Calculate the slope of the inverse function at \(x=3\) using the formula: \(\left(f^{-1}\right)'(3) = \frac{1}{f'(0)} = \frac{1}{\frac{4}{3}}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Function and Its Differentiability

An inverse function f^(-1) reverses the effect of the original function f, such that f(f^(-1)(x)) = x. For the inverse to be differentiable at a point, the original function must be continuous and have a non-zero derivative at the corresponding point. This ensures the inverse function's slope exists and can be calculated.
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Derivative of the Inverse Function

The derivative of the inverse function at a point x = c is given by (f^(-1))'(c) = 1 / f'(f^(-1)(c)). This formula relates the slope of the inverse function to the slope of the original function at the corresponding point. It requires finding the input value where f(x) = c and then using the derivative of f at that input.
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Using Tabulated Values to Find the Slope

Given a table of values for x, f(x), and f'(x), you can find the slope of the inverse function at x = c by first locating the x-value where f(x) = c. Then, use the corresponding f'(x) value to compute the inverse slope using the formula. This approach is practical when explicit functions are not provided.
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Related Practice
Textbook Question

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

1. c. tan^(-1)(1/√3)

Textbook Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:


c. Find the equation for the tangent line to f at the specified point (x_0, f(x_0)).


70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

Textbook Question

3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?

c. √(x^4 + x^3)

Textbook Question

9. True, or false? As x→∞,

c. x = O(x+5)

1
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Textbook Question

c. Find the slopes of the tangent lines to the graphs of h and k at (2, 2) and (−2, −2).

Textbook Question

23. Human evolution continues The analysis of tooth shrinkage by C. Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years.

c. What will be our descendants’ tooth size 20,000 years from now (as a percentage of our present tooth size)?