Transcript 2004 AP Calculus Exam

2004 AP Calculus
Exam
Free Response Problem #6
6. Consider the differential equation:
dy
 x 2 ( y  1)
dx
a) On the axes provided, sketch a slope field for the differential equation at the
twelve points indicated.
6. Consider the differential equation:
dy
 x 2 ( y  1)
dx
a) On the axes provided, sketch a slope field for the differential equation at the
twelve points indicated.
Next, use the values for
dy/dx to graph the
slopes at each point.
Approximate the angles
of the slopes.
First, create a chart to
find dy/dx
b) While the slope field in part (a) is drawn at only twelve points, it is defined at every
point in the xy-plane. Describe all points in the xy-plane for which the slopes are
positive.
First, study the graph of dy/dx at each of the
twelve points.
What pattern do you observe where the
slopes are positive?
Click when you’re ready for the answer!
dy
dx
is positive for all points where y > 0 and x ≠ 0
c) Find the particular solution y= f(x) to the differential equation with the initial condition
f(0) = 3. Use
Rewrite
Rewrite
U-Substitution
this
the
expression
natural
(in
log
this
as
ay
an
multiple
u =solve
y-1)
of two
to
equation
exponential
du/u,
whichfor
is C
cthe
Rewrite
Since
Make
Use
the
the
e
initial
is
equation
aequation
constant,
condition
equal
equal
it as
(f(0)=3)
can
tocase,
to
be
y exponential
using
rewritten
to
the
for
value
asget
C
C
you
found
Combine
Start
like
by
terms
Integrate
using
(x
implicit
on
one
differentiation
side
and
y
on
the
other)
Substitute
y-1 inbase
for u
equal
expressions
withto
e as
ln|u|
the
with
base
the same
3
dy
 x 2 ( y  1)
dx
dy  x 2 ( y  1)dx
dy
 x 2 dx
y 1
dy
2

x
 y  1  dx
du x 3
 u  3 C
x3
ln u 
C
3
x3
ln y  1 
C
3
e
e





x
3


C 


 x3 


 3 


Ce
Ce
 y 1
eC  y  1
 x3 


 3 


 x3 


 3 


 y 1
1  y
 03 


 3 


1  3
Ce
C 1  3
C 2
y  2e
x3
3
1
Congratulations!
You have completed the AP
Calculus Free Response
Problem #6!
With the help of:
Anna Woodbury and Sara Buchanan