AP Calculus

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Table of Contents Special Focus: The Fundamental Theorem of Calculus Developing and Understanding the Fundamental Theorem of Calculus Caren Diefenderfer ..............................................................................................................................3 Stage One: Exploring and Developing Interesting Results That Lead to Conjectures Introduction .....................................................................................................................................................6 Key Ideas that Help Students Understand the Fundamental Theorem Steve Olson ...............................................................................................................................................7 Functions Defined by a Definite Integral Mark Howell ......................................................................................................................................... 16 Stage Two: Refining the Level of Understanding and Making Conjectures Introduction .................................................................................................................................................. 25 From Riemann Sums to Net Change Ray Cannon ........................................................................................................................................... 27 The Integral Function—Class Worksheet Benita Albert ........................................................................................................................................ 33 The Integral Function—Teacher Notes J. T. Sutcliffe ........................................................................................................................................... 41 Stage Three: Assessing and Improving Student Understanding and Confidence Introduction .................................................................................................................................................. 57 Functions Defined by Integrals Ray Cannon ........................................................................................................................................... 58 Exploring the FTC from Numerical and Graphical Points of View Mark Howell ......................................................................................................................................... 62 Using the Fundamental Theorem of Calculus in a Variety of AP Questions Larry Riddle .......................................................................................................................................... 76 AP® Calculus: 2006–2007 Workshop Materials 1
Table of Contents Stage Four: Confirming Conjectures with a Formal Proof Introduction .................................................................................................................................................. 97 Proving the Fundamental Theorem of Calculus Lisa Townsley ....................................................................................................................................... 98 Stage Five: Historical Background and More Advanced Examples Introduction ............................................................................................................................................... 104 Why Do We Name the Integral for Someone Who Lived in the Mid-Nineteenth Century? David M. Bressoud ......................................................................................................................... 105 Examples to Reinforce Concepts That Are Connected to the Fundamental Theorem of Calculus Steve Kokoska ................................................................................................................................... 112 Contributors ............................................................................................................................................... 126 Contact Us ................................................................................................................................................... 128 Important Note: The following set of materials is organized around a particular theme, or “special focus,” that reflects important topics in the AP Calculus course. The materials are intended to provide teachers with resources and classroom ideas relating to these topics. The special focus, as well as the specific content of the materials, cannot and should not be taken as an indication that a particular topic will appear on the AP Exam. 2 AP® Calculus: 2006–2007 Workshop Materials
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AP Calculus

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