SECTION 2.8 THE DERIVATIVE AS A FUNCTION
GROUP WORK 3: THE DERIVATIVE FUNCTION
Give each group of between three and five students the picture of all eight graphs. They are to sketch the derivative functions by first estimating the slopes at points, and plotting the values of f x . Each group should also be given a large copy of one of the graphs, perhaps on acetate. When they are ready, with this information they can draw the derivative graph on the same axes. For closure, project their solutions on the wall and point out salient features. Perhaps the students will notice that the derivatives turn out to be positive when their corresponding functions are increasing. Concavity can even be introduced at this time. Large copies of the answers are provided, in case the instructor wishes to overlay them on top of students‘ answers for reinforcement. Note that the derivative of graph 6 (y e x ) is itself. Also note that the derivative of graph 1 (y cosh x ) is not a straight line. Leave at least 15 minutes for closure. The whole activity should take about 45–60 minutes, but it is really, truly worth the time.
If a group finishes early, have them discuss where f is increasing and where it is decreasing. Also show that where f is increasing, f is positive, and where f is decreasing, f is negative.
ANSWER (larger answer graphs are included after the group work)
Graph 1
Graph 2
Graph 3
Graph 4
Graph 5
Graph 6
Graph 7
Graph 8
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