MATH 2210 - Final Exam - 12/11/03
Note: Each problem is worth 10 points. For full credit, explain your work and justify your
answers.
Name:
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(1) Let u = ha, bi be a vector. Find the projection P of u in the direction of h1, 1i in terms of a
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and b. Let N = u − P : Show that N is perpendicular to P .
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(2) Calculate the following limits:
2x2 y
.
(a) lim
(x,y)→(0,0) x4 + y 3
(b)
x3 − y 3
.
(x,y)→(0,0) x2 + y 2
lim
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(3) Find the plane passing through the point (0, 1, 0) and perpendicular to the two following
planes:
• The tangent plane to f (x, y) = sin(xy) + 3xy 2 − 2exy at (0, 1).
• The plane with normal vector h1, 1, 1i passing through the point (1, 1, 1).
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(4) Let f (x, y, z) = x2 + y 2 + z 2 ,
9x
x+y+z
the composition g ◦ f at the point (1, 1, 1).
and g(x, y) = (y − x, 2xy 2 ). Find the derivative of
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(5) Let f (x, y) = xy 2 − 5x2 y. Find the maximum and minimum value of f on the region above
the graph of y = x2 and below the line y = 4.
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(6) Find the maximum volume of a rectangular box with no top that can be built with 1200 square
feet of material.
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(7) Calculate the double integral
(1, 1).
RR
R
y 2 dA where R is the triangle with vertices (2, 0), (0, 1), and
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RRR
(8) Set up the triple integral
f (x, y, z) dV where K is the region below the graph of the
K
function 4 − x2 − y 2 and above the x, y-plane.
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(9) Let C be the circle of radius 2 and center (0, 1, 0) in the y, z-plane. Show that
Z
Z
xyz
2
2
4xy dx + e dx − y dx − 2x y dy + y dz − x dz =
2xy 2 dx + z dx + 2xy dy − 2x2 dy − z dy.
C
C
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(10) Let f (x, y, z) be a function. Calculate div(grad(f )) in terms of f and its partial derivatives.
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(Extra-credit) Let A be the 3 by 3 matrix


1 2 0
A = 0 2 1  .
0 0 −1
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(a) Find all the numbers a such that A v = a v for some non-zero vectors v . [Hint: There are
three such numbers a]
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(b) For each a that you found in part (a), find a non-zero vector v such that A v = a v .
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