PH 8
Page 1
PH 8
Page 2
#2:
@Y
@Y
= aL (1
)K =) K
= KaL (1
@K
@K
@Y
@Y
= a L 1 K 1 =) L
= La L 1 K 1
@L
@L
)K
= aL (1
)K 1
= a L K1
Add the two expresssions:
K
@Y
@Y
+L
=
@K
@L
=
=
=
aL (1
)K 1
+ a L K1
aL K 1
aL K 1
Y
(1
+)
which is what we were supposed
PAPER HOMEWORK 6
21-256: Multivariate Analysis, Fall 2012
Due: Monday, October 22nd, in lecture
The use of solutions provided to students who took the course in previous years
is strictly forbidden. You must show your complete work for credit. Answers not
PAPER HOMEWORK 7
21-256: Multivariate Analysis, Fall 2012
Due: Friday, October 26th, in lecture
NO LATE HWS ACCEPTED!
The use of solutions provided to students who took the course in previous years
is strictly forbidden. You must show your complete work f
Learning goals for section 10.1:
At the end of this lecture you should:
- know how to plot points in a 3d coordinate system
- know the equations for the coordinate axes, coordinate planes, planes
parallel to the coordinate planes
- know the distance formu
11/ 9/12
Math 32A: Solution to Pg. 811, Problem 25 in Stewart
pg. 811, #25
The temperature T in a metal ball is inversely proportional to the distance from the center of the ball,
which we take to be the origin. The temperature at the point (1, 2, 2) is 1
Math 263 Practice Problem Set 1 Solutions
1. Find parametric equations for the tangent line to the curve of intersection of the paraboloid z = x2 +y 2 and the ellipsoid 4x2 +y 2 +z 2 = 9
at the point (1, 1, 2).
Solution. Let the parametric representation
Math 2433 Homework #5
Solutions
Section 15.6.
8, 12, 16, 20, 22, 24, 28, 32
Find the gradient of f , evaluate the gradient at point P and nd the rate of change of f
at P in the direction of the vector .
u
2
1
8. f (x, y ) = y , P (1, 2), u = 3 2, 5 .
x
So
Math 2433 Homework #5
Solutions
Section 15.6.
8, 12, 16, 20, 22, 24, 28, 32
Find the gradient of f , evaluate the gradient at point P and nd the rate of change of f
at P in the direction of the vector .
u
2
1
8. f (x, y ) = y , P (1, 2), u = 3 2, 5 .
x
So
What are matrices used for?
Organize information:
Store 1 of a three store chain has 3
refrigerators, 5 stoves, 3 washing machines,
and 4 dryers in stock.
Store 2: no refrigerators, 2 stoves, 9 washing
machines and 5 dryers
Store 3: 4 refrigerators, 2 st
Math 21-256, Multivariate AnalysisI, Fall 12
Squeeze Theorem Example
Here is an example of how to use the squeeze theorem.
x2 y 3
(x,y )(0,0) 2x2 + y 2
lim
0
Plugging in (0,0) yields 0 . We rst try to see if the limit diverges. We try the
path along x = 0
MULTIVARIATE ANALYSIS (21-256)
FALL 2012
Instructor:
Dana Mihai
8128 Wean Hall
dmihai@andrew.cmu.edu
Lecture Hall:
DH 2210
Course website:
http:/www.cmu.edu/blackboard
Office hours:
Will be posted on Blackboard
-LEARNING OBJECTIVES
These objectives should
Learning goals for section 10.2 (Vectors):
At the end of this lecture you should:
- know how to characterize a vector
- be able to recognize equivalent/equal vectors
- understand the concepts of components and decomposition
- know examples of vectors: dis
FAQ and Course Policies
EXAMS:
Are there any requirements for how the work should be presented on the exam?
o
Can I use a calculator, cell phone or laptop during an exam?
o
You are strongly advised to use the restroom before the exam. However, you are all
PAPER HOMEWORK 10
21-256: Multivariate Analysis, Fall 2012
Due: Wednesday, November 28th, in lecture
NO LATE HOMEWORK ACCEPTED
The use of solutions provided to students who took the course in previous years
is strictly forbidden. You must show your comple
PAPER HOMEWORK 8
21-256: Multivariate Analysis, Fall 2012
Due: Friday, November 9th, in lecture
The use of solutions provided to students who took the course in previous years
is strictly forbidden. You must show your complete work for credit. Answers not
PAPER HOMEWORK 9
21-256: Multivariate Analysis, Fall 2012
Due: Friday, November 16th, in lecture
The use of solutions provided to students who took the course in previous years
is strictly forbidden. You must show your complete work for credit. Answers no