AP Calculus
Name
CHAPTER 7 WORKSHEET
INVERSE FUNCTIONS
Seat #
Date
Slope Fields
Match the differential equation with its slope field.
1

a) y'  y
c) y'  x  y
b) y '  x  1
2
1.














I)


















 





 

 


2.



 
 
 
 







IV)















(A)
dy
 1 x
dx
(B)






Shown below is the slope
field for which of the following differential equations?





May the Slope
be with you!



III)
x
y
 
II)


d) y '  
dy
 x2
dx
(C)

dy
 x y
dx
(D)
dy x

dx y
(E)
dy
 ln y
dx

3.
On the axes provided, sketch a slope field for each differential equation. Then use it to sketch a
solution curve that passes through the given point.
a) y'  y  2 x ;
point (1, 0)
b) y'  1  xy ;
point (0, –1)
y
;
x
point (–1, 1)
c) y'  x  y ;
point (0, 1)
4.
d) y '  
The figure below shows the slope field for the differential equation
dy
x
.

dx 2 y


















a) Show that you understand the meaning of slope field by calculating dy/dx at the points (3, 5)
and (–5, 1) and by showing that the results agree with the figure.
b) Sketch the graph of the particular solution of the differential equation that contains the point
(1, 2).
c) Sketch the graph of the particular solution that contains the point (4, 1). Remember: your
solution-graph must be a function…
d) Solve the differential equation algebraically. Find the particular solution that contains the
point (4, 1). Express your final answer as a function y  f x  . What is the domain of this
function?
5.
The figure below shows the slope field for the differential equation
dy
2x
 .
dx
y


















a) Sketch the graph of the particular solution of the differential equation that contains the point
(2, 1). Remember: your solution must be a function…
b) Sketch the graph of the particular solution that contains the point (–1, –1). Again: this must be the
graph of a function…
c) Solve the differential equation algebraically. Find the particular solution that contains the point
(–1, –1). Express your final answer as a function y  f x  . What is the domain of this function?
6.
Which one of the functions shown below could be a solution for the given slope field?
(A)
7.
y  x2
(B)
y  x3
(C)
y  sin x
(D)
y   tan x
(E)
y  ex
TRUE OR FALSE: (Justify your answer.) y  tan x  2 is a solution for the slope field shown on
question #6.
AP Calculus
ANSWER KEY
CHAPTER 7 WORKSHEET
INVERSE FUNCTIONS
Slope Fields
1.
a) to IV
2.
(C)
b) to I
c) to III
3.0
d) to II
3.0
2.0
3.
b)
a)
2.0
1.0
1.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
-4.0
-1.0
4.0
-3.0
5.0
-2.0
-1.0
1.0
2.0
3.0
4.0
5.
-1.0
-2.0
-2.0
-3.0
d)
-3.0
c)
3.0
3.0
2.0
2.0
1.0
1.0
-4.0
-4.0
-3.0
-2.0
-1.0
1.0
2.0
3.0
4.0
-3.0
-2.0
-1.0
1.0
5.0
-1.0
-1.0
-2.0
-2.0
-3.0
-3.0
2.0
3.0

4.
dy
3
dy
5
 ;

dx ( 3,5) 10
dx ( 5,1)
2
b) A branch of a hyperbola.
a)



















c) Half of a branch of yet another hyperbola.
















d) y  
1 2
x  7 with domain
 x  14
2











5.

a) Half of an ellipse































b) Half of a smaller ellipse















c) y   3  2 x 2 with domain: 
3
x
2
3

2
6.
(B)
7.
False. The graph of y  tan x  2 seems to “fit” the slope field, but
dy
 sec2  0   1 . This
dx x 0
means that the graph of y  tan x  2 has a slope of 1 at x = 0 and the slope field shows slopes 0
at x = 0.