Teaching Concavity

As you’ve probably noticed, different authors use different definitions based on how they plan to present topics later in their texts. So, the same concept seems to have different definitions. A definition in one book is a theorem in another. Students should be aware of this; looking at different definitions and related theorems about the…

Summer Fun

Every Spring I have a lot of fun proofreading Audrey Weeks’ new Calculus in Motion illustrations for the most recent AP Calculus Exam questions. These illustrations run on Geometers’ Sketchpad. In addition to the exam questions Calculus in Motion (and its companion Algebra in Motion) include separate animations illustrating most of the concepts in calculus…

Graphing with Accumulation 2

Accumulation 4: Graphing Ideas in Accumulation – Concavity In the last post we saw how thinking about Riemann sum rectangles, RΣR, moving across the graph of the derivative made it easy to see when the function whose derivative was given increased and decreased and had its local extreme values. Today we will consider concavity. Suppose…

Reading the Derivative’s Graph

A very typical calculus problem is given the equation of a function, to find information about it (extreme values, concavity, increasing, decreasing, etc., etc.). This is usually done by computing and analyzing the first derivative and the second derivative. All the textbooks show how to do this with copious examples and exercises. I have nothing…