MATH 2450, QUIZ 7 (solution), 28 OCT., 2010
(1) Let B be the lower semidisk cfw_x, y : x2 + y 2 1, y 0, and let f (x, y) =
x2 (y y 3 ). Without calculating, decide whether the integral B f dA is positive, negative, or zero. (Justify your answer.)
On B we
MATH 2450, QUIZ 1 (solutions), 25 MAY, 2011
(1) Find an equation for the sphere of radius 5, centered at the point 1, 2, 3.
(x 1)2 + (y 2)2 + (z + 3)2 = 25
(2) Sketch the graph of the equation x2 + z 2 = 4, and briey describe the surface
in words.
All cro
MATH 2450, QUIZ 9 (solutions), 11 NOV., 2010
(1) Calculate
C
(2 r, where C is the straight line path from 0, 0 to 5, 0.
i+3j)d
(2 + 3 d = C 2 dx + 3 dy. C can be parameterized as x = t,
i
j) r
y = 0, 0 t 5. Then dx = dt and dy = 0, so the integral is
5
2
MATH 2450, QUIZ 11 (solution), 02 DEC., 2010
(1) Find (3x2 sin(xy) + xez
i
j
k).
This is the divergence of a vector eld, and so a scalar function.
(3x2 )
( sin(xy) (xez )
(3x2 sin(xy) + xez =
i
j
k)
+
+
=
x
y
z
6x x cos(xy) + xez .
r
(2) Use the diverge
MATH 2450, QUIZ 3 (solutions), 31 MAY, 2011
(1) (a) (2 points) If = 3 and = + 5 nd 2 + 4 .
u
i j
v
i
j,
u
v
2 +4 = 2(3
u v
i j)+4( = 6 4
i+5j)
i2j i+20 = 2
j
i+18
j.
(b) (3 points) List all vectors in the plane with magnitude 5 and rst component equal to
MATH 2450, EXAM 2 (solutions), 17 JUNE, 2011
(Each of the following six problems is worth 10 points, with parts of problems weighted
equally. Be sure to show all work; and, when in doubt, DRAW A PICTURE.)
(1) Let f (x, y) = x3 y + xy 2 .
(a) Does f achiev
MATH 2450, QUIZ 7 (solutions), 13 JUNE, 2011
Refer to the function f (x, y) = x3 3x + y 3 3y in answering the following.
(1) Find all four critical points of f , and classify each as local maximum, local minimum, or
saddle point.
First solve f (x, y) = (3
MATH 2450, QUIZ 9 (solutions), 21 JUNE, 2011
(1) Find parametric equations for the straight line that contains the points 1, 5, 2
and 5, 0, 1.
One possible parallel vector is = (1 5) + (5 0) + (2 (1) =
a
i
j
k
+ 5 + 3 So a parameterization using and the
MATH 2450, QUIZ 11 (solutions), 27 JUNE, 2011
(1) Use Greens theorem to calculate C xy dx + x2 dy, where C is the positivelyoriented perimeter of the rectangle cfw_x, y : 0 x 1, 0 y 2.
Where P (x, y) = xy and Q(x, y) = x2 , Greens theorem tells us that
1
MATH 2450, QUIZ 5 (solutions), 07 JUNE, 2011
Suppose z = xy, where x = u cos v and y = uv.
(1) Calculate z/v by rst eliminating the intermediate variables, and then calculating partial derivatives in the usual way.
z = (u cos v)(uv) = u2 v cos(uv), so
u2