Purpose: To develop the product rule and understand the need for it.
Classroom Procedure: Begin with students working in small groups on Q1 and then follow with a class discussion to establish the need for developing a rule to take the derivative of products of functions. After this students should work on investigating the rule in small groups and the practice applying it. It is a good idea to either have a whole class discussion to establish the rule or if students are in very different places to ask them to check with you before moving on to the practice portion.
Ideas this Activity Builds On: This activity builds on their previous investigations of derivative rules and the derivatives they are already able to find symbolically.
Introduction/Motivation of the Activity: The motivation for this activity is really included in Q1 - we are currently unable to find a symbolic derivative of g(x) and therefore need to develop a rule for that.
Need to Establish by the End of Activity/Wrap-Up: The product rule.
Additional Notes: In the synthesis question 3, students find d very difficult despite having completed a-c correctly. It is good for them to struggle with this however as they have to identify that the way to discover where the largest area occurs is by using the derivative.