AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
series we sometimes find collapsing sums, here we have a difference of just two terms that
expands instead. If we have a partition
a = x < x < x < ... < x < x = b
0 1 2 n−1
of [ a, b ] into n subintervals of width ∆x, then
F ( b ) − F ( a ) = F ( xn ) − F ( x0 ) = F ( xn ) − F ( xn−1 ) + F ( xn−
1 ) − F( xn−2) + ... + F( x1) − F( x0
).
Use the Mean Value Theorem on each pair to replace the difference F ( xi
) − F ( xi−1 ) with
F′ ( ci )( xi − xi−
1) = f ( ci
) ∆ x , where c i
is in the interval ( x i−1
, xi)
. Summing these and
applying the limit as n → ∞, which is equivalent to ∆x → 0, yields the definite integral form:
n
b
∑ i ∫
∆x→ 0
a
i = 1
F( b) − F( a) = lim f ( c ) ∆x = f ( x) dx.
A nice feature of proving the evaluation part of the Fundamental Theorem first is
that the students’ confusion about the dummy variable t versus the independent
variable x is postponed until you approach the antiderivative part. The students may
encounter difficulty, however, in interpreting an expanding sum and understanding
the introduction of an index i and an indeterminate upper limit n of the sum.
Once you have completed the proof of the evaluation part of the Fundamental Theorem,
you have several options on how to handle the antiderivative part of the theorem.
One approach is discussed in the article “Functions Defined by Integrals.” A formal
proof can also be given using the definition of the derivative as explained above.
Here are some questions to ask in class after the completion of the proofs of the two
parts of the Fundamental Theorem.
• At what point(s) in the proofs did we need the fact that f is continuous?
(We need f continuous in the evaluation of limits when using the
Squeeze Theorem.)
• At what point(s) in the proof did we use the Mean Value Theorem?
(For most proofs of the evaluation part of the FTC, you use the Mean
ValueTheorem on F ( xi
) − F ( xi−1 ) in the Riemann sum, or you use the
corollary about functions with the same derivative to deduce that two
n
102
AP® Calculus: 2006–2007 Workshop Materials