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 8
Due Thursday 26th June 2014
. . . but if you want feedback before the test, you should submit it on Wednesday
2
4
4
2
y 3 e2x dx dy.
y 3 e2x dy dx and verify that it is equal to
1. Compute
0
0
0
2. Find the average value of the function
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
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 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: [email protected]
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 Homework 6
Updated Tuesday 10th June 2014
Due Friday 13th June 2014
1. Find
f when f (x, y) = ln(x6 + 3x2 y 4 + y 6 ) .
2. Find
g when g(x1 , x2 , , xn ) = x1 + 2x2 + 3x3 + + nxn .
n
2
3
3. Find and classify the critical points of the function f (x
21-256 Homework 7
Updated Thursday 19th June 2014
Due Monday 23rd June 2014
Use the method of Lagrange multipliers to solve the following problems.
1. Maximize exy subject to x3 + y 3 = 16.
2. Minimize x2 + y 2 + z 2 subject to x + y + z = 12.
3. By using
Nico Slate
[email protected]
Phone: 412-268-1408
Baker Hall 365
Office Hours: Mon. 2:30-3:30
and by appointment
Global Histories: Innovation and Social Change
History 79104, Fall 2016, Monday / Wednesday 12:30-1:20 or 1:30-2:20, Porter Hall 100
If you wanted