Graphing f '(x):

Increasing, Decreasing, Concavity

This activity is appropriate for a calculus or analysis class. It is designed to follow the activities From Secant Lines to Tangent Lines and Slopes and Derivatives, but can be easily adapted to stand on its own.

Using a dynamic tangent line and the value of the tangent line's slope, students analyze the graph of a function, determining its absolute and local extrema, intervals of increasing and decreasing, concavity, and inflection points. Throughout the activity, as students gain more information about the nature of the first derivative of the function, they draw and refine a sketch of f '(x).

Mathematics:  Students investigate the meanings of and relationships between a function's first derivative, absolute and local extrema, intervals of increasing and decreasing, intervals of concave up and concave down, and inflection points.

Mathematical Thinking:  Students make conjectures and predictions and use graphical reasoning skills.

Technology:  The Geometer's Sketchpad 4.0.