Calculus - How Can One Graph Help Describe Another Graph?

This activity incorporates the graphing calculator to help students understand how the behavior of one graph can describe the behavior of another function.

Three related equations are entered in y1, y2, and y3 in the graphing calculator.

• The original function is entered in y1.

• An expression that represents the derivative of y1 is entered in y2.

• An expression that represents the derivative of y2 is entered in y3.

After graphing y1, y2, and y3 in the same window, a series of questions are asked about those graphs to help the students realize how the various graphs are interrelated.

• When is y1 increasing? Decreasing? Changing from decreasing to increasing?

• When is y2 positive? Negative? Changing from being negative to positive?

• When is y1 concave up? Concave down?

• When is y3 positive? Negative?

• What do you notice about the slope of y2? How is this reflected in the graph of y3?

• Study the slope of y1 for given domain. As you move left to right, how is the slope changing? How is this reflected in the graph of y2?

• Study the slope of y1 for a given domain. As you move left to right, how is the slope changing? How is this reflected in the graph of y2?

• Where does a minimum occur in y1? How can you use the graph of y2 to predict the location of this minimum in y1?

This is an extreme useful skill that all calculus students should understand and be able to use.

Comments from Buyers:

• I like using graphing calculator explorations, but they take so long to make. Thanks for sharing!

• Good