Activity Overview
In this Derive™ activity, students relate the behavior of the graph of an exponential function to the signs of its first two derivatives. They realize that if the signs of the first two derivatives over a given interval are known, a fairly accurate picture of the graph of the function can be obtained.
Before the Activity
See the attached DFW file for detailed instructions for this activity
Print pages from the attached DFW file for your class
During the Activity
Distribute the pages to the class.
Follow the Activity Procedures:
Set up a 2D plot window
Set the plot range and graph an exponential function
Insert annotations where the curve rises or falls
Use the Derive™ software to differentiate the function, and graph the result
Label where the graph is positive or negative
Test the sign of the derivative of the function with the help of Derive's 'If' syntax
Evaluate the second derivative of the function, and graph it along with the function in the plot window
Label the function where it is concave or convex
Label the graph of the second derivative where it is positive or negative
Check if the regions on each graph coincide
Combine the first and second derivative tests using a logical AND operator
Record the intervals on the x-axis where the graphs are positive or negative
Observe the points where the derivatives are zero
After the Activity
Students answer questions listed on the activity sheet.
Review student results:
As a class, discuss questions that appeared to be more challenging
Re-teach concepts as necessary