AP Calculus
Name: _____________________
Review Unit 8 Differential equations
SHOW ALL WORK! On the test, you will be required to show all of your work. The test will be divided into calculator and no
calculator parts. This review is not comprehensive. Please look back over your notes, your homework, and your quizzes to
help you study for the test.
Topics to study:

Differential Equations

Euler’s Method

Applications of differential equations

Logistic Equations

Slope Fields
Solve the differential equation using the initial conditions given.
1.
dy
 15  3 y, y (0)  0
dx
2.
dP
 Pt , P (1)  2
dt
3.
dy
1
 (2 x  1) y 2 , y (2) 
dx
2
4.
dy x  y
e , y (0)  3
dx
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Solve the differential equation for the general solution.
5.
2 y '  xe y
6.
dy
 xy sin( x 2 )
dx
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7. The change in price of a popular toy was proportional to the square root of the number of weeks it had been on
the market. Calculator allowed.
(A) Write a differential equation to find the price, p, of the toy t weeks after it was on the market.
(B) What is the instantaneous rate of change of the price of the toy for p (12)  7.08 in terms of k, the
proportionality constant?
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8. When an object is removed from a furnace and placed in an environment with a constant temperature of 80 °F,
its core temperature is 1500 °F. One hour after it is removed, the core temperature is 1120 °F. Find the core
temperature 5 hours after the object is removed from the furnace.
9. The rate of change of the population of a newly built high school with respect to time is directly proportional to
the difference between 200 and the current population.
(A) At t  0 , the school had 1300 students. Three years later at t  3 , the student population had risen to 1500.
Find the particular solution for the differential equation you solved in the previous problem.
(B) When will the school’s population exceed 1800 students? Justify your answer.
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dy x 2  1
10. Consider the differential equation
.

dx
y
y

(A) On the axes provided, sketch a slope field
for the given differential equation at the twelve
points indicated.


(B) Let y  f ( x) be the particular solution to the given

differential equation with the initial condition f (3)  4 .
Write an equation for the tangent line to the graph of f at x = 3.
x



(C) Find the particular solution to the given differential equation with the initial condition f (3)  4 .
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11. The graph is a slope field for which of the following differential equations?
dy
 y  2x
dx
dy
 (1  x)( y  2)
(C)
dx
(A)
dy
 1 x  y
dx
dy
 xy2
(D)
dx
(B)
(E)
dy
 ( x  1) y 2
dx
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12. Use Euler’s Method with step size 0.25 to estimate y(0.5), where y(x) is the solution of the initial-value
problem y '  y  xy, y (0)  1
13. A particle moving along the x-axis has position
x  t  at time t with the velocity of the particle v(t )  esin t .
At time t = 1, the particle’s position is (4, 0).
(A) Use Euler’s Method with step size 0.5 to estimate x ( 2) .
(B) Use the Fundamental Theorem of Calculus to write an expression for
x(t ) and evaluate that expression at
t  2.
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14. The table gives selected values of the derivative of a function f . What is the approximation of f (1) using
Euler’s method with step size 0.5 ?
x
0
0.5
1
f (x) 4
f ' ( x) 3
7
2
7
4
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15. Suppose a rumor is spreading through a dance at a rate modeled by the logistic differential equation
dP
P 

 P3
.
dt
2000 

(A) What is
lim P  t  ?
t 
What does this number represent in the context of this problem?
(B) What is the fastest growth rate of the rumor?
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16. The population of a small ecosystem with little resources is given by
the logistic differential equation
dP
 3P  3P 2 , where P is the
dt
population in thousands, and the slope field is shown.
(A) On the slope field shown, sketch three solution curves showing
different types of behavior for the population P.
(B) Find
lim P (t ) for each of the solution curves sketched in part (A).
t 
(C) Describe the meaning of the shape of the solution curves for the population.
(D) What is the fastest growth rate of the population?
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17. The rate of change, dP , of the number of mountain lions in a state park is modeled by a logistic differential
dt
equation. Due to resources, the population of mountain lions will not exceed 500. In 2006, the number of
mountain lions is 250 and is increasing at the rate of 15 mountain lions per year. Write a logistic differential
equation that describes the situation.