AP Calculus

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Special Focus: The Fundamental Theorem of Calculus Stage Four: Confirming Conjectures with a Formal Proof Introduction Goals of Stage Four • Determine one or more suitable proofs to discuss with your students • Celebrate an important mathematical (and cultural) achievement with your students Article in Stage Four • “Proving the Fundamental Theorem of Calculus” by Lisa Townsley Teachers need to choose whether or not to prove the Fundamental Theorem in their classes. Furthermore, those who do choose to prove the theorem have many options to consider. This chapter suggests some strategies and choices for those who wish to share a formal or informal proof with their students. Some teachers may prefer to give a proof in class that is different from the one in their text. This might illustrate an alternative approach or a teacher’s favorite method of proof. AP® Calculus: 2006–2007 Workshop Materials 97
Special Focus: The Fundamental Theorem of Calculus Proving the Fundamental Theorem of Calculus Lisa Townsley Benedictine University Lisle, Illinois For the purposes of this paper, we refer to the Fundamental Theorem by its parts: • Antiderivative part of the FTC: d x If f is continuous on an open interval containing a, then f t dt f x dx ∫ ( ) = ( ). a • Evaluation part of the FTC: If f is continuous on [a,b], and F is any antiderivative of f, b then f ( x) dx = F( b) − F( a). ∫ a Students sometimes employ circular reasoning when trying to understand the Fundamental Theorem. They see the antiderivative part of the FTC as saying, “Well, you start with a function, take an antiderivative and evaluate at x, then differentiate with respect to x. Naturally, you end up back at the original function!” Here they have absorbed the evaluation part of the theorem as a process and think of the antiderivative part in terms of that process. Of course, you need the very theorem you are proving to know that there is indeed an antiderivative to employ, but the student usually misses the circular nature of this reasoning. There are several choices on how to prove the two parts of the Fundamental Theorem. An informal survey of 11 frequently used texts reveals two basic proofs for each part of the theorem. Many proofs of the Fundamental Theorem invoke the Mean Value Theorem at some point. In this paper, we offer a proof of the antiderivative part of the FTC with the evaluation part of the FTC as its consequence and also an alternative proof of the evaluation part of the FTC. In order to prove the antiderivative part of the FTC, it is worthwhile to spend some class time on understanding what the theorem is saying. A discussion of all the “letters” x present in the expression F( x) = ∫ f ( t) dt will help students to understand F(x) as a a function. Some questions to ask are: • What is the input variable for F(x)? • What is the domain of the function F? • What is a picture of a computation of F(x) for a specific f, a, and x? • Where is t in this picture? • As x varies, how does the picture change? • Why is f required to be continuous? 98 AP® Calculus: 2006–2007 Workshop Materials
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