MATH230
Differential Equations
Homework assignment 1 (Part 2)
Due Thursday, 1/24/08
Spring 2008
dP
P
 0.5P(1 ) , which has been created to provide
dt
40
predictions about the future population of rabbits over time.
3. Consider the rate of change equation
(a) For what values of P is P(t) increasing? Explain your reasoning.
(b) For what values of P is P(t) decreasing? Explain your reasoning.
dP
(c) For what values of P is
neither positive nor negative? What does this imply about
dt
P(t)?
4. In a previous problem, you used the rate of change equation dP/dt = 0.8P to make predictions
about and draw some population-over-time curves for a particular fish population. A group
of scientists is studying a different fish population and modified this equation to get
dP
 P 
 0.8P 1 , where P represents thousands of fish in a lake, and t is in years.
dt
5
(a) Plot by hand a number of possible population-over-time curves for this rate of change
equation that you believe show the important features / possibilities, and describe what
those important features are.
(b) What does this rate of change equation predict about the long-term outcome of the fish
population if the initial population is 2 (i.e., P = 2 at t = 0)? How about if P = 6 at t = 0?
(c) Why are the predictions you made in part (b) reasonable (or not) for a fish population?
Explain.
5. Suppose two students are memorizing a list according to the rate of change equation
dL
 0.5(1 L) , where L represents the fraction of the list that is memorized at any time t.
dt
(a) If one of the students knows one-third of the list at time t=0 and the other student knows
none of the list, which student is learning most rapidly at this instant?
(b) According to the rate of change equation, will the student who starts out knowing none of
the list ever catch up to the student who knows one-third of the list? Explain your
reasoning.
(c) What does the rate of change equation predict for someone who starts off with the list
completely memorized?