Math 221-501
Solutions Final Exam
December 12, 2003
1. (30) Dene the following:
(a)
lim
(x,y )(2,1)
f (x, y ) = 5
This means that for any
then |f (x, y ) 5| < .
> 0 there is a > 0 such that if 0 < (x, y ) (2, 1) < ,
(b) the directional derivative of f at
Math 221-502
1.
Solutions Final Exam
(20) Define the following:
a.
lim
f x, y, z 7
x,y,z
2,1,3
Means that for any 0 there is a 0 such that if
|f x, y, z 7 | .
b.
h
c.
d.
x 2, y
the directional derivative of f at the point 2, 3 in the direction
D N f lim
2
Math 221-501
Solutions Exam 3
December 2, 2003
1. (15) Dene the following:
(a)
lim
(x,y )(2,1)
f (x, y ) = 5
This means that for any > 0, there is a > 0 such that if 0 <
, then |f (x, y ) 5| < .
(x + 2)2 + (y 1)2 <
(b) State the mean value theorem
Let f b
Math 221-501
Solutions to Exam 2
November 4, 2003
1. (15) Dene the following
(a) f (x, y ) is dierentiable at the point (1, 2).
This means that both of f s partial derivatives exist at the point
(1, 2), and for x and y small enough we have
f (1 + x, 2 + y
Math 221-501
Solutions to Exam 1
October 7, 2003
1. (20) Dene the following
(a)
lim
(x,y )(2,5)
f (x, y ) = 2.
This means that for any
if 0 <
> 0 there is a > 0 such that
(x 2)2 + (y 5)2 < , then |f (x) 2| < .
(b) f (x, y ) is dierentiable at the point (1
Math 221-500
Solutions to Final Exam
May 7, 1999
1. Let f (x, y, z ) = 3x2 2y + z 2 . Let S be the surface which is the locus
of points which satisfy the equation f (x, y, z ) = 0. Find an equation
for the plane tangent to S at the point (1, 2, 1).
Since
Math 221
1.
Exam 3 Solutions
November 29, 2006
(10)
a.
Define
lim
x,y
2,3
f x, y 7,
This means that for any 0, there is a 0 such that if
0
b.
x2
2
y
2
3
, then |f x, y
state the Divergence Theorem.
Let E be a bounded region in R 3 whose boundary S consis
Math 221-500
Solutions to Exam 3
April 21, 1999
1. (30) Let R = cfw_(x, y ) : x2 + y 2 1, and 0 y . Let F (x, y ) =
x+
y2
, 2x + xy .
2
Hint: remember Greens theorem.
(a) Find the integral of the tangential component of F along R,
where R is traversed in
Math 221
1.
Solutions Exam 2
November 1, 2006
(15) Define the following
a. f x, y is differentiable at the point 1, 2 .
This means there are numbers i such that
f
f 1 x , 2 y f 1, 2
x
x 1, 2
and
lim
x, y
0,0
f
y
y 1 x 2 y,
1, 2
i 0 for i 1, 2.
b.
f x, y
Math 221-500
Solutions to Exam 2
March 26, 1999
1. (20) Show that the equation r = 2 sin is a circle. Find its center and
radius.
y
r = 2 =
r
r 2 = 2y =
x2 + y 2 2y + 1 = +1 =
x2 + (y 1)2 = 1
Center at (0, 1) and radius 1.
2. (20) What are the cylindrical
Math 221-502
1.
Exam 1 Solutions
(20) Define the following:
a.
lim f x, y 3,
x,y
0
b.
1,2
Means that for any 0, there is a 0 such that if
x, y
1, 2 , then |f x, y 3 |
f is differentiable at the point 1, 2 ,
Means there are numbers 1 and 2 such that
f
f
x
Math 221-500
Solutions to Exam 1
February 17, 1999
Be sure to show and/or explain your work, bald answers are not worth
much.
1. (20) Let (t) = (cos t, sin t, 8 t/2) for 0 t 8, be the path of a y
in our classroom.
(a) Where will the y be located when t =