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Math 192, Prelim
2
April 14, 2005
1) a) (7 points) Give an example of a function
f
(
x, y
) that has a local maximum at (0
,
0).
Show that your example is valid.
b) (7 points) Give an example of a function
f
(
x, y
) that has a saddle point at (0
,
0).
Show
that your example is valid.
2) (11 points) Determine all the maxima and minima of the function
f
(
x, y
) =
x
2
+
xy
on
the region of points (
x, y
) satisfying
x
2
+
xy
+
y
2
≤
1.
3) (11 points) Find the average distance from a point in the interior (or on the boundary)
of a circle of radius
R
to the center of the circle.
4) (11 points) Let
C
be a circle of radius 2 +
√
2
2
centered at the origin.
Find the area of
the region
outside
C
but
inside
the region bound by
r
= 2 + cos
θ
.
5) The questions below are true/false. You need not give reasons.
You will be penalized for wrong answers, just like the SAT’s. So do not guess!!
a) (
±
3 points) If
f
(
x, y
) is given and
±u
is a unit vector in two dimensions then the directional
derivative of
f
in the direction of
±u
at (
x
0
, y
0
) is
f
x
(
x
0
, y
0
)
±
i
+
f
y
(
x
0
, y
0
)
±
j
.
b) (
±
3 points) If
f
(
x, y, z
) is a scalar function then the expression grad(div(
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This note was uploaded on 10/31/2008 for the course MATH 1920 taught by Professor Pantano during the Fall '06 term at Cornell University (Engineering School).
 Fall '06
 PANTANO
 Math

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