Name___________________________________
Period____
Calculus
Review 6.4 and 6.5
1. What are all the horizontal asymptotes of all the solutions of the logistic differential equation
dy
y 

 y 8 
?
dx
 1000 
2. Suppose P(t) denotes the size of an animal population at time t and its growth is described by the differential
dP
 0.002 P 1000  P  . At what population is the population growing fastest?
equation
dt
3. Which of the following statements characterize the logistic growth of a population whose limiting value is L?
I. The rate of growth increases initially.
II. The growth rate attains a maximum when the population equals L/2.
III. The growth rate approaches 0 as the population approaches L.
4. Find the carrying capacity for a population growth rate modeled by
dP
 6 P  0.012 P 2
dt
5. A population P of wolves at time t years ( t  0 ) is increasing at a rate directly proportional to 600 - P, where
the constant of proportionality is k.
(a) If P(0) = 200, find P(t) in terms of t and k.
(b) If P(2) = 500, find k.
(c) Find lim P (t ) .
t 
1991 BC 6
A certain rumor spreads through a community at the rate
dy
 2 y (1  y ) , where y is the proportion of the
dt
population that has heard the rumor at time t.
(a) What proportion of the population has heard the rumor when it is spreading the fastest?
(b) If at time t = 0 ten percent of the people have heard the rumor, find y as a function of t.
(c) At what time t is the rumor spreading the fastest?
1998 BC 26
The population P(t) of a species satisfies the logistic differential equation
dP
P 

 P2
 , where the initial
dt
5000 

population P (0)  3000 and t is the time in years. What is lim P (t ) ?
t 
(A) 2,500
1973 AB 37
(A) 4e 4 x
(B) 3,000
If
(C) 4,200
(D) 5,000
(E) 10,000
dy
 4 y and if y = 4 when x = 0, then y =
dx
(B) e 4 x
(C)
3  e4 x
(D) 4  e 4 x
(E) 2 x 2  4