AP CALCULUS AB WORKSHEET 1 ON DIFFERENTIAL EQUATIONS Do not use your calculator. Solve for y as a function of x. 1. 6. 2. 7. 3. 8. 4. 9. 5. 10. Find a curve in the xy-plane that passes through the point (0, 3) and whose tangent line at a point has slope . __________________________________________________________________________________________ Write a differential equation to represent the following: 11. The rate of change of a population y, with respect to time t, is proportional to t. 12. The rate of change of a population P, with respect to time t, is proportional to the cube of the population. 13. Let represent the number of wolves in a population at time t years, where . The rate of change of the population , with respect to t, is directly proportional to 14. Water leaks out of a barrel at a rate proportional to the square root of the depth of the water at that time. 15. Oil leaks out of a tank at a rate inversely proportional to the amount of oil in the tank. () 2 4 and 0 1 dy xy y dx = - = () ln and 1 2 dy xy x y dx = = () 3 and 2 5 dy x y dx y - = = - () 2 sec and 2 2 y x y y p ¢ = = - () 2 and 2 25 y x y y ¢ = = () 2 and 0 0 y y y xe e y ¢- = = ( ) 2 2 4 sec 2 and 1 8 dy y x y dx p æ ö = = ç ÷ è ø ( ) () 2 3 2 sin and 0 1 dy xy x y dx = = - () 2 3 and 1 4 dy x y dx y = = - ( ) , x y 2 2 x y () Pt 0 t ³ () Pt () 500 . Pt -
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AP CALCULUS AB WORKSHEET 1 ON DIFFERENTIAL EQUATIONS Do not use your calculator. Solve for y as a function of x.
1. 6.
2. 7.
3. 8.
4. 9.
5.
10. Find a curve in the xy-plane that passes through the point (0, 3) and whose tangent line at a
point has slope .
__________________________________________________________________________________________ Write a differential equation to represent the following: 11. The rate of change of a population y, with respect to time t, is proportional to t. 12. The rate of change of a population P, with respect to time t, is proportional to the cube of the population. 13. Let represent the number of wolves in a population at time t years, where . The rate of change
of the population , with respect to t, is directly proportional to 14. Water leaks out of a barrel at a rate proportional to the square root of the depth of the water at that time. 15. Oil leaks out of a tank at a rate inversely proportional to the amount of oil in the tank.
( )24 and 0 1dy xy ydx
= - = ( )ln and 1 2dyxy x ydx
= =
( )3 and 2 5dy x ydx y
-= = - ( )2 sec and 2
2y x y y p¢ = = -
( )2 and 2 25y x y y¢ = = ( )2 and 0 0y yy xe e y¢ - = =
( )2 24 sec 2 and 18
dy y x ydx
pæ ö= =ç ÷è ø
( ) ( )232 sin and 0 1dy xy x ydx
= = -
( )23 and 1 4dy x y
dx y= = -
( ),x y 2
2xy
( )P t 0t ³
( )P t ( )500 .P t-
CALCULUS AB WORKSHEET 2 ON DIFFERENTIAL EQUATIONS Work these on notebook paper. Do not use your calculator. Solve for y as a function of x.
1.
2.
3.
4.
5.
6.
7.
8.
9. Find an equation of the curve that satisfies and whose y-intercept is
10. The rate at which a kitten gains weight is proportional to the difference between its adult weight and its current weight. At time t = 0, when the kitten is first weighed, its weight is 2 pounds. If is
the weight of the kitten in pounds, at time t days after it is first weighed, then .
Let be the solution to the differential equation with initial condition Use
separation of variables to find , the particular solution to the differential equation with initial
condition
2 given 5 when 3dy x y x
dx y= = - =
( )26 and 0 4dy x y ydx
= =
( )21 and 0 4dy y
dx y= =
( )1 and 2 9dy x ydx y
+= =
( ) ( )23 5 and 1 9dy x y ydx
= + - = -
( )2 1 and 14
dy xy ydx
= - = -
( ) ( )4 3 and 1 4dy x y ydx
= - = -
( )4 ln and 9
y xdy y edx x
= =
34dy x ydx
= 7.-
( )W t
( )1 203
dW Wdt
= -
( )y W t= ( )0 2.W =
( )y W t=
( )0 2.W =
CALCULUS AB WORKSHEET 3 ON DIFFERENTIAL EQUATIONS Work these on notebook paper. Do not use your calculator. Solve for y as a function of x.
1. 4.
2. 5. If then y = ?
3.
__________________________________________________________________________________________ Problems 6 and 7 are multiple choice. All steps must be shown.
6. If when x = 1, then when x = 2, y =
(A) (B) (C) 0 (D) (E)
__________________________________________________________________________________________
7. If , then y could be
(A) (B) (C) (D) (E)
__________________________________________________________________________________________
8. Consider the differential equation Find the particular solution to the differential
equation with the initial condition , and state its domain.
9. Consider the differential equation Find the particular solution to the
differential equation with the initial condition , and state its domain. 10. The rate at which a population of bears in a national forest grows is proportional to , where t is
the time in years and is the number of bears. At time t = 0, there are 200 bears in the forest. The
rate of change of the population of bears is modeled by the differential equation and
is the solution to the differential equation with initial condition (a) Is the number of bears growing faster when there are 300 bears or when there are 400 bears? Explain your reasoning.
(b) Find in terms of B. Use your answer to determine whether the graph of B is concave up or concave
down when there are 450 bears in the forest. (c) Use separation of variables to find , the particular solution to the differential equation with initial
condition
( )2 22 , 1 4dy x y x ydx
= + - = ( )3 and 4 5dy x xy ydx
= + = -
( )2
3 , 0, and 4 0dy y x ydx x
-= ¹ = cos and 3 when 0,dy y x y x
dx= = - =
( ) ( )22 cos 3 and 32
dy y x ydx
pæ ö= - =ç ÷è ø
22 and if 1dy y ydx
= = -
23
-13
-13
23
2dy x ydx
=
3ln3xæ ö
ç ÷è ø
3
3 7x
e +3
32x
e 23 xe3
13x
+
3.dy xydx
= ( )y f x=
( ) 2 1f = -
5 , 0.dy y xdx x
+= ¹ ( )y f x=
( ) 3 1f - =
( )600 B t-
( )B t
( )1 600 ,2
dB Bdt
= -
( )y B t= ( )0 200.B =
2
2
d Bdt
( )y B t=
( )0 200.B =
CALCULUS AB REVIEW WORKSHEET ON DIFFERENTIAL EQUATIONS Work these on notebook paper. Do not use your calculator. Solve for y as a function of x.
1. , 6.
2. 7.
3. , 8.
4. 9.
5.
__________________________________________________________________________________________
10. Consider the differential equation Find the particular solution to the differential
equation with the initial condition , and state its domain. 11. The rate at which a population of moose in a national park grows is proportional to , where t is
the time in years and is the number of moose. At time t = 0, there are 300 moose in the forest. The
rate of change of the population of moose is modeled by the differential equation and
is the solution to the differential equation with initial condition
(a) Find in terms of M. Use your answer to determine whether the graph of B is concave up or
concave down when there are 400 moose in the park. (b) Use separation of variables to find , the particular solution to the differential equation with initial
condition
( )2 1dy x ydx
= - ( )0 3y = ( )2 24 sec 2 and 18
dy y x ydx
pæ ö= =ç ÷è ø
( )2 18 and 13
dy xy ydx
= - = - ( )2 26 18 , 1 8dy x y x ydx
= + = -
2dy xdx y
= - ( )1 1y = - ( )ln and 1 4dyxy x ydx
= = -
( )5 and 3 1dy x ydx y
-= = - ( ) ( )232 sin and 0 1dy xy x y
dx= = -
( )6 and 1 16dy x y ydx
= =
.dy xdx y
= ( )y f x=
( ) 5 1f - = -
( )800 M t-
( )M t
( )1 800 ,2
dM Mdt
= -
( )y M t= ( )0 300.M =2
2
d Mdt
( )y M t=
( )0 300.M =
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