Textbook Question
91. [Technology Exercise] 91. The continuous extension of to (sin x)^x to [0, π]
b. Verify your conclusion in part (a) by finding lim(x→0⁺)f(x) with l’Hôpital’s Rule.
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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 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 7.1.47b
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
Verified step by step guidance1
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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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'.
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06:35Derivatives 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.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 67–74 in terms of
b. natural logarithms.
71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))
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Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
4. b. arcsin(-1/√2)
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
2. b. tan^(-1)(√3)
Textbook Question
1. Express the following logarithms in terms of ln 2 and ln 3.
b. ln(4/9)
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:
b. Solve the equation y=f(x) for x as a function of y, and name the resulting inverse function g.
68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2