d) Complete the following conjecture, where
a
,
b
, and
c
are real numbers:
If
f
(
x
)
is a
continuous function on
[
a, b
]
, then
b
a
f
(
x
)
dx
=
b
+
c
a
+
c
dx
.

140
The AP CALCULUS PROBLEM BOOK
5.5
The Logistic Curve
1187.
The graph of a function of the form
P
(
t
) =
M
1 +
Ce
−
rMt
, where
M
,
r
, and
C
are constants,
is called a
logistic
curve. Graph the function
y
(
x
) =
8
1 + 10
e
−
0
.
9
x
in the window
−
1
≤
x
≤
10,
−
1
≤
y
≤
9. What value does
y
approach as
x
→ ∞
? What appears to be the
y
-value of the
point where
dy/dt
is changing the fastest?
1188.
The solution to the di
ff
erential equation
dP
dt
=
r
(
M
−
P
)
P
is a logistic curve, where
C
is determined by the initial condition.
Can the values found in the previous problem be
found without solving the di
ff
erential equation?
In other words, in the equation
dP/dt
=
0
.
001(100
−
P
)
P
, what does
P
approach as
x
→ ∞
? What appears to be the
P
-value of the
point where
dP/dt
is changing the fastest?
1189.
A 2000 gallon tank can support no more than 150 guppies. Six guppies are introduced
into the tank. Assume that the rate of growth of the population is
dP/dt
= 0
.
0015(150
−
P
)
P
,
where
t
is in weeks. Find a formula for the guppy population in terms of
t
; then, determine
how long it will take for the guppy population to be 100.
1190.
A certain wild animal preserve can support no more than 250 gorillas.
In 1970, 28
gorillas were known to be in the preserve.
Assume that the rate of growth of population is
dP/dt
= 0
.
0004(250
−
P
)
P
, where
t
is in years.
Find a formula for the gorilla population
in terms of
t
; then, determine how long it will take for the gorilla population to reach the
carrying capacity of the preserve. What is the gorilla population when the rate of change of
the population is maximized?
1191.
Solve the di
ff
erential equation
dP/dt
=
kP
2
for constant
k
, with initial condition
P
(0) =
P
0
. Prove that the graph of the solution has a vertical asymptote at a positive value of
t
. What
is that value of
t
? (This value is called the
catastrophic solution
.)
1192.
Given a di
ff
erential equation of the form
ay
′′
+
by
′
+
y
= 0, find constants
a
and
b
so that
both
y
=
e
x
and
y
=
e
2
x
are solutions.
1193
(AP)
.
At each point (
x, y
) on a certain curve, the slope of the curve is 3
x
2
y
. If the curve
contains the point (0
,
8), then its equation is
A)
y
= 8
e
x
3
B)
y
=
x
3
+ 8
C)
y
=
e
x
3
+ 7
D)
y
= ln(
x
+ 1) + 8
E)
y
2
=
x
2
+ 8
The simplest schoolboy is now familiar with facts for which Archimedes would have sacrificied his life.
—Earnest Renan

CHAPTER 5.
APPLICATIONS OF INTEGRALS
141
5.6
Differential Equations, Part Two
1194.
You are driving along the highway at a steady 60 mph (88 ft/sec) when you see an
accident ahead and slam on the brakes. What constant decceleration is required to stop your
car in 242 ft?
1195.
The rate of change in the number of bacteria in a culture is proportional to the number
present. AP Biology students at Rockdale discovered that there were 3000 bacteria initially,
and 90,000 bacteria after two hours.