MATH 125, FINAL EXAM (common)
December 2005
1.
(6 points each) Calculate the following limits.
a) lim
x
→
4
x

4
√
x

2
.
b) lim
x
→
1
e
x

e
x

1
.
c) lim
x
→
0
3
x
+
x
2
sin
x
.
2.
(7 points each) Find
dy
dx
.
a)
y
= ln(2
x
2

3
x
).
b)
y
=
2
x

3
e
x
+1
.
c)
y
=
x
sin
x
(Use logarithmic differentiation).
d)
y
=
tan
x
0
√
t
2
+ 4
dt
for

π
2
≤
x
≤
π
2
.
3.
(8 points each) Evaluate the following integrals:
(a)
2
1
x
2
(
x

2)
2
5
dx
.
(b)
sin
√
x
√
x
dx
.
(c) Find the area of the region bounded by the curves
y
= 0,
y
=
xe
x
2
,
x
= 0 and
x
= 1.
4.
Consider the following function and its first and second derivative:
f
(
x
) =
x
x
2
+ 4
f
(
x
) =

x
2

4
(
x
2
+ 4)
2
f
(
x
) =
2
x
(
x
2

12)
(
x
2
+ 4)
3
.
a) (5 points) Find the critical numbers of
f
.
b) (10 points) Determine where
f
is increasing, where
f
is decreasing,
and find the local maxima and minima of
f
.
c) (5 points) Find the asymptotes of
f
.
f
(
x
) =
x
x
2
+ 4
f
(
x
) =

x
2

4
(
x
2
+ 4)
2
f
(
x
) =
2
x
(
x
2

12)
(
x
2
+ 4)
3
.
d) (5 points) Find the inflection points of
f
and determine where
f
is
concave upwards and downwards.
e) (8 points) Draw a rough sketch of the graph of
f
.
1
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5.
(12 points) a) Find the absolute maximum and minimum of
f
(
x
) =
xe

3
x
on the interval [

1
,
1].
b) (8 points) For what values of
c
is the function
f
(
x
) continuous
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 Fall '07
 Tuffaha
 Math, Calculus, Limits

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