MATH 101 Final, Fall 2007
American University of Beirut · Calculus and Analytical Geometry I · 10 pages
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1
AMERICAN UNIVERSITY OF BEIRUT
FINAL EXAM-MATH 101 Jan.,2008
INSTRUCTOR: Z. KHACHADOURIAN
NAME:…………………………………… SECTION: 4 5 6 7
Ques. 1
2
3
4
5
6
7
8
9
10
Exam Gr.
Pts.
1. Solve the two independent parts: (4+4=8points)
a) Let
.
8cos
1
5
3
2
x dx
I ∫
+
=
Without attempting to integrate, show that the value of the above integral lies
between 2 and 6. Hint: Use the Min-Max inequality.
b) Find the average value of the function
,
2
y = sin x
over the interval
4
,0 π
.
2
2. (4+4+4=12points)Consider the region in the first quadrant limited by the y-axis
and the curves :
.4
2
=
=
and y
x
y
Answer the following three questions:
a) Draw the region and find its area.
b) Find the volume of the solid generated when the region is revolved about
the x-axis. (Drawing is not a must)
c) The cross sections of a solid perpendicular to the y-axis are equilateral
triangles with a base running from the y-axis to the curve
y = x2
.
Find the volume of this solid. (Drawing is not a must).
3
3. (4+4+4=12points)Evaluate the integrals: a)
∫
−
+
=
1
0
2
1
5
4
)
8
1(
dx
x
x
I
b)
x dx
J
(5 )
sin
5
0
2
∫
=
π
c)
K ∫ x dx
−
=
3
2
4
4. (3+3+3+3=12points)Find dx
dy in the following :
a)
∫
+
=
1
4
3
6
x
t dt
y
b)
4 .
tan( ) ;
3 ;
4
2
= π
=
+
−
=
at x
x
u
u
u
y
c)
sec(4 )
1
x
y
+
=
5
d)
5
3
2
2
=
−
+
y
x
xy
5. (6points)
Express the limit :
4,3 ].
[
,
int
,
)
5
(
lim
3
1
0
−
∆
−
∑
=
=
→
egral where P is a partitionof
as a definite
x
c
c
k
k
k
n
k
k
P
6
6. (3+3+3+3=12points) a) Solve the inequality:
3
5
3
2
− x >
.
b) Given the function:
x
x
y
2
− 1
=
. Find the points on the curve where the
tangent is parallel to the line
x
y
= 9
.
c) For what values of a and b will
≥
+
−
>
=
2
3 ;
2
;
)
(
2
x
bx
ax
x
ax
f x
be differentiable for all values of .x
7
d) Find an equation of the normal line to the curve whose parametric equation is:
.2
3
1
1 ;
1
2
=
= −
= +
at t
t
y
t
x
7. (6points)
( )
.0
0
)
sin(
=
= −
and y
x
dx
dy
Find y as a function of x if
π
π
8
8. (4+4+4=12points)Given the function :
x
x
f x
y
−
=
=
3
( )
a) What is the domain? Find the value(s) of x for which the derivative is zero.
b) For what values of x is the function i) increasing?
ii)decreasing?
c) Sketch the graph of the function. Show important steps.
9
9. (4+4+4=12points)Evaluate the limits:
a)
x
x
x
x
5
2
3
lim
+
→∞
b)
−
→ +
5
cos
8
lim
5
x
x
x
π
c)
2
0
1
sec
lim
x
x
x
−
→
10
10. Solve the two independent parts: (4+4=8points)
a) Assume that f is an even function and g is an odd function.
).
(
.
, ,
7 ( ))
3 ( )
(
( )
,
( )
,
)
(
5
2
5
2
5
2
2
0
Explain
g x dx intermsof a b and c
f x
Find
c
f x dx
b
g x dx
a
f x dx
Let
∫
∫
∫
∫
−
−
=
=
=
b) (Independent part) Evaluate:
(
)
2
.....
3
1 2
lim
n
n
n
+
+
+
+
→∞