AP Calculus

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Special Focus: The Fundamental Theorem of Calculus 0 ⎛ 2 2 T ⎞ 1 ⎛ T ⎞ Note that the first element of L4 represents Y3( 0) = ∫ cos⎜ ⎟ dT = − cos⎜ 1 2 ∫ ⎟ dT . 0 ⎝ ⎠ ⎝ 2 ⎠ 11. Why is the graph of L4 versus L1 (Plot2) below the graph of L3 versus L1 (Plot1)? (Remember, the only difference between the two is the lower limit of integration.) _______________________________________________________________ _______________________________________________________________ 12. It appears that Plot1 and Plot2 differ by a constant, i.e., that Y3(X)-Y2(X) is the same for all values of X. Find the value of this constant. _______________________________________________________________ (You could calculate this in several different ways. If you get stuck, try expressing Y3(X) - Y2(X) in terms of an integral.) 13. Explain why plots 1 and 2 have the same shape. _______________________________________________________________ _______________________________________________________________ Look at the table of values for L3 and L4, and notice that the locations of the extrema are the same for both functions (as you would expect from an inspection of the scatterplots). 14. Look at the locations of the local minima and the local maximum on the graph of Y1(X). At these values of X, describe what appears to happen with the concavity of the graphs from Plot1 and Plot2. _______________________________________________________________ _______________________________________________________________ AP® Calculus: 2006–2007 Workshop Materials 67
Special Focus: The Fundamental Theorem of Calculus 15. Write an equation with a derivative on one side that shows how Y1(X) and Y2(X) are related. _______________________________________________________________ 16. Recall that the derivative of a function f at x=a can be defined by f x − f a f ′( a) = lim ( ) ( ) . x→a x − a Approximate Y2'(1) by evaluating the expression(Y2(1.1)–Y2(1))/0.1. _______________________________________________________________ Your answer should be close to the value of Y1(1). Why? _______________________________________________________________ _______________________________________________________________ 17. Approximate Y3'(1)by evaluating the expression (Y3(1.1)–Y3(1))/0.1. _______________________________________________________________ Again, your answer should be close to the value of Y1(1). Why? _______________________________________________________________ _______________________________________________________________ 68 AP® Calculus: 2006–2007 Workshop Materials
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AP Calculus

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