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Math 41 Fall 2005 Practice Final Exam
1. Evaluate each of the following limits. If there is an inﬁnite limit, then state whether
the limit is
∞
or
∞
(a) lim
x
→∞
(ln
x
)
2
x
(b) lim
t
→
0
2
te

t
(c) lim
x
→
0
+
cos
x
x
2. Diﬀerentiate each of the following functions
(a)
y
=
x
(
e
x
)
(b)
f
(
y
) =
e
y
2

3
sin(
πy
)
(c)
g
(
x
) =
±
x
2

2
e
cos
t
+7
t
2
dt
3. Evaluate each of the following integrals.
(a)
±
e
1
(ln
x
)
2
dx
(b)
±
1

1
sin
x
x
2
+ 2
dx
(c)
±
t
t

1
dt
(d)
±
tan

1
x dx
(e)
±
2
1
x
(
x
2

1)
dx
(f)
±
2
s
2
+
√
s
s
3
/
2
ds
4. The following table shows the velocity
v
(
t
) (in m/s) after
t
seconds of an accelerating
car on the entrance ramp to a highway.
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View Full Document5. Find an equation for the tangent line to the graph of the equation
x
2
+ 2
y
2
=
xe
y
at the point (1
,
0).
6. Find the absolute maximum and minimum values of
f
(
x
) =
2
x
x
2
+ 1
on the interval
[

2
,
2].
7. A beacon that makes one revolution every 10 s is located on a ship anchored 4 km from
a straight shoreline. How fast is the beam moving along the shoreline when it makes
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