precisely when they are inverse functions. This activity is intended to explore the relation between the derivative of a function at the point (
), and the derivative of the inverse function at the point (
).
1. Inverse functions.
There are several ways to characterize when two functions are inverses to each other. The graphs of two functions will be reflections of each other across the line
precisely when they are inverse functions. This activity is intended to explore the relation between the derivative of a function at the point (
), and the derivative of the inverse function at the point (
).
Submission:
(a) Plot the functions
and
on the same axes, over the domain x=0..5 and range y=0..5. In order to get the proper aspect ratio, you should choose the option
scaling=CONSTRAINED
to the plot command. How do these graphs illustrate that these are inverse functions?
(b) Find the slope of the tangent line to the curve
at the points (
)=(1,1), (
), (
), (
).
Then find the slope of the tangent line to the curve
at the points (
)=(1,1), (
), (
), (
).
What is the relation between the slopes of these tangent lines?
(c) Plot the tangent line to the curve
at (
) and the tangent line to the curve
at (
) on the same axes, again using a
CONSTRAINED
scaling. What is the relationship between these two tangent lines?
Submission worksheet: