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SLOPE FIELDS ASSIGNMENT #15 NAME_________________
31
1.
The calculator drawn slope field for the
differential equation dy xydx
= is shown in the
figure below. The solution curve passing through the point (0, 1) is also shown.
2. The calculator drawn slope field for the differential
equation dy x ydx
= + is shown in the figure below.
(a) Sketch the solution curve through the point (0,
2).
(a) Sketch the solution curve through the point (0, 2).
(b) Sketch the solution curve through the point (0, -
1)
(b) Sketch the solution curve through the point (0, -2)
Draw a slope field for each of the following differential equations. Show a segment at each indicated point.
3. 1dy
xdx
= + 4. 2dyy
dx= **Note the
scales!!!
5. dy ydx x
= −
(a) Sketch a solution curve
which passes through the
point (1, 0)
(a) Sketch a solution curve which
passes through the point (0, -
1).
(a) Sketch a solution curve which
passes through the point (2, -1).
For problems 6-8, find the equations of the solution curves you sketched in problems 3-5. Each equation should be
expressed in the form of ( )y f x= . Use your graphing calculator to graph each of your equations for problems 6-8
to see if those graphs match your solution curves drawn in problems 3-5.
−4 −2 2 4
−4
−2
2
4
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
x
y
−3 −2 −1 1 2 3
−2
−1
1
2
x
y
−4 −2 2 4
−4
−2
2
4
−3 −2 −1 1 2 3
−1.0
−0.5
0.5
1.0
•
SLOPE FIELDS ASSIGNMENT #15 NAME_________________
32
6. 1dy
xdx
= + 7. 2dyy
dx= 8. dy y
dx x= −
9. At right is a slope field for the differential equation
xdye
dx−= .
(a) Sketch the solution curve passing through the point (0, 0)
(b) Find a particular solution in the form of ( )y f x= to the
differential equation xdye
dx−= .
(c) Without the use of a calculator (instead use transformations
to the graph of xy e= ), determine whether your equation
from part b represents the function which you graphed in
part a.
10. The slope field for a differential equation is shown at the
right. Which statement is true for solutions of the
differential equation?
I. For 0x < all solutions are decreasing
II. All solutions level off near the x-axis.
III. For 0y > all solutions are increasing
(a) I only (b) II only (c) III only (d) II and III only (e) I, II and III
SLOPE FIELDS ASSIGNMENT #15 NAME_________________
33
11. The slope field for the differential equation
2 2
4 2dy x y ydx x y
+=
+ will have horizontal segments when
(a) 2 ,y x= only (b) 2 ,y x= − only (c) 2 ,y x= − only (d) 0y = , only (e) 0y = or 2y x= −
12.
Which one of the following could be the graph of the solution of
the differential equation whose slope field is shown at right.
(E) (A) (B)
(C) (D)
13.
Which statement is true about the solutions y(x), of a differential
equation whose slope field is shown at the right.
I. If y (0) > 0 then lim ( ) 0.xy x
→∞=
II. If 2 (0) 0y− < < then lim ( ) 2.xy x
→∞≈ −
III. If y (0) < -2 then lim ( ) 2.xy x
→∞≈ −
(A) I ONLY (B) II ONLY (C) III ONLY (D) II and III ONLY (E) I, II, and III
SLOPE FIELDS ASSIGNMENT #15 NAME_________________
34
14. Shown at the right is the slope field for which of the
following differential equations?
(A) 1dyx
dx= + (B) 2dy
xdx
= (C) dy x ydx
= +
15. Consider the differential equation given by 2
dy xydx
= .
(a) On the axes provided below, sketch a slope field for the given differential equation at the nine points
indicated.
−1 1
1
2
3
x
y
(b) Find the particular solution ( )y f x= to the given differential equation with the initial condition
(0) 3f = .
SLOPE FIELDS ASSIGNMENT #15 NAME_________________
35
16.
Consider the differential equation 2 4dyy x
dx= − .
−1 1
−2
−1
1
2
x
y
(a) The slope field for the given differential equation is
provided. Sketch the solution curve that passes
through the point (0, 1) and sketch the solution curve
that passes through the point (0, -1).
(b) Find the value of b for which 2y x b= + is a solution to the given differential equation. Justify your
answer.
(c) Let g be the function that satisfies the given differential equation with the initial condition g (0) =0.
Does the graph of g have a local extremum at the point (0, 0) ? If so, is the point a local maximum
or a local minimum? Justify your answer.
17. Consider the differential equation given by ( )21dyx y
dx= − .
(a) On the axes provided, sketch a slope field for the given differential equation at the eleven points
indicated.
−2 −1 1 2
−1
1
x
y
(b) Use the slope field for the given differential equation to explain why a solution could not have the
graph shown below.
−2 −1 1 2
−1
1
x
y
SLOPE FIELDS ASSIGNMENT #15 NAME_________________
37
Match each slope field with the equation that the slope field could represent.
(A)
(B)
(C)
(D)
(E) (F)
(G)
(H) 18. 1y =
20. 2
1y
x=
22. 2y x=
24. cosy x=
19. y x=
21. 316
y x=
23. siny x=
25. lny x=
Match the slope fields with their differential equations.
(A)
(B) (C)
(D)
(E) 26. cosdy
xdx
=
28. dy xdx y
= −
30. dy ydx
=
27. dy x ydx
= −
29. 1 12
dyx
dx= +