AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
The minimum value of the function F occurs at what x-value(s)?
The minimum value for F is 0, and it occurs at both x=0 and at x=4.
An example of a student’s analysis using velocity:
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. The original displacement is 0 because
the object hasn’t yet moved anywhere. At the end of 4 seconds, the object
would be back where it started because it moved the same distance forward
as backward. This means that once again, the displacement would be 0. For
all other values of x between 0 and 4, the displacement would be positive.
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. When x=0, I am evaluating
0
∫ f ( t) dt = 0.
When x=4, I have added positive products to get larger
0
values until I reach the sum of 8/3 for x=2. Then I begin to subtract values
because the products are negative. From x=2 to x=4, I have subtracted off
8/3 because of the symmetry of the graph. This means that I have gotten
back to a net sum of 0. Since I never subtracted off more than I had added
on, my answer was never negative.
Function F increases on what interval(s)? Decreases on what interval(s)?
An example of a student’s analysis using velocity:
If I think of f as a velocity function, an object is moving forward (velocity
positive) between t=0 and t=2. This means that the function describing
displacement of the moving object increases on this interval. Another way
to say this is that if the object is moving along a horizontal line, it is moving
further and further to the right for these two seconds. However, between
t=2 and t=4, the object is moving in reverse because the velocity is
negative. So, the displacement function is decreasing on [2,4]; between t=2
and t=4 the object is moving left on the horizontal line I am imagining.
AP® Calculus: 2006–2007 Workshop Materials 49