Justify conclusions about the behavior of function based on its derivative.
I can consider both relative extrema and endpoints to determine absolute extrema.
I can justify an absolute maximum or minimum value.
Quick Lesson Plan
Students will formalize the circumstances for and the process required when investigating function values at endpoints. Today’s lesson is straightforward and intuitive for most students. After identifying critical values along the x-axis and finding the associated relative extreme values, students investigate the idea of analyzing function behaviors at endpoints to determine absolute (global) extreme values.
Study the scoring guidelines for free response question 2018 AB 5c to better teach students the best way to structure their solutions and justifications. Clarify for students that the Candidates Test is used for locating absolute (global) extreme values on a closed interval. (The first derivative test is reserved for relative extrema.)
An excellent example of the Candidates Test is found in 2018 AB 5c. It is interesting to note that students are able, even at this early point in their studies, to complete the entire AP question!
The need for correct and precise language is evident in these last lessons of Unit 5. Students who write “x = 3 is a local min because the slope changes there,” or similar vague statements, miss the opportunity to earn full points on our assignments and assessments. A careful statement is required. “The function f(x) has a local minimum of 6 at x = 3 where f’(x) changes sign from negative to positive” accurately describes the function value in addition to the x-value where that value occurs. Terms such as “the slope” or “the function” are no longer allowed: students must specifically name all functions and their derivatives.