Calculate derivatives of inverse functions
I can determine when two functions are inverses
I can verbalize the relationship between slopes at inverse points
I can calculate derivatives of inverse functions
Quick Lesson Plan
This lesson was designed to address the required skills on the AP Calculus Test. Finding an inverse function and then differentiating that inverse is not a component of the test. Instead, students need to have knowledge of the relationship between a function and its inverse (domain and range values, for example) and of the relationship between their derivative values at reflected points (the derivatives are reciprocals). In today’s lesson, students advance through the basic characteristics of inverse functions (comparing graphs and predicting ordered pairs) and finally discover on their own the relationship between the derivative of a function and the derivative of its inverse.
Materials: Each student receives a card containing a graph and a table with selected values of the derivative. The student’s partner receives a card containing the inverse graph and a table with selected values of the derivative of the inverse. Students need both cards to complete the activity.
After completing today’s activity, students will have uncovered the relationship between derivative values of a function and its inverse. We have postponed using formal notation in the activity and desired, instead, to develop an intuitive understanding of the reciprocal relationship between the derivatives.
Formal notation is introduced in the Important Ideas section. Careful attention must be paid to the input values for each expression. A graphical representation of a function and its inverse can help students understand why the input values are different and where they originated.
The first problem in Check Your Understanding has an inverse that is relatively easy to find. Students who are unsure of how to use the formula for derivatives of inverses could determine the inverse function and its derivative. Then, using a point on the original graph (x = 0 and y = 6, for example) and its reflected point (x = 6 and y = 0), they could evaluate both derivatives and confirm the reciprocal relationship.
Great care was given to the problems chosen for the Check Your Understanding section. These examples are designed to mimic expectations on the AP Test. Your students will be better prepared for the test if they can confidently answer these questions. Access question 3(d) from the 2007 AP Calculus AB Test.
The notation for inverses and derivatives that is presented in textbooks and online sources can be challenging for students. Precise language on the teacher’s part can clear confusion caused by complex expressions. Reinforcing often the origin of each input value will increase student fluency navigating between the original function and the inverse function.