AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
Now take the limit and use the Squeeze Theorem as before to obtain
F
lim ( x ) − F ( x − k ) f ( t)
dt
x−k
= lim
=
+ k
+ k
k→0 k→0
∫
x
f ( x).
Putting the two results together shows that
F x + h − F x
F′ ( x) = lim ( ) ( ) = f ( x).
h→0
h
The proof of the evaluation part of the Fundamental Theorem follows directly from
the antiderivative part, although it is more symbolic than geometric, which can
x
make it less satisfying to students. We begin with the function F( x) = ∫ f ( t) dt,
a
which we just proved is an antiderivative for f, and notice that we have
b
a
b
F( b) − F( a) = ∫ f ( t) dt − f ( t) dt f ( t)
dt
a ∫ =
a ∫ . Now we want to show that this
a
relationship is also true for any other antiderivative G(x) of f (so G′ ( x) = f ( x)). By the
antiderivative part of the theorem, we know F′ ( x) = f ( x) = G′
( x). Both F and G are
continuous on [ a, b ] since they are differentiable, and so F( x) = G( x)
+ C by a direct
corollary of the Mean Value Theorem (functions with the same derivative differ by a
constant). Now since F( b) − F( a) = ∫ f ( t)
dt , and also
b
a
( ) − ( + ) = − ,
F( b) − F( a) = G( b) + C G( a) C G( b) G( a)
b
a
we finally arrive at ∫ f ( t) dt = G( b) − G( a)
as desired for any antiderivative of f.
Stewart’s Calculus: Concepts and Contexts 1 is an example of one text that proves the
evaluation part of the FTC before a formal discussion of the Fundamental Theorem.
Since students easily grasp the procedural nature of this theorem, this approach is
certainly valid. The proof of the antiderivative part of the Fundamental Theorem
follows later in the text. In this case, the proof of the evaluation part starts by building
a Riemann sum from F(b) – F(a) (recall in this exposition that F represents some
unknown antiderivative of f and hence is continuous since it is differentiable). The proof
then requires the clever addition of 0, many times. Whereas in the study of infinite
1
James Stewart, Calculus: Concepts and Contexts, 3 rd ed., Pacific Grove, California:Brooks/Cole, 2005.
AP® Calculus: 2006–2007 Workshop Materials 101