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MAC 2311
PM Feb 27, 2009
Exam II and Key
Prof. S. Hudson
1) [30 pts] Compute the derivative,
f
0
(
x
) (shortcuts such as the Product
Rule are OK for these, but show all work);
a)
f
(
x
) =
x
sin(
x
2
)
b)
f
(
x
) = tan

1
(2
x
)
c)
f
(
x
) =
√
x
+1
x
d)
f
(
x
) = ln(ln(
x
))
e)
e
sec(
x
)
f) Compute the second derivative,
f
00
(
x
), given that
f
(
x
) = sin(3
x
+1).
2) [10 pts] Find the derivative using logarithmic diﬀerentiation:
y
=
x
x
.
3) [10 pts] A 6foot man is walking towards a 15foot lamppost at 3 feet
per second. How fast is the length of his shadow decreasing ?
4) [10 pts] Find the slope of the tangent line to
y
=
x
2
at
x
= 4, using the
deﬁnition of
m
tan
. So, you will need to compute a limit. Do not use any
methods beyond Ch 3.1, such as derivative shortcuts.
5) [15 pts] Answer True or False. You do not have to explain.
For all
x
, ln(
e
5
x
) = 5
x
.
If
f
is diﬀerentiable, then
f
0
is continuous.
[using notation from the LLA section]
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 Calculus, Derivative

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