MATH 203
Solutions
Week 6
In exercises 1,2,3,6,7,8, find and classify the critical points of the given functions.
Exc 13.1 no 1) f (x, y) = x2 + 2y 2 4x + 4y
fx (x, y) = 2x 4 = 0 when x = 2. fy (x, y) = 4y + 4 = 0 when y = 1
(2, 1) is the critical point.

MATH 203
Solutions
Week 2x 1
p
(2 0)2 + (1 0)2 + (2 0)2 = 4 + 1 + 4 = 9 = 3
p
Exc 10.1 no 2) d = (1 (1)2 + (1 (1)2 + (1 (1)2 = 4 + 4 + 4 = 12 = 2 3
Exc 10.1 no 1) d =
Exc 10.1 no 6) if we let A=(1,2,3); B=(4,0,5); C=(3,6,4), then the lengths of the three

Math 203
Solution
week 8
Exercise 14.2
2) calculate
R1Ry
0
0
(xy + y 2 )dxdy.
solution:
for questions 6,8,12,14, evaluate the double integrals by iteration.
6)
ZZ
x2 y 2 dA
R
, where R is the rectangle 0 x a, 0 y b.
solution:
8)
ZZ
(x 3y)dA
T
, where T is

MATH 203
Solutions
Week 5
1
Exc 12.5 no 6) Method 1:
u
x
y
sest + p
s2 sint
=p
t
x2 + y 2
x2 + y 2
xsest ys2 sint
p
=
x2 + y 2
Method 2:
u=
p
e2st + (1 + s2 cost)2
u
2se2st 2s2 sint(1 + s2 cost)
p
=
t
2 e2st + (1 + s2 cost)2
=
x2 s ys2 sint
p
x2 + y 2
Ex

MATH 203
Solutions
Week 11
In exercises 2,4,6,8 find divF and curlF for the given vector fields
Exc 16.1 no 2) F = yi + xj
divF = x
(y) + y
(x) + z
(0) = 0 + 0 = 0
i
j
k
curlF = x
y
z = (1 1)k = 0
y
x
0
Exc 16.1 no 4) F = yzi + xzj + xyk
divF = x
(yz)

Chapter 1
Math 203 Hw 9
1.1
14.5
* Evaluate the triple integrals over the indicated region. Be alert for simplifications and auspicious orders of iteration for 2, 4, 6.
RRR
2)
xyzdV , over the box B given by 0 x 1, 2 y 0, 1 z 4.
B
Z Z Z
Z
xyzdV =
Z
y
Z
xd

MATH 203
Solutions
Week 2
Exc 10.4 no 2) The plane equation passing through the point (0, 2, 3) and normal to the vector
< 4, 1, 2 > is: 4(x 0) 1(y 2) 2(z (3) = 0. Or more simply:
4x y 2z 4 = 0.
Exc 10.4 no 3) The plane equation passing through the origin

MATH 203
Solutions
Week 7
Find the maximum and minimum values of:
Exc 13.2 no 1) f (x, y) = x x2 + y 2 on the rectangle 0 x 2 , 0 y 1.
fx (x, y) = 1 2x = 0 and fy (x, y) = 2y = 0 gives ( 12 , 0) is the critical point inside R (the rectangle)
and f ( 21 ,

Math 203 Hw 4
Spring 2015
Chapter 1
Math 203 Hw 4
1.1
12.2
* Evaluate the indicated limit or explain why it does not exist for 1, 3, 5, 6.
1) lim(x,y)(2,1) xy + x2
= 2(1) + 22 = 2
3) lim(x,y)(0,0)
x2 +y 2
y
It does not exist. If (x, y) (0, 0) along x = 0,

Math 203
Solution
week 3
Exc 11.3 no 5) the cylinders z = x2 and z = 4y 2 intersect in two curves, one of which passes through the
point (2,-1,4). Find a parametrization of that curve using t = y as parameter.
solution:
if t = y, then z = 4y 2 = 4t2 and x

MATH 203
Solutions
Week 10
Exc 15.1 no 2)
F = xi + yj.
The field lines satisfy
dy
dx
=
x
y
Thus,
lny = lnx + lnC
or
y = Cx
The field lines are straight half lines emanating from the origin.
1
Exc 15.1 no 8)
F = cosyi cosxj.
The field lines satify
dx
dy
=