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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 7.1.47b
Chapter 7, Problem 7.1.47b

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
b. x=2
Table showing values of x, f(x), and f prime(x) at points 0 to 4, used to find the slope of the inverse function at x=2.

Verified step by step guidance
1
Identify the value of \(b\) for which we want to find the slope of the inverse function \(f^{-1}(x)\). Here, \(b = 2\).
Recall the formula for the derivative of the inverse function: \(\left(f^{-1}\right)'(b) = \frac{1}{f'(a)}\) where \(a = f^{-1}(b)\), meaning \(f(a) = b\).
From the table, find the value of \(a\) such that \(f(a) = 2\). Looking at the \(f(x)\) row, \(f(4) = 2\), so \(a = 4\).
Find \(f'(a)\) from the table. For \(a = 4\), \(f'(4) = \frac{1}{7}\).
Use the formula to find the slope of the inverse function at \(x = 2\): \(\left(f^{-1}\right)'(2) = \frac{1}{f'(4)} = \frac{1}{\frac{1}{7}}\).

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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 one-to-one 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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Derivatives of Other Inverse Trigonometric Functions

Derivative of the Inverse Function

The derivative of the inverse function at a point x = b is given by (f^(-1))'(b) = 1 / f'(a), where a = f^(-1)(b). This formula relates the slope of the inverse function to the slope of the original function at the corresponding point, allowing us to find the slope of the inverse using known values of f and f'.
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Using Tabulated Values to Find the Slope

Given a table of values for x, f(x), and f'(x), we identify the x-value a such that f(a) = b. Then, we use the derivative f'(a) to compute the slope of the inverse at x = b. This method applies the inverse derivative formula directly using discrete data points.
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Related Practice
Textbook Question

Suppose that the function g and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.

Assuming the inverse function g^(-1) is differentiable, find the slope of g^(-1)(x) at

a. x=1

Textbook Question

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

7. a. sec^(-1)(-√2)

Textbook Question

1. Express the following logarithms in terms of ln 2 and ln 3.

b. ln(4/9)

Textbook Question

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

67. ∫(from 0 to 2√3)dx/√(4+x²)

1
views
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).

Textbook Question

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

2. y' = y²

b. y = -1/(x+3)