AP Calculus

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Special Focus: The Fundamental Theorem of Calculus antiderivatives differ only by a constant. A proof of the antiderivative part can also be given that uses the Mean Value Theorem for Definite Integrals. See Larson’s Calculus of a Single Variable 2 .) • Looking at each part of the theorem, can you explain why we say differentiation and integration are inverse processes? ∫ x ( d f ( t) dt = f ( x) , so differentiation after integration yields the original dx a b function. In reverse, we have ∫ F ′( x ) dx = F ( b ) − F ( a ), so by integrating the a derivative F ′ we get back to the function F (up to a constant).) • Can we solve the differential equation dF = f for every continuous function f? What about the initial value? dx x (Yes, the solution is F( x) = F( a) + ∫ f ( t) dt , where F( a) is the given a initial condition.) • How is the concept of limit, the foundation of both the integral and differential calculus, used in these proofs? (When we prove the antiderivative part, we use a limit as h → 0 to compute the derivative using difference quotients, and for the evaluation part via the Mean Value Theorem, we compute the limit of a Riemann sum as n → ∞.) Whichever proofs you choose, the consideration of ∫ f ( t) dt as a function can provide a opportunities for rich discussion. Identifying the domain of a particular function ∫ x a f ( t) dt, understanding that the Fundamental Theorem says that continuous functions are in fact derivatives, and realizing that the theorem shows that we can find solutions to initial value differential equations even when a closed form of the antiderivative is not available—each of these requires a level of understanding that goes beyond the rote. Helping students understand the proofs you choose can contribute to the goal of helping them reach the level of understanding and appreciation we hope for them to achieve. x 2 Ron Larson, et al., Calculus of a Single Variable: Early Transcendental Functions, 3 rd ed., Boston: Houghton Mifflin, 2003. AP® Calculus: 2006–2007 Workshop Materials 103
Special Focus: The Fundamental Theorem of Calculus Stage Five: Historical Background and More Advanced Examples Introduction Goals of Stage Five • Provide some historical background for the FTC • Provide additional examples of using the FTC later in the course Articles in Stage Five • “Why Do We Name the Integral for Someone Who Lived in the Mid-Nineteenth Century?” by David Bressoud • “Examples to Reinforce Concepts That Are Connected to the Fundamental Theorem of Calculus” by Steve Kokoska David Bressoud’s article provides some important background information about the historical importance of the FTC. Even experienced teachers may gain new insights about the relationship between familiar theorems and the definitions that mathematicians use. Steve Kokoska presents a variety of examples that show how the FTC occurs in subsequent calculus, physics, probability, and statistics classes. These examples may be used as the basis for individual or group projects. 104 AP® Calculus: 2006–2007 Workshop Materials
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AP Calculus

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