Math 259 Solutions to Homework #4 Problems from Pages 521-522 (Section 10.1) 6.
Winter 2009
(a) In R2, x = 4 represents a vertical line. In R3 this represents a plane that is parallel to the yz plane. The sketches are shown below.
y x=4 z
x
x=4
x
y
(b)
In
Math 259 Solutions to Homework #9 Problems from Pages 658-659 (Section 11.8) 10.
Winter 2009
Given f(x, y, z) = x4 + y4 + z4 and the constraint g(x, y, z) = x2 + y2 + z2 - 1 = 0, the three equations that we get by setting up the Lagrange multiplier equati
Math 259 Solutions to Homework #7 Problems from Pages 599-601 (Section 11.1) 22.
Winter 2009
The contour plot on the left corresponds to the paraboloid and the contour plot on the right corresponds to the cone. You can tell because the bottom (or top) of
1
MATH 259 FINAL EXAM
Friday, May 8, 2009.
NAME:
Alexandra
Klimova
Oleksii
Mostovyi
Reshma
Ramadurai
Stephen
Russel
William
Boney
A
B
C
F
D
E
G
H
Instructions:
1.
2.
3.
4.
5.
6.
7.
Do not s eparate the p ages of the exam.
Please read the instructions for
Warning: The following is a practice exam. Do not expect the actual exam to necessarily consist
of the same or similar problems.
1. Reparameterize the curve r(t) = h3
t; 2t; 4i, 0
t
1, with respect to arc length.
2. Find the maximum directional derivative
Carnegie Mellon University
Department of Mathematical Sciences
21-259 Calculus in 3 Dimensions
Review Exam 3, Fall, 2010
Your exam shall consist problems similar to the homework and quiz problems. Please prepare by practicing
these problems. Wednesday, No
Note Title 10/13/2010
1. Find the limil. or Show it does not. exist, Should the limit. exist. you must determine the limit
and provide the appropriate argument. Il the limit. does not. exist. you must Clearly explain why.
shorting your work.
rug
l .
2:;.r. + If} 3 = 2
A A [at +¢~L£A m
e 6Iru-X'W Yukon V of 1 WC C < 4, 0>
:. 3 '
35? «<z.-t.z>x4-Z»'I'> )
\I - l >
t :7 .1 <1 29>)
(Sea-WE ) W 3 1
:, \
a) Z
3? ' 3
7 1) W «A. Can'- Lo)\, \ )
)0
4 . \
. 4L _. . a
M Z ' z °
nib} Find the point.
Math 259 Solutions to Homework #3 Problems from Pages 502-504 (Section 9.3) 18.
Winter 2009
We will substitute for x and y in the linear equation and then solve for r. x+y=9 rcos() + rsin() = 9 r(cos() + sin() = 9
r=
28.
9 . cos(" ) + sin(" )
A sketch of