I use this activity around the same time I teach graphing (how derivatives shape a graph) to help students solidify the relationship between f (x), f '(x) and f "(x). You’ll notice that many of the questions are actually the same concepts asked in different was. For example,
• For what x- value(s) is f (x) concave up?
• For what x- value(s) is f '(x) increasing?
• For what x- value(s) is f "(x) positive?
Students who recognize that these are the same questions will have a stronger grasp of these intricate relationships. Also, students will get practice with evaluating different functions. How many times do we see students looking at the given graph as if it was f (x) when actually the graph is of f '(x) ? This practice will help to address these issues.