MAT 321 Exam I Sample
Professor Sormani
1. (20 points) Let A = [2, 1] [5, 3]. Let B = B(3,4) (5). Let C = cfw_(x, y ) : x2 + y 2 100 and let D = cfw_(x, y ) : y = x 5. a) Draw all of these sets on one chart shading them lightly if they have area. Use dash
AMath 567, Autumn 2011
Assignment 4. Due Fri., Oct. 28.
Reading: Secs. 2.4-2.5, Secs. 5.1-5.4.
1. Exercise 2.2 on p. 58.
2. Exercise 2.5 on p. 59.
3. Suppose y : R R and z : R R satisfy
y (t) = f (t, y(t)
y(0) = y0
z (t) = f (t, z(t)
,
z(0) = y0 + 0
where
AMath 567, Autumn 2011
Assignment 6. Due Mon., Nov. 21.
Reading: Secs. 5.1-5.4.
1. Exercise 5.7 on p. 122 in the text.
2. Exercise 5.8 on p. 122 in the text.
3. Let k k and k k be two norms on a vector space V . Suppose that all sequences
cfw_vn V which
AMath 567, Autumn 2011
Sample Solutions for Assignment 1.
Due Friday, Oct. 7.
Reading: Ch 1.
1. If x, y, and z are points in a metric space (X, d), show that
d(x, y) |d(x, z) d(y, z)|.
By the triangle inequality,
d(x, y) + d(y, z) d(x, z) = d(x, y) d(x, z
AMath 567, Autumn 2011
Assignment 7.
Due Wednesday, Nov. 30.
Reading: Rest of Ch 5. Ch. 6.
1. Exercise 5.2 on pp. 120-121 in the text. NOTE: There are typos in this exercise.
In part (a), show that
xi =
n
X
ji xj .
L
n
X
Lij j .
j=1
In part (b), show tha
AMath 567, Autumn 2010
Assignment 1.
Due Friday, Oct. 7.
Reading: Ch 1.
1. If x, y, and z are points in a metric space (X, d), show that
d(x, y) |d(x, z) d(y, z)|.
2. Let X = Cn and let A be a nonsingular n by n matrix. Define a function kkA : X R
by
kxkA
AMath 567, Autumn 2011
Assignment 3.
Due Friday, Oct. 21.
Reading: Secs. 3.1-3.3 and 3.5. Also Sec. 2.4-2.5.
1. The Weierstrass approximation theorem says that the set of polynomials is dense in
(C[a, b], k k ). Let S = cfw_xp(x) : p is a polynomial; i.e.
AMath 567, Autumn 2011
Assignment 2.
Due Friday, Oct. 14.
Reading: Finish Ch 1. Read Secs 2.1-2.3 and Sec. 3.1.
1. Do ONE of the following two problems:
(a) Use MATLAB or another programming language to fit a polynomial of degree 12
to the Runge function
AMath 567, Autumn 2011
Assignment 8. Due Wed., Dec. 7.
Reading: Sec. 7.1-7.4. Sec. 8.1-8.2.
1. Apply Parsevals relation to the function f (x) = x on (, ) to evaluate
2. Exercise 7.3 on p. 183.
3. Exercise 7.7 on p. 184.
4. Exercise 7.10 on p. 184.
5. Exer
Quantiers
Professor Sormani A supplement for MAT175
Quantiers are special symbols that are used to make it easier to think about mathematical statements. There are three quantiers: means for all for every means there exists there is a ! means there exists
MAT 321 Exam II Sample Answers
Professor Sormani
1. (20 points) Let A = [2, 1] [5, 3). Let B = B(3,4) (5). Let C = cfw_(x, y ) : x2 + y 2 100 and let D = cfw_(x, y ) : y x 5. b) Write down the closures of each of these sets: Cl(A) = [2, 1] [5, 3] Cl(B ) =
Problem 1.1.
(a)We just need to build an array containing all rational numbers without repeat to show it is a one-to-one map to N.
1 1 2 2 1 2 1 2 3 3 3 3
My build is 01 , 11 , 1
1 , 2 , 2 , 1 , 1 , 3 , 3 , 3 , 3 , 1 , 2 , 1 , 2 , . . . Thus, it is a bije