AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
antiderivatives differ only by a constant. A proof of the antiderivative part
can also be given that uses the Mean Value Theorem for Definite Integrals. See
Larson’s Calculus of a Single Variable 2 .)
• Looking at each part of the theorem, can you explain why we say differentiation
and integration are inverse processes?
∫
x
( d f ( t) dt = f ( x)
, so differentiation after integration yields the original
dx a
b
function. In reverse, we have ∫ F ′( x ) dx = F ( b ) − F ( a ), so by integrating the
a
derivative F ′ we get back to the function F (up to a constant).)
• Can we solve the differential equation dF = f for every continuous function f?
What about the initial value?
dx
x
(Yes, the solution is F( x) = F( a) + ∫ f ( t)
dt , where F( a) is the given
a
initial condition.)
• How is the concept of limit, the foundation of both the integral and differential
calculus, used in these proofs?
(When we prove the antiderivative part, we use a limit as h → 0 to compute the
derivative using difference quotients, and for the evaluation part via the Mean
Value Theorem, we compute the limit of a Riemann sum as n → ∞.)
Whichever proofs you choose, the consideration of ∫ f ( t)
dt as a function can provide
a
opportunities for rich discussion. Identifying the domain of a particular function
∫
x
a
f ( t)
dt, understanding that the Fundamental Theorem says that continuous functions
are in fact derivatives, and realizing that the theorem shows that we can find solutions
to initial value differential equations even when a closed form of the antiderivative is
not available—each of these requires a level of understanding that goes beyond the rote.
Helping students understand the proofs you choose can contribute to the goal of helping
them reach the level of understanding and appreciation we hope for them to achieve.
x
2
Ron Larson, et al., Calculus of a Single Variable: Early Transcendental Functions, 3 rd ed., Boston:
Houghton Mifflin, 2003.
AP® Calculus: 2006–2007 Workshop Materials 103