AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
Using the Fundamental Theorem of Calculus in a Variety
of AP Questions
Larry Riddle
Agnes Scott College
Decatur, Georgia
Most students have no problem using the Fundamental Theorem of Calculus to evaluate
a definite integral when they are able to find an antiderivative. Indeed, for many of them
this is just an algorithm, not a “theorem.” In the examples below, therefore, we use the
Fundamental Theorem to solve other types of problems.
We use the version of the Fundamental Theorem in the form
b
a
∫ f ′( t) dt = f ( b) − f ( a),
or, in words, that the definite integral of a rate of change gives the total change. An
equivalent expression is
b
a
f ( b) = f ( a) + ∫ f ′( t)
dt ,
which allows us to compute the value of a function given an initial value and the rate of
change. These two ideas appear frequently in AP Calculus questions. Compute position
given velocity, or compute velocity given acceleration, or find the change in the amount
of water in a tank given the rate at which the water has leaked from the tank, or find the
temperature of an object given the rate at which the object is cooling or heating.
The second version of the FTC says that
d t
f x dx f t
dt
∫ ( ) = ( ).
a
In both versions, we assume all necessary differentiability and continuity conditions.
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AP® Calculus: 2006–2007 Workshop Materials