AP Calculus

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Special Focus: The Fundamental Theorem of Calculus 4. From the list of data values stored in L3, for what value of X (stored in L1) X ⎛ 2 T ⎞ does Y2( X) = ∫ cos⎜ ⎟ dT appear to reach its first local maximum? 0 ⎝ 2 ⎠ _______________________________________________________________ 5. Explain why the twentieth value in L3 is smaller than the nineteenth value in 1 L3. That is, why is . 9 ⎛ 2 T ⎞ 1 cos dT . 8 ⎛ 2 T ⎞ ∫ ⎜ ⎟ < cos dT 0 2 ∫ ⎜ ⎟ ? 0 ⎝ ⎠ ⎝ 2 ⎠ _______________________________________________________________ _______________________________________________________________ _______________________________________________________________ 6. From the list of data values stored in L3, for what positive value of X does X ⎛ 2 T ⎞ Y2( X) = ∫ cos⎜ ⎟ dT appear to reach its first local minimum? 0 ⎝ 2 ⎠ _______________________________________________________________ Select 1:Plot1 from the STAT PLOT menu and define Plot1 as a scatterplot of L3 versus L1 as shown in the screen shot below. Select Plot1 and Y1 for graphing. 7. Select 2:zero from the CALC menu and find the two smallest positive X-intercepts on the graph of Y1, indicated by A and B in the screen shot above. (A)_________________________ and (B) ________________________ AP® Calculus: 2006–2007 Workshop Materials 65
Special Focus: The Fundamental Theorem of Calculus 8. Compare the answers to 7(A) and 7(B) with the answers to numbers 4 and 6 respectively. Explain why they are similar. _______________________________________________________________ _______________________________________________________________ 9. Find the next positive X-intercept of the graph of Y1(X) (close to X = 4)_____ 10. What happens with Y2(X) at the point found in number 9? _______________ _______________________________________________________________ (Verify your answer numerically by looking at the table of values for Y2(X)in L3.) Go back to the Y= editor and define Y3=fnInt(Y1(T),T,1,X,.1), thus changing X ⎛ 2 T ⎞ the lower limit of integration from 0 to 1. So we have Y3( X) = ∫ cos⎜ ⎟ dT . Evaluate 1 ⎝ 2 ⎠ Y3 at all the inputs in L1, and store the results in L4. See the screen shots below. Define Plot2 as a scatterplot of L4 versus L1, using the plus symbol as the Mark. Select Plot1, Plot2, and Y1 for graphing. Study the graph and table: 66 AP® Calculus: 2006–2007 Workshop Materials
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AP Calculus

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