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Robert B. Davis Institute for Learning. (2003). B45, The fundamental theorem of calculus, Session 1 of 2: Discussing the meaning of the theorem (student view), College, June 25, 2003, raw footage [video]. Retrieved from

B45-20030625-KNWH-SV-IFML-GR15-CALC-FTC-RAW

This video presents one camera view from a post-high school session conducted with seven students who participated in the Rutgers longitudinal study. The session, the first of two in which the students discuss the Fundamental Theorem of Calculus, took place during the summer of 2003, four years after the students had taken an AP Calculus course in high school. It was conducted by researcher Ralph Pantozzi who had taught the participating students AP Calculus. In this view, three of the students, Angela, Magda, and Romina, work together on the problem described in the following prompt:A current student of calculus has asked you to help them understand the Fundamental Theorem of Calculus. The student, knowing that you have taken calculus in the past, is interested in what the theorem means, what the theorem is "for," and why the theorem is true. You have agreed to the student's request. In preparing to respond to the student, you can use any materials that you feel would be helpful to you, including textbooks, other calculus resource materials, specific calculus questions, calculators, and computers. You may also discuss your ideas with the other students here with you today. In this session, the group discusses the meaning of the Fundamental Theorem of Calculus. They note the theorem defined in a provided textbook, identifying the function f as the derivative of the function g. The students then discuss the meaning of statement, using the term “integral” in several ways: to refer to the graph of a function depicting the accumulated area between the function f and the x axis, to refer to the area under the graph of a function, and as the antiderivative of a function. They explain the notation for the left side of the equation denoting the definite integral on the interval [a, b] as the area under a graph of a function from point a to point b, and interpret the notation for the right side of the equation denoting the difference between the antiderivatives of b and a as the difference between two areas as well as the difference between two y values of the “integral” function depicting accumulated area under the function f. To check their understanding of the theorem, the group plots sample equations to check that they behave in the ways they predict. They plot the accumulated area under the graph for the function f(x) = x^2 and confirmed that the equation y=1/3x^3 is valid for representing its accumulated area. They explain that the Fundamental Theorem of Calculus means that the area under the graph of f(x) can be calculated by finding the difference between two values of its antidervative function.

doi:10.7282/T3HQ3X26

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.