AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
4 ⎛ 1
f ( 4) f ( 0) f ( x) dx 8 ( )
2 2 2 ⎞
− = ∫ ′ = − −
8 2
0
⎝
⎜ π
⎠
⎟ = − + π,
and so f ( 4) = f ( 0)
− 8 + 2π = − 5 + 2π .
2003 AB3
t
(minutes)
R(t)
(gallons per minute)
0
30
40
50
70
90
20
30
40
55
65
70
The rate of fuel consumption, in gallons per minute, recorded during an airplane flight
is given by a twice-differentiable and strictly increasing function R of time t. The graph
of R and a table of selected values of R(t), for the time interval 0 ≤ t ≤ 90 minutes, are
shown above.
(d) For 0 < b ≤ 90 minutes, explain the meaning of R t dt in terms of fuel
consumption for the plane. Indicate units of measure in [the] answer.
The function R(t) is the rate of change of the amount of fuel with units of gallons per
minute. Therefore the FTC tells us that the definite integral of this rate of change is the
total change in the amount of fuel, or more specifically in this particular question, the
total amount of fuel in gallons consumed for the first b minutes.
A similar interpretation question was asked in 1999 AB3/BC3 and in 2004 (Form B)
AB3/BC3.
1976 AB6
3
x
(a) Given 5x
+ 40 = ∫ f ( t)
dt
c
(i) Find f(x).
(ii) Find the value of c.
3
16
(b) If F( x) = ∫ 1+ t dt, find F′
( x)
.
x
b
∫ 0
( )
AP® Calculus: 2006–2007 Workshop Materials 81