AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
Graph of f(t)
Question 1
To answer question 1 correctly, students must understand that
∫
3
∫
4 4
f ( t) dt = f ( t)
dt = − . This level of understanding was indicated as a prerequisite
2
3 3
b
above (i.e., ∫ f ( x)
dx takes on a negative value that is the additive inverse of area when
a
f ( x) ≤ 0 on the interval [a,b]), but teachers should be alert for students who have not yet
achieved this level of understanding. If a student needs additional help with this concept,
teachers have several options. Among them are:
• If the class has investigated a velocity context such as the one Ray Cannon
introduced, f could be identified as a velocity function, and students could
rethink why these integrals have a negative value. For example: During the first 2
seconds, a car moves away from a point of reference, thus increasing its distance
from that point. In these last 2 seconds, the car moves back toward the point of
reference, and therefore its distance from that reference point is decreasing.
• If the class has investigated Riemann sums, the students could partition the
interval [0,4] into eight subintervals and sketch a left- or right-hand Riemann
sum. It should then be clear that the products of the form f ( x) × ∆ x are
positive on [0,2] and negative on [2,4].
AP® Calculus: 2006–2007 Workshop Materials 43