AP Calculus

Recommendations

Info

Special Focus: The Fundamental Theorem of Calculus Stage Two: Refining the Level of Understanding and Making Conjectures Introduction Goals of Stage Two • Distinguish between displacement and distance traveled • Investigate more complicated integral functions • Discover and reinforce properties of the definite integral • Use the FTC to solve initial value problems Articles in Stage Two • “From Riemann Sums to Net Change” by Ray Cannon • “The Integral Function—Class Worksheet” by Benita Albert • “The Integral Function—Teacher Notes” by J. T. Sutcliffe Ray Cannon’s article, “From Riemann Sums to Net Change,” extends the simple setting of Stage One in several ways. The velocity is given by a formula, and the formula he considers, v( t) = 80 − 32 t on [0,3], gives both positive and negative velocity values. His exploration leads students to discover that definite integrals can give either displacement or distance traveled, depending on whether the integrand is velocity or speed. “The Integral Function—Class Worksheet” encourages students to experiment with functions defined by definite integrals. Students will sketch graphs and investigate x properties of functions of the form F( x) = ∫ f ( t) dx where f is given geometrically and a not analytically. Students will change the constant lower limit in the definite integral and create new, although similar, functions. Reflecting on the similarities will reveal important properties of the definite integral. Some teachers will use “The Integral Function—Class Worksheet” in order to help students discover properties of the definite integral using a geometric approach. Other teachers may prefer to discuss properties of the definite integral first and use this activity to reinforce the concepts. “The Integral Function—Teacher Notes” gives teachers some classroom guidelines and hints on using “The Integral Function—Class Worksheet.” In Steve Olson’s article in Stage One, the material in “Antiderivative Step 2” from the “Antiderivative Part of the Fundamental Theorem” section gives an important variation on the FTC that is helpful in solving initial value problems. AP® Calculus: 2006–2007 Workshop Materials 25
Special Focus: The Fundamental Theorem of Calculus Stage Two focuses on a variety of contexts, and each one is more sophisticated than the Stage One settings. It is important to notice the similarities between the simple examples of Stage One and the new settings in Stage Two. 26 AP® Calculus: 2006–2007 Workshop Materials
Page 1 and 2: AP ® Calculus 2006-2007 Profession
Page 3 and 4: The College Board: Connecting Stude
Page 5 and 6: Table of Contents Stage Four: Confi
Page 7 and 8: Special Focus: The Fundamental Theo
Page 27: Special Focus: The Fundamental Theo
Page 79 and 80:
Special Focus: The Fundamental Theo
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Contact Us College Board Regional O
show all

AP Calculus

Create successful ePaper yourself

Delete template?

Save as template?