] at which the
graph of
f
has a horizontal tangent line when
f
(
x
) =
x
+ 2 cos(
x
).
(b)
The following limit represents
f
′
(
a
) for some function
f
and some
number
a
.
Using that information, evaluate the limit.
lim
x
→ π
sin(3
x
)
0
x
π
______________________________________________________________________
7. (10 pts.) (a)
Using complete sentences and appropriate notation,
provide the precise mathematical definition for the derivative,
f
′
(
x
), of a
function
f
(
x
).
(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
)
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______________________________________________________________________
8. (5 pts.)
Determine whether the following function is differentiable
at x = 1.
______________________________________________________________________
9.
(5 pts.)
Compute f
″
(x) when
f(
x
)
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
??]
______________________________________________________________________
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.
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 Fall '08
 STAFF
 Calculus, Continuous function, 5 pts, 10 pts, 15 cm, 9 cm, 6 G

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