MATH
221
EXAM
3
DECEMBER
3,
1992
(You
must
show
all
work
to
receive
full
credit.)
[15]
1.
Find
the
Taylor
series
of
f(x) =
1
+
x"
at
x =
0
by using
suitable
operations
on
the
Taylor
series
of
: -
.
[10]
2.
If
f(x)
has
Taylor
series
2
+
3x
+
4x
2
+
5x
3
+
....
what
is
f"(0)?
[15]
3.
Solve
the
differential
equation
y'
=
3
-
y,
y(0)
=
1
[15]
4.
Let
y
= f(t) be
the
solution
of
y'
=
t
y,
y(0) =
-2.
(a)
Use
Euler's
method
with
n
=
2
on
the
interval
0 s
t
s
1
to
approximate
the
solution
f(t).
[Give
the
points
(ti
(
yi)
produced
by
Euler's
method. Do not
make a sketch..]
(b)
Estimate
f(l)
and
f'(l).
[20]
5.
Consider
the
differential
equation
y'
=
g(y)
where
the
graph
of
g(y)
is
shown
on
the
right.
(a)
What
are
the
constant
solutions?
(b)
For
what
initial
values
y(0) will
the
solution
of
the
differential
equation
be
an increasing
function?
(c)
Sketch
the
graph
of
the
solution
with
initial
value
y(0)
=
2.
o
[15]
6.
A
savings
account
earns
5%
interest
per
year,
compounded
continuously,
and
continuous deposits
are
made
into
the
account
at
the
rate
of
$1000
per
year.
Set
up
a
differential
equation
satisfied
by
the
amount
f(t)
of
money
in
the
account
at
time
t.
Sketch
typical
solutions
of
the
differential
equation.
[10]
7.
The
cumulative
distribution
function
for a
random
variable
X
on
the
interval
1
^
x
^
5 is
F(x)
=
-
V
x
-
1
.
Find
the
corresponding
density
function.