WORKSHEET 24  Fall 1995
1.
a) Find three distinct functions
f
(
x
) such that
f
0
(
x
) = 0 for all
x
.
b) Find three distinct functions
g
(
x
) such that
g
0
(
x
)=4
x
for all
x
.
c) Find three distinct functions
h
(
x
) such that
h
0
(
x
)=3
x
−
x
2
for all
x
.
2. Let
f
1
(
x
) be a function such that
f
0
1
(
x
)=s
in(
x
2
). (We do not yet know that there is such a function,
but we will see that there is in about a week. We will then describe how to compute it, but we will never
be able to write it down as a combination of any functions that you already know.) In the following
problems, your answer will be in terms of
f
1
(
x
).
a) Find
f
2
(
x
) such that
f
0
2
(
x
x
2
)and
f
2
(
p
π/
2) = 10
.
b) Find
f
3
(
x
) such that
f
0
3
(
x
x
2
f
3
(
p
4) =
−
18
.
c) Draw the graph of
f
2
(
x
) as best you can from the information known.
(Hint: consider the interval [
√
kπ,
p
(
k
+1)
π
] for each integer
k
.)
3. Solve the following di±erential equations subject to the prescribed initial conditions.
a)
dy
dx
=4(
x
−
7)
3
,x
=8
,y
=10
b)
dy
dx
=
x
2
+1
x
2
=1
c)
dy
dx
=
x
2
y
2
=0
d)
dy
dx
=
4
p
(1 +
y
2
)
3
y
.
(Hint: For c) and d), move anything involving the variable
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 Grether
 Continuous function, Inverse function, Graph of a function, Leonhard Euler

Click to edit the document details