AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
Evaluation Step 2: Understanding that the limiting value of a Riemann sum is a
definite integral
After the idea of summing products to approximate total change has had a few days
2
to sink in, I begin a class by giving the students the formula s( t) = t + t + 20 for the
position of a car in feet as a function of time t, for t=0 to t=10 seconds, and have the
students answer the following questions.
(a) What is the position of the car at t=0?
(b) What is the position of the car at t=10?
(c) What is the change in position of the car from time t=0 to time t=10?
(d) Were these questions easy or difficult for you?
Most students find these questions easy to answer and wonder why these questions were
asked. I ask for their patience and suggest that we try to generalize the processes they
have been using to answer recent textbook questions.
We start by reviewing the Step 1 exploration. This allows us to have a concrete example to
refer to when necessary. Next, however, I guide them through a more general analysis of
what we did. The details follow:
We began by estimating the change of position from time t=0 to t=10 by dividing the
time interval into five subintervals, each with ∆t = 2 . Using general notation, we wrote
Change in position ≈ v( t ) ⋅ ∆t + v( t ) ⋅ ∆t + v( t ) ⋅ ∆t + v( t ) ⋅ ∆t + v( t ) ⋅ ∆t .
1 2 3 4 5
We can express this even more generally by partitioning the time interval into n
pieces, writing
Change in position ≈ v( t1 ) ⋅ ∆t + v( t2
) ⋅ ∆t + ⋅⋅⋅ + v( tn
) ⋅ ∆ t , where ∆t = 10 .
n
Next, I usually ask how we make our estimate even more accurate, and almost always
someone suggests a limit. We agree to write
Change in position = lim v ( t ) ⋅ ∆t + v ( t ) ⋅ ∆t + ⋅⋅⋅ + v ( t ) ⋅ ∆ t .
n
( )
1 2
n
→∞
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AP® Calculus: 2006–2007 Workshop Materials