is called the derivative of
f
with respect to
x
.
The domain of
f
′
consists
of all
x
in the domain of
f
for which the limit above exists.//
(b)
Using only the definition of the derivative as a limit, show all steps
of the computation of
f
′
(
x
) when
f
(
x
) = 1/
x
.
f
(
x
)
lim
h
→
0
f(
x
h
)
f(
x
)
h
lim
h
→
o
(
x
h
)
1
x
1
h
lim
h
→
0
x
(
x
h
)
hx
(
x
h
)
lim
h
→
0
1
x
(
x
h
)
1
x
2
for every real number x
≠
0.
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Page 4 of 4
______________________________________________________________________
8. (5 pts.)
Determine whether the following function is differentiable
at x = 1.
Since
lim
x
→
1
f
(
x
)
lim
x
→
1
3
x
3
and
,
lim
x
→
1
f
(
x
)
lim
x
→
1
(
x
2
x
2)
4
f
is not continuous at
x
= 1.
Consequently,
f
is cannot be differentiable
there either.
[Check continuity first, folks.
There was a red herring
here!]
______________________________________________________________________
9.
(5 pts.)
Compute f
″
(x) when
f(
x
)
sin(2
x
3
).
f
(
x
)
cos(2
x
3
)
(6
x
2
)
6
x
2
cos(2
x
3
).
f
(
x
)
12
x
cos(2
x
3
)
6
x
2
sin(2
x
3
)
(6
x
2
)
12
x
cos(2
x
3
)
36
x
4
sin(2
x
3
).
______________________________________________________________________
10.
(10 pts.) A spherical balloon is to be deflated so that its radius
decreases at a constant rate of 15 cm/min.
At what rate must air be
removed when the radius is 9 cm.?
[
V
= (4/3)
π
r
3
??]
Let
r
(
t
) denote the radius of the sphere at time
t
in minutes.
Then
the volume is given by
V
(
t
) = (4/3)
π
(
r
(
t
))
3
.
We are told that
r
′
(
t
) = 15
(cm/min) for any
t
.
The question is, what is
V
′
(
t
0
) at the moment
t
0
when
r
(
t
0
) = 9 cm?
Evidently,
V
′
(
t
) = 4
π
(
r
(
t
))
2
r
′
(
t
) for any
t
.
Substituting
t
0
into this last equation, using the know values of
r
and
r
′
at
t
0
, and doing
the boring multiplication by hand yields
V
′
(
t
0
) = 4860
π
cubic cm/min.
______________________________________________________________________
Silly 10 point Bonus Problem:
Explain completely how to obtain the limit
lim
x
→
0
x
sin
1
(
x
)
1
from
lim
x
→
0
sin(
x
)
x
1.
Say where your work is, for it won’t fit here. (Found on the bottom of Page
2 of 4.)
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 Fall '08
 STAFF
 Calculus, Derivative, Continuous function, lim g

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