AP Calculus

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Special Focus: The Fundamental Theorem of Calculus You could repeat this extension, using the outputs from the function defined in L4 instead of L3. That is, form the difference quotients for L4 this time and store to L6. The results should be identical, since L4 and L3 differ by a constant. In fact, you could calculate just what that constant is in two different ways: by subtracting the lists or by evaluating Y2(1) (which is, of course, an integral). See the screen shot below. AP® Calculus: 2006–2007 Workshop Materials 75
Special Focus: The Fundamental Theorem of Calculus Using the Fundamental Theorem of Calculus in a Variety of AP Questions Larry Riddle Agnes Scott College Decatur, Georgia Most students have no problem using the Fundamental Theorem of Calculus to evaluate a definite integral when they are able to find an antiderivative. Indeed, for many of them this is just an algorithm, not a “theorem.” In the examples below, therefore, we use the Fundamental Theorem to solve other types of problems. We use the version of the Fundamental Theorem in the form b a ∫ f ′( t) dt = f ( b) − f ( a), or, in words, that the definite integral of a rate of change gives the total change. An equivalent expression is b a f ( b) = f ( a) + ∫ f ′( t) dt , which allows us to compute the value of a function given an initial value and the rate of change. These two ideas appear frequently in AP Calculus questions. Compute position given velocity, or compute velocity given acceleration, or find the change in the amount of water in a tank given the rate at which the water has leaked from the tank, or find the temperature of an object given the rate at which the object is cooling or heating. The second version of the FTC says that d t f x dx f t dt ∫ ( ) = ( ). a In both versions, we assume all necessary differentiability and continuity conditions. 76 AP® Calculus: 2006–2007 Workshop Materials
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AP Calculus

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