Determine critical points of implicit relations.
Justify conclusions about the behavior of an implicitly defined relation based on evidence from its derivatives.
I can find the critical point of an implicit relation.
I can use derivatives to justify conclusions about implicitly defined functions.
Quick Lesson Plan
This lesson provides an engaging exploration of the meaning and application of the derivatives of implicitly-defined relations. The activity is initiated with the graphical interpretation of an interesting relation (that is definitely not a function!), and then guides students to connect the first and second derivatives to the behavior of the relation.
Encourage groups to attempt all six questions in the activity (if needed, be ready to guide student thinking on questions 4 - 6 toward a discussion of behavior in a quadrant). They should easily recall that a point on a function where the first derivative equals zero or does not exist is a critical point of the function. By comparing their analytic work with the graphed relation, students are led to discover that this relationship exists within an implicit relation, as well! As written in the margin notes, be sure students define values for x and y, when necessary, to describe where an implicit relation is increasing or decreasing (or concave up, concave down, etc.). Before listing critical values, however, students need to verify that the intended point exists on the given relation (see Activity question 4 and Check Your Understanding question 1a).
Implicit differentiation appears on both parts (multiple choice and free response) of the AP Test. For an excellent overview of skills related to implicit differentiation, have students attempt these questions from previous AP Tests: 2016 BC 4a, 2015 AB 4b, 2015 AB 6 (all parts)
Although mystifying to many calculus teachers, it is quite possible that your students are now more adept with their calculus skills than other necessary algebra and trigonometry skills. Solving for dy/dx may be a bigger challenge than actually developing a first or second derivative! Of course, accurate algebra skills are a must. Continue to demand that students flex their Algebra 2 and Precalculus skills!