AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
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 F is increasing on the
interval [0,2], and F is decreasing on the interval [2,4].
Question 6
Without the aid of the antiderivative part of the Fundamental Theorem, it is only
hoped that students will notice that the graphs differ (if at all) in their vertical
placement on the grid; that is, they are vertical displacements of one another. One or
more students may even propose that the functions have the same slope values; that
is, F′ ( x) = G′ ( x) = H′
( x). If this happens, the student has provided the perfect segue
into a more formal discussion of the antiderivative part of the Fundamental Theorem.
Here again, teachers may want to return to this worksheet to help their students see and
understand the conceptual groundwork being laid in these six questions.
Question 7
This question allows students to extend their understanding of the Fundamental
Theorem, but it also requires that they understand composite functions. The correct
graph and a brief commentary for each follows.
(a) Since F ( x) = f ( t) dt = − f ( t) dt = − F( x),
1
0
x
x
∫ ∫ students should be able to
determine the graph of F 1
shown below.
0
The graph of F 1
on [0,4]
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AP® Calculus: 2006–2007 Workshop Materials