Department of Mathematics
The City College of New York
Math 203
Final Exam
Spring 2008
No calculators permitted. Answers may be left in terms of radicals,
π
,
e
, etc and do not
need to be simpliﬁed unless stated otherwise. Each question is worth 10 points.
Part I. Answer all 7 questions
1.
(a) A curve is given parametrically by the equations
x
= sin(4
t
);
y
= cos(4
t
);
z
= 2
t
3
/
2
.
Compute the parametric equations of the tangent line at the point when
t
= 1.
(b) Write an iterated integral, using polar coordinates, whcich can be used to ﬁnd the surface
area of the portion of the graph of
z
=
xy
3
which is above the region in the ﬁrst quadrant
of the xyplane inside the circle
x
2
+
y
2
= 1. YOU DO NOT HAVE TO EVALUATE THE
INTEGRAL.
2.
For parts (a),(b) and (c) let
f
(
x,y
) =
x
2

e
xy
.
(a) At the point (3
,
0) compute the unit vector in the direction of maximum increase of the
function
f
and compute the rate of increase in that direction.
(b) Compute the directional derivative of the function
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 Spring '11
 Ocken
 Radicals, right circular cylinder, circle x2

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