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
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 Spring '08
 Grether
 Continuous function, Inverse function, Graph of a function, Leonhard Euler

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