AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
12. (1997 BC22)
(C) By the Fundamental Theorem, g′ ( x) = f ( x). The only critical value of g on
(a, d) is at x = c, where g′
changes from positive to negative. Thus the absolute
maximum for g occurs at x = c.
13. (1997 BC88)
2
(C) By the FTC and the chain rule, f ′( x) = 2xsin( x ). For the average rate of
change of f we need to determine f(0) and f ( π ). We have f(0) = 0 and
π
∫ 0
f ( π ) = sin t dt = 2 . The average rate of change of f on the interval is
2
therefore . See how many points of intersection there are for the graphs of
π
2
y = 2xsin( x ) and y = 2 on the interval ⎡0, π ⎤. There are two.
π
⎣ ⎦
14. (1997 BC89)
(D) Both statements below follow from the Fundamental Theorem of Calculus:
x
2
t
4
2
t
f ( x) = ∫ dt; f ( 4) = ∫ dt = 0.
376, or
1 5
5
1+
t
1
1+
t
f ( 4) x
f ( 1) 4
2
= + ∫
1 x dx = 0 . 376.
1 5
+
15. (1998 AB9)
(D) Let r(t) be the rate of oil flow as given by the graph, where t is measured in
hours. The total number of barrels is given by
∫
24
0
r( t)
dt
. This can be
approximated by counting the squares below the curve and above the
horizontal axis. There are approximately five squares with area 600 barrels.
Thus the total is about 3,000 barrels.
16. (1998 AB11)
b
(A)
∫ f ′′( x) dx = f ′( b) − f ′( a) = 0 since the derivative is constant for a linear
a
function. (Alternatively, f ′′( x) = 0 for a linear function, so the value of the
definite integral is 0.)
94
AP® Calculus: 2006–2007 Workshop Materials