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

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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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주요 개념

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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.

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'.

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.

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) atb. x=2