Justify conclusions about the behavior of function based on its second derivative
I can use a second derivative to determine the concavity of a function.
I can find points of inflection using the second derivative.
I can use the Second Derivative Test to determine if a critical point is a maximum, minimum, or neither.
Quick Lesson Plan
Today’s activity illustrates the connections between function behavior and characteristics of the first and second derivatives. After the Activity and Debrief segments, students will have developed a conceptual understanding of first derivative relative extrema, inflection points, concavity, and the associated relative extrema on the original function. Additionally, students will see how concavity and the first derivative converge in the second derivative test for relative extrema.
Encourage students to graph the function f(t) (which counts the number of people who have contracted the flu) simultaneously with its first and second derivative. Students will see both a numerical and graphical representations of the interplay between a function and its derivatives. This visual reference will help students make conjectures that are confirmed and formalized later.
Consider having your students work through free response question #6 from the 2007 AB Calculus Test. The question requires discussion of inflection points and relative extrema justified with the second derivative test. This is an excellent vehicle to assess student understanding!
One of the most commonly mistaken justifications produced by students involves the identification of an inflection point (a POI). Simply stating a second derivative equals zero is not sufficient to prove the existence of a POI. Students must show that the second derivative changes sign to justify a POI. This is similar to the false argument that relative extrema must exist wherever the first derivative equals zero. Remind students often that they must show the appropriate sign change to name a relative max or min as well as a POI.