AP Calculus

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Special Focus: The Fundamental Theorem of Calculus Developing and Understanding the Fundamental Theorem of Calculus Caren Diefenderfer, Editor Hollins University Roanoke, Virginia Numerous problems involving the Fundamental Theorem of Calculus (FTC) have appeared in both the multiple-choice and free-response sections of the AP Calculus Exam for many years. AP Calculus students need to understand this theorem using a variety of approaches and problem-solving techniques. Before 1997, the AP Calculus questions regarding the FTC considered only a limited number of variations. Traditional problems are often procedural: area, volume, or distance questions or those that check a student’s mechanical ability to find the derivative of a function defined by an integral. In addition, we now assess the theorem from a wider variety of settings. These materials provide different ways that teachers can help students understand these settings. We hope the following chapters will help AP Calculus teachers think about how to organize and present the material on the FTC in ways that will encourage students to discover important principles, foster critical thinking, and build confidence. The 2006–2007 AP Calculus Course Description includes the following item: Fundamental Theorem of Calculus • Use of the Fundamental Theorem to evaluate definite integrals. • Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined. Calculus texts often present the two statements of the Fundamental Theorem at once and refer to them as Part I and Part II, although even the most popular texts do not agree on which statement is Part I and which is Part II. In these materials, we refer to the two parts of the Fundamental Theorem of Calculus in the following manner: Evaluation part of the FTC: If f is continuous on [a, b], and F is any antiderivative of f, then ∫ b a f ( x) dx = F( b) − F( a). AP® Calculus: 2006–2007 Workshop Materials 3
Special Focus: The Fundamental Theorem of Calculus Antiderivative part of the FTC: d x If f is continuous on [a, b], then f t dt f x dx ∫ ( ) = ( ) for every x in [a, b] a In addition to explaining and expanding on these two parts, we showcase the approaches that several experienced calculus teachers successfully use in their classrooms. We discuss multiple approaches for introducing the FTC and include some calculator activities that will help students to learn important concepts that are connected to the FTC. Experienced teachers agree that important topics require careful planning and subtle repetition of key concepts. The primary goal of these materials is to assist AP Calculus teachers in creating classroom activities that will help students to probe, discover, question, and master the FTC and its applications. During the process of writing and editing these workshop materials, we found it necessary to articulate some organizing principles to present the work of individual writers in an order and manner that will be helpful to teachers of AP Calculus. J. T. Sutcliffe articulated these developmental stages, and we have used her suggestions to guide this work. We thought carefully about the stages through which many students pass between being introduced to a new concept and achieving competency with the concept. Although we strongly urge each teacher to create lesson plans that meet the needs of his or her students, we selected the following scenario to organize our materials. Stage One: Exploring and Developing Interesting Results That Lead to Conjectures Stage Two: Refining the Level of Understanding and Making Conjectures Stage Three: Assessing and Improving Student Understanding and Confidence Stage Four: Confirming Conjectures with a Formal Proof Stage Five: Historical Background and More Advanced Examples 4 AP® Calculus: 2006–2007 Workshop Materials
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AP Calculus

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