AP Calculus

Recommendations

Info

Special Focus: The Fundamental Theorem of Calculus Teacher Notes and Answers Class Management This activity takes less than one full class period to complete (30 to 35 minutes). Teachers should work through the activity once before doing it with students. Students could work independently or in groups. Calculus Prerequisites Before doing this activity, students should know: • That a definite integral is a number • The relationship between an integral and the area under the graph of a positive function Calculator Prerequisites A calculator is not required for this activity. If your students have never used their calculators to make a scatterplot, however, this activity could be used to introduce them to that functionality. This approach is illustrated in the answers. Preliminary Comments The primary purpose of this activity is to develop students’ familiarity with and understanding of a function defined by a definite integral, where the independent variable of the function is a a limit of integration. If f ( t) > 0 and a > 0, then ∫ f ( t) dt represents the area between the 0 x graph of y = f ( t) and the t-axis from t = 0 to t = a. When x > 0, A( x) = ∫ f ( t) dt can be 0 interpreted as the “area so far” function. It represents the area under the graph of f and above the t-axis from 0 up to the variable limit of integration, x. A secondary purpose is to foreshadow the Fundamental Theorem. See the answers to questions 13 and 14. The integrands in this activity are the simplest kind: constant functions. This enables students to evaluate the integrals by simply finding areas of rectangles. It would be useful at the end of the activity to ask students to express in words the connection they observe between the slope of the f and g functions and the integrand. For each unit that x increases, the area increases by an amount equal to the value of the integrand. AP® Calculus: 2006–2007 Workshop Materials 21
Special Focus: The Fundamental Theorem of Calculus Extending These Ideas If time permits, this activity can easily be extended by introducing simple linear functions as the integrands. Consider f ( x) = x, 2x, x + 2, 2x + 3, or similar linear functions with negative slopes. Students can evaluate f x x 0 ( ) = ∫ f ( t) dt and f x x f t dt 0 2 ( ) = ∫ ( ) for 2 specific values of x by computing the areas of triangles and trapezoids. Answers 1. 0 2. 1.5 3. 3 4. x f0 ( x ) = ∫ 1. 5 dt 0 x 0 0 1 1.5 2 3 3 4.5 5. 22 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 23: Special Focus: The Fundamental Theo
Page 75 and 76:
Special Focus: The Fundamental Theo
Page 77 and 78:
Page 79 and 80:
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?