# CA III.3 Analysis of Functions

**Purpose**: To formalize our work on optimization by introducing students to some of the language that surrounds these problems and asking them to specifically use the first derivative and second derivative to justify their optimization.

**Classroom Procedure**: It is important that each question in this activity is discussed in class. It generally takes two days to complete in class with students completing some work between classes. Students struggle with Q1 and need time and discussion on this question.

**Ideas this Activity Builds On**: Students have been completing optimization questions informally throughout the previous module. This is activity builds on that informal development and pushes students a little further than it to make sure that they are considering all of the subtleties.

**Introduction/Motivation of the Activity**: To further our work on optimization and introduce some language that they will need going further.

**Need to Establish by the End of Activity/Wrap-Up**: The language of optimization, critical points, inflection points, global max/min, local max/min and emphasizing how to test if something is a max or min.

**Additional Notes**: There are no synthesis questions for this activity. In many ways the next activity serves as the synthesis for this activity.