Skip to main content
Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.1.47b
Capitolo 7, Problema 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.

Guida verificata passo dopo passo
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 \(x = 2\): \(\left(f^{-1}\right)'(2) = \frac{1}{f'(4)} = \frac{1}{\frac{1}{7}}\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
06:35
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'.
Video consigliato:
06:35
Derivatives of Other Inverse Trigonometric Functions

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.
Video consigliato:
04:09
Understanding Slope Fields Example 2
Pratica correlata
Domanda del libro di testo

31. The incidence of a disease (Continuation of Example 4.) Suppose that in any given year the number of cases can be reduced by 25% instead of 20%.

b. How long will it take to eradicate the disease—that is, reduce the number of cases to less than 1?

9
views
Domanda del libro di testo

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

7. a. sec^(-1)(-√2)

26
views
Domanda del libro di testo

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:


b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.

67. y= √(3x-2), 2/3 ≤ x ≤ 4, x_0=3

34
views
Domanda del libro di testo

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

20
views
Domanda del libro di testo

[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.

139. a. y=arctan(tan x)

31
views
Domanda del libro di testo

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

67. ∫(from 0 to 2√3)dx/√(4+x²)

27
views