Thursday, February 18, 2010
calculus: cumulative area function
Download Sketch (GSP5 file): Calculus_CumulativeArea_7.zip
Download Sketch (GSP4 file): Calculus_CumulativeArea_8.zip
Download Worksheet (.doc file): GSPActivity_CumulativeArea.doc
This sketch offers a look at a simple antiderivative. The sketch shows a shaded region below a continuous "function" defined by seven line segments. The region can be adjusted by moving any of the eight independent points, and the arrangement of segments will always be a function. That is, the seven segments which define the area connect the eight independent points according to the order of their abscissae. Each segment represents a quadratic or linear increase in the cumulative area function, which is represented as a continuous string of parabolic loci.
My calculus students seemed to enjoy this activity. I took them to the computer lab to work with this sketch and worksheet after a discussion of an AP Calculus free-response question (2005B, #4) which deals with similar concepts.
Update 1: I was not initally able to make this GSP4-compatible since the construction relied on a new feature of GSP5, namely the ability to find an intersection of a line and a function. I was able to find a way around it and the GSP4 file is now available too.
Update 2: I have now included a worksheet (.doc file) in the zip file for the GSP5 version. The GSP5 sketch now includes a set of buttons to view eleven different graph overlays. Try to move the points around so that the graph of the cumulative area function matches the picture.
Update 3: I have put the graph overlay feature in the GSP4 file as well.
Update 4: I have now put a Hide/Show button for the cumulative area function in both versions of the sketch. Additionally, I have changed two of the questions on the worksheet.
Posted by Nate