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Transcript and student work are also available.

Robert B. Davis Institute for Learning. (2003). B47, The fundamental theorem of calculus, Session 2 of 2: Discussing why the antiderivative can be used to calculate the area under a graph (student view), College, July 24, 2003, raw footage [video]. Retrieved from

B47-20030724-KNWH-SV-IFML-GR15-CALC-FTC-RAW

This video presents one view of the second and last session in which seven post-high school students discuss the Fundamental Theorem of Calculus. This session took place approximately one month after the first session of this series. In this view, three of the students, Angela, Magda, and Romina, continue discussing ideas related to the problem introduced in the previous session. Specifically, they discuss why antiderivatives can be used to calculate the area under a graph.The students begin by drawing graphs of the function y = x^2 and its antiderivative y=(1/3) x^3. They discuss the fact that the slope of the integral graph increases as the function value increases because larger and larger areas are being accumulated for each successive interval, while if the function is positive but decreasing, the slope of the integral function is positive but decreasing since smaller and smaller areas per interval are being accumulated. The students suggest that accumulation may be the reason why an integral is a "higher power" than its derivative function.After the researcher displays a graph of a function and its accumulated area function using Geometer’s Sketchpad, the students discuss the difference between antiderivatives and integrals. Magda explains that antiderivatives can be vertically translated, demonstrating translations using varying “initial conditions” of antiderivative functions. The students note that the derivative of the integral on the interval [a, x] is always the same, irrespective of the value of a, since the slope of the accumulated area graph is not affected by its vertical position. They further investigate the relationship between accumulated area and antiderivatives by recreating a graph of the sine function which they recognize as the function of negative cosine. Next, the students discuss why the slope of the graph of the integral on the interval [a, x] represents the rate at which the area under f is accumulating. Interpreting the integral as a graph of accumulated area and explaining that the rate of accumulation of area determines the slope of the integral graph, they argue that the slope of the graph of the integral is determined by the value of f since the graph of the integral increases by the value of the area under the graph of f. Therefore, they explain that the average rate of change of the integral is determined by the value of f.

doi:10.7282/T3HM56MD

The video is protected by copyright. It is available for reviewing and use within the Video Mosaic Collaborative (VMC) portal. Please contact the Robert B. Davis Institute for Learning (RBDIL) for further information about the use of this video.