HOMEWORK 1
Multivariate Analysis and Approximation
Due date: September 3rd
Total points: 30
Only 5 problems are graded in detail, the others are graded for completion
Stewart:
10.1: 10, 16
10.2: 12, 18
10.3: 18, 26
7. A cat is sitting on the ground at the
21-256 Homework 3
Due Wednesday 28th May 2014
1. Use the scalar triple product to verify that following three vectors are coplanar:
u = i + 5j 2k,
v = 3i j,
w = 5i + 9j 4k
2. Find the acute angle between the lines 2x y = 3 and 3x + y = 7 in R2 .
3. Find t
21-256 Homework 4
Due Monday 2nd June 2014 (extra credit for early submission, see below)
1. Show that if u, v, w are linearly dependent vectors in R3
1
T 3B, AB and BA when A = 1
2. Compute A + B, 2A
3
then [u, v, w] = 0.
2 1
1 1 0
0 2 and B = 1 0 1.
1 1
21-256 Homework 2 (solutions)
Due Friday 23rd May 2014
1. Find the equation of a sphere if one of its diameters has end-points (2, 1, 4) and (4, 3, 10).
The center of the sphere is the midpoint of the given points, i.e. (3, 2, 7). The radius is the
distan
Multivariate Analysis (21-256)
Clive Newstead, Summer I 2014
Class info
Instructor info
Time: Every weekday at 10:30am11:50am
Name: Clive Newstead
Location: Wean Hall 4623
Oce: Wean Hall 8205
Units: 9 units
Email: cnewstead@cmu.edu
Website: http:/math.cmu
Quiz 14.7
(If you don know the answers to these questions, you should look them
t
up in your lecture notes, and textbook, and you should know them before
you start working problems for the exam).
1. How do you determine if a matrix is positive denite, neg
14.8: Lagrange Multipliers
o Maximize the production of a firm under a budget constraint.
0 Let f(x, y) be the production function, x, y: quantities of two raw materials
fay) : x2r3y113
lfx and y are purchased at prices p1 and p2 thousands of dollars per
21-256: Dot and cross products
Clive Newstead, Thursday 22nd May 2014
This is a summary of the important results about dot and cross products that you should know.
Dot product
The dot product v w of two n-dimensional vectors v and w is a scalar, dened by
21-256: Tangent planes and linear approximation
Clive Newstead, Thursday 5th June 2014
Tangent planes
Equations involving three variables all describe surfaces in R3 ; moreover, any such equation can
be rearranged to take the form f (x, y, z) = 0, just by
21-256: Applications of integration to probability
Clive Newstead, Monday 23rd June 2014
Intuitively, a (real, continuous) random variable is a real number quantity whose precise value
is unknown until it is observed. The probability distribution function
21-256: Matrices
Clive Newstead, Thursday 29th May 2014
This is a summary of the important results about matrices that you should know.
Operations on matrices
Matrix addition. If A and B are both m n matrices then A + B is the m n matrix
dened by
(A + B)
21-256: Lagrange multipliers
Clive Newstead, Thursday 12th June 2014
Lagrange multipliers give us a means of optimizing multivariate functions subject to a number of
constraints on their variables. Problems of this nature come up all over the place in rea
21-256: Integration
Clive Newstead, Monday 23rd June 2014
b
Recall that if f is a function of a single variable then
f (x) dx is equal to the area of the region
a
under the curve y = f (x) lying above the x-axis between x = a and x = b (minus the area lyi
21-256 Homework 5
Due Friday 6th June 2014
1. Find and sketch the largest possible domain of the bivariate function f dened by
f (x, y) =
2. Compute
x2 y
3f
f 2 f
and
,
when f (x, y, z) =
x x2
xyz
3. Find the partial derivatives of u(x, y) =
4. Compute
1
21-256 Homework 2
Due Friday 23rd May 2014
1. Find the equation of a sphere if one of its diameters has end-points (2, 1, 4) and (4, 3, 10).
2. Describe in words the region of R3 represented by the equation x = z.
3. Find v + w, 2v + 3w, v and v w when v
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PAPER HOMEWORK 9
21-256: Multivariate Analysis, Spring 2014
Due: Wednesday, April 23rd (No late homework accepted)
1. Find the local maximum and minimum values (and the points at which
they occur) and the saddle point(s) of the function.
f (x; y) = xy (1
21-256 Multivariate Analysis: HW4 Solutions
Hanif Joey Cheung
October 3, 2016
Graded problems: Chapter 14.2, Problem 16 (3 points); Chapter
14.2, Problem 38 (4 points); Chapter 14.3, Problem 50 (3 points).
See grading scheme following each problem for det
21-256 Multivariate Analysis: HW1 Solutions
Hanif Joey Cheung
September 8, 2016
Graded questions: Ch 12.1 Ex 20 (3 points), Ch 12.2 Ex 24 (3 points),
Ch 12.3 Ex 24 (4 points). See grading scheme following each question
for details.
1
Ch 12.1, Ex 20
Show t
21-256 Assignment 3, Solutions
Disclaimer: This sheet contains summarized answers to the homework problems, not the full homework. Students are expected to show all the relevant
work in their homework assignments.
1. For simplicity we only show explicitly