MAT 203 Midterm, Fall 2009
American University of Science and Technology · Calculus III · 1 page · Ms. Hiba Othman
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MAT 203
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I. (20 pts- 5 pts each)
a) Find the Maclaurin series for
xe .
b) Find the radius and interval of convergence of the Maclaurin series for
xe .
c) Find the Maclaurin series for
e
1 ?
d) Evaluate
e
1 with an error of magnitude less than
!7
1 ?
II. (10 pts – 5 pts each) Consider the polar curve
2
cos
r2
a) Sketch the curve of the lemniscates.
b) Find the area enclosed by one loop of the above lemniscate.
III. (5 pts ) Sketch the following surface :
18
3
2
2
2
2
x
z
y
IV. (10 pts) Find the volume which lies inside of the sphere centered at the origin and of
radius19 and between the cones
6
&
4
V. (10pts) Find all local maxima, minima, saddle points of the equation:
3
30
2
5
,
2
2
xy
xy
x y
f x y
VI. (5 pts each) Evaluate the following integral :
2
0
2
4
2
2
1
1
dy
dx
y
y
y
x
VII.
(10 pts) Evaluate the volume of the region in space bounded by the paraboloid
2
2
1
y
x
z
and the plane
z 0
VIII. (10 pts) Find the spherical coordinate limits for the integral that calculates the
volume of the solid formed by the intersection of the inside of the sphere
25
3
2
2
2
x
z
y
and the inside of the cylinder
4
2
2
y x
IX. (5 pts) Find the directional derivative of
2
cos
, ,
z
xy
f x y z
at
2 1,
,1
0
p
in the direction of vector
k
i
v
3
4
X. (5 pts) Maximize the function
xyz
f x y z
, ,
subject to the constraints
64
2
2
2
xz
yz
xy
XI. (5pts) Find the Fourier series of
x
where
x
f x
4
2
XII.
(5pts) Find using the Maclaurin series the following limit:
9
3
sin
0
lim
x
x
x
x