MATH 2450, QUIZ 7 (solutions), 11 JUNE, 2012
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, EXAM 3, 01 JULY (solutions), 2011
(Each of the following ve problems is worth 12 points, with parts of problems weighted
equally. Be sure to show all work ; and, when in doubt, draw a picture.)
(1) Consider the vector-valued position function r
MATH 2450, QUIZ 11 (solutions), 25 JUNE, 2012
(1) Find a potential (=primitive) function z = f (x, y) for the path-independent
vector eld F (x, y) = (x y) + (y 2 x). Use this function to evaluate
r
F d, where C is any piecewise smooth curve from 2, 0 to
MATH 2450, QUIZ 1 (solutions), 23 MAY, 2012
(1) Find an equation for the sphere, centered at the point 1, 2, 3 and tangent to
the xz-plane.
(x 1)2 + (y 2)2 + (z + 3)2 = 4
(2) There is just one level curve (= contour) for the function f (x, y) = 3x2 + y 2
MATH 2450, QUIZ 5 (solutions), 06 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
= u2 (cos v v sin v) =
u
u2 cos v u2 v
MATH 2450, QUIZ 3 (solutions), 29 MAY, 2012
(1) (a) (2 points) If = 3 and = + 5, nd 2 + 4 .
u
v
u
v
2 + 4 = 2(3 ) + 4( + 5) = 6 2 4 + 20 = 2 + 18.
u v
(b) (3 points) Find all vectors in the plane with magnitude equal to 5 and
with
-component equal
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, EXAM 2 (solutions), 15 JUNE, 2012
(Each of the following ten questions is worth 6 points. Be sure to show your work. And, when in doubt,
draw a picture.)
(1) Find the second-order Taylor polynomial Q(x, y) for f (x) = xey , close to the point 2
MATH 2450, EXAM 1 (solutions), 01 JUNE, 2012
(Each of the following ten questions is worth 6 points. Be sure to show your work. And, when in doubt,
draw a picture.)
(1) Describe precisely the domain of the function f (x, y) =
the plane look like?
x+y
4 (x
MATH 2450, QUIZ 9 (solutions), 19 JUNE, 2012
(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, EXAM 1 (solutions), 03 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) =
1 (x2 + y 2 ).
(a) What is the