AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
Students who have been successful in correctly completing the tables of questions 1, 2, and
3 and in plotting points on the grids will probably have little difficulty in completing this
table. Whether they are doing more than “connecting the dots” on the graphs will become
apparent as they try to defend some of their answers for question 4 in the next question.
Question 5
During class discussion or when walking around the room as students attempt to
explain their reasoning, teachers may discover that some students have correct answers
to question 4 alone because they have considered only integer values of x and thus are
not worrying about what happens between the integers. Teachers should help students
understand, for example, that just because the greatest table value occurred at x=2
doesn’t explain why the maximum value for F on [0, 4] must occur at x=2.
Students are only asked to support their answers for function F since they can provide
reasonable arguments to support their answers based upon the properties they have
learned previously or that they discovered as they worked through this exploration. They
have not been asked to support their answers for functions G and H because it is not
assumed that they have learned the antiderivative part of the Fundamental Theorem,
d x
which says that f t dt f x
dx
∫ ( ) = ( ). If the class has already studied the antiderivative
k
part of the Fundamental Theorem, they should be asked to support their answers for G
and H as well. Teachers of classes that have not yet considered the antiderivative part of
the Fundamental Theorem may want to return to this part of the worksheet after doing so.
The maximum value of the function F occurs at what x-value(s)?
An example of a student’s analysis using velocity:
The maximum value of 8/3 occurs at x=2. I thought of f as a velocity
function. The object moved forward (positive velocity) for 2 seconds
then moved backward (negative velocity) the same distance for the final
2 seconds. This means the maximum displacement occurred when the
direction of the object changed from moving forward to moving backward.
An example of a student’s analysis using Riemann sums:
I pictured a Riemann sum with tiny ∆t values. For F, I was adding positive
value products from t=0 to t=2, but then I started adding on negative
value products from t=2 to t=4. This means that the greatest sum would
have occurred at t=2.
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AP® Calculus: 2006–2007 Workshop Materials