AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
2002 AB2/BC2
The rate at which people enter an amusement park on a given day is modeled by the
function E defined by
E t
( ) =
15600
.
2
( t − 24t
+ 160)
The rate at which people leave the same amusement park on the same day is modeled by
the function L defined by
( ) =
L t
9890
.
2
( t − 38t
+ 370)
Both E(t) and L(t) are measured in people per hour, and time t is measured in hours
after midnight. These functions are valid for 9 ≤ t ≤ 23, the hours during which the park
is open. At time t = 9 there are no people in the park.
(a) How many people have entered the park by 5:00 P.M. (t = 17)? Round answer
to the nearest whole number.
t
( ) = −
( )
(c) Let H t ∫ E( x) L( x)
dx for 9 ≤ t ≤ 23. The value of H(17) to the
9
nearest whole number is 3725. Find the value of H' (17) and explain the
meaning of H(17) and H' (17) in the context of the park.
This is again a Fundamental Theorem of Calculus question. How can we tell? We are
given information about rates of change (rate at which people enter a park and rate at
which people leave the park), and we are asked about the change in the number of people
in the park. Notice the units of E(t) and L(t) are people per hour. Thus the definite
integral of E(t) will give the change in the number of people who enter the park, so in
17
part (a) we need to evaluate ∫ E ( t ) dt numerically with the calculator. We again need
9
to use the FTC in part (c) to explain the meaning of H(17) and to compute H' (17). The
second version of the FTC tells us immediately that
d t
H′ ( t) = ∫ ( E( x) − L( x)) dx = E( t) − L( t)
,
dt 9
and so H′ ( 17) = E( 17) − L( 17 ). Moreover, once we recognize that E(t) – L(t) represents
the rate of change of the number of people in the park, the FTC also allows us to
conclude that H(t) is the total change in the number of people in the park between
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AP® Calculus: 2006–2007 Workshop Materials