Explanation Given information: y = − 2 x + 12 is the equation of the tangent line to the graph y = f ( x ) at x = 4 . The function g ( x ) is the inverse of the function f ( x ) . Formula used: The derivatives of the inverse functions can be obtained by, g ′ ( x ) = 1 f ′ ( x ) where g ( x ) = f − 1 ( x ) Slope-Intercept form of a line is y = m x + c , where m is the slope and c is the y-intercept. Calculation: Let us consider the equation of the tangent line. y = − 2 x + 12 Comparing the equation with y = m x + c , we get m = − 2 The slope of tangent line to a graph y = f ( x ) is given by the derivative of the function of the graph. Hence, at the point x = 4 , f ′ ( x ) = − 2 Since, y = g ( x ) is the inverse of the function y = f ( x ) , g ( x ) = f − 1 ( x ) To find the tangent line g ' ( x ) of the graph y = g ( x ) at x = 4 , we can use the theorem of the derivatives of the inverse function

4th Edition

ISBN: 9781319055844

Author: FRANZOSA

Publisher: VST

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Chapter 7, Problem 41CRE

To determine

To find: The tangent line to y=g(x) at x=4 .

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Chapter 7, Problem 40CRE

Chapter 7, Problem 42CRE

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The tangent line to y = g ( x ) at x = 4 .

Solution Summary: The author calculates the slope of the tangent line to a graph y=f(x).

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To find: The tangent line to y=g(x) at x=4 .

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