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Fall 2007 Math2253 Fianl Exam Review
Dr. Taixi Xu
Department of Mathematics, SPSU
Math2253  Review Problems for Final Exam
Chpater 2: Limits and Continuity
1.
If
2
2
)
(
2
+

=
x
x
x
f
, find
(a)
)
3
(
f
(5)
(b)
)
3
(
h
f
+
(
5
4
2
+
+
h
h
)
(c)
h
f
h
f
)
3
(
)
3
(

+
(
4
+
h
)
2.
Find the average rate of change of the function over the given interval or intervals.
1
)
(
3
+
=
x
x
f
;
a)
]
3
,
2
[
(19)
b)
]
2
,
2
[
h
+
(
2
6
12
h
h
+
+
)
3.
Find the limit.
a)
4
2
lim
2
2

+

→
x
x
x
(1/4)
b)
)
8
5
3
(
lim
2
2
+

→
x
x
x
(10)
c)
7
lim
2
3
+
→
x
x
(4)
d)
x
x
x
3
9
lim
0

+
→
(1/6)
e)
1
1
lim
3
1


→
x
x
x
(1/3)
f)
12
tan
lim
3
x
x
π
→
(1)
g)
x
x
x
4
5
sin
lim
0
→
(5/4)
h)
x
x
x
tan
lim
0
→
(1)
i)
x
x
x
sin
cos
1
lim
0

→
(0)
j)
+
≤

=
2
,
3
2
,
1
2
)
(
2
x
x
x
x
x
f
)
(
lim
2
x
f
x

→
(3)
)
(
lim
2
x
f
x

→
(7)
k)
x
x
x
2
sin
lim
∞
→
(0)
l)
1
5
3
2
lim
+

∞
→
x
x
x
(2/5)
Page 1 of 8
Fall 2007 Math2253 Fianl Exam Review
Dr. Taixi Xu
Department of Mathematics, SPSU
m)
2
1
lim
2

+
→
x
x
(
∞
)
4.
Discuss the continuity of each of the following functions.
a)
1
2
3
)
(
2

+

=
x
x
x
x
f
(continuous everywhere but x=1)
b)
+
≤

=
2
,
3
2
,
1
2
)
(
2
x
x
x
x
x
f
(continuous everywhere but x=2)
5.
Determine the value of c such that the function is continuous on the entire real
line.
+
≤
+
=
4
,
6
4
,
3
)
(
x
cx
x
x
x
f
(c = ¼)
6.
Find the vertical asymptote (if any)
1
)
(
2
2

=
x
x
x
f
(x = 1 and x = 1)
7.
The function
f
is defined as follows
0
,
2
tan
)
(
≠
=
x
x
x
x
f
a)
Find
)
(
lim
0
x
f
x
→
(if it exits)
(2)
b)
Can the function
f
be define at
0
=
x
such that it is continuous everywhere?
If so, how?
8.
Find an equation of the tangent line to the curve of
2
x
y
=
at the point (1, 1).
(y=2x+5)
9.
Graph the following functions using horizontal, vertical, or slant asymptotes if
possible.
a)
1
1
2
)
(

+
=
x
x
x
f
b)
2
1
)
(
2


=
x
x
x
f
Chapter 3: Differentiation
1.
Find the derivative of the function by the limit process.
a)
3
2
)
(
2
+

=
x
x
x
f
(
2
2

x
)
b)
1
)
(
+
=
x
x
f
x
2
1
2.
Find an equation of the line that is tangent to the graph of
)
7
)(
8
(
)
(
2
2


=
x
x
x
f
at (3, 2).
(
52
18
+
=
x
y
)
Page 2 of 8
Fall 2007 Math2253 Fianl Exam Review
Dr. Taixi Xu
Department of Mathematics, SPSU
3.
Find the derivative of each of the following functions.
(a)
6
3
2
)
(
2
3
4

+

+
=
x
x
x
x
x
f
(
29
3
2
6
4
2
3
+

+
x
x
x
(b)
x
x
x
f
4
)
(
2

=
+
2
4
1
x
(c)
3
6
)
(
x
x
x
f

=

3
2
2
2
1
x
x
(d)
3
2
1
)
(


=
x
x
x
f


2
)
3
2
(
1
x
(e)
4
2
)
1
3
(
)
(
+

=
x
x
x
f
4 (
x
K
3)
3
(
x
2
C
1)
4
K
8 (
x
K
3)
4
x
(
x
2
C
1)
5
4.
Find an equation of the line that is tangent to the graph of
2
2
3
)
(
x
x
x
f
+

=
and
parallel to the line
3
2
4
=
+
y
x
.
(
3
2
+

=
x
y
)
5.
Find an equation of the line that is tangent to the graph of
2
3
)
(
x
x
f
+
=
and
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