we have defined open sets in R: for any a ∈ R, there is sigma > 0 such that (a − sigma, a + sigma) ⊆ A.

(i) Let A and B be two open sets in R. Show that A ∩ B is open.

(ii) Let {Aα}α∈I be a family of open sets in R. Show that ∪(α∈I)Aα is open. Hint: Follow the definition of open sets.

Please be specific and rigorous! Thanks!

Find two power series solutions of the following differential equations.

y'' - xy' = 0

Let B = (p0, p1, p2) be the standard basis for P2 and B = (q1, q2, q3) where:

q1 = 1 + x , q2 = x + x
2 and q3 = 2 + x + x
2
1. Show that S is a basis for P2.
2. Find the transition matrix PS→B
3. Find the transition matrix PB→S
4. Let u = 3 + 2x + 2x
2
.
Deduce the coordinate vector for u relative to S

Show that the inverse of an invertible matrix A is unique. That is, suppose that B is any matrix such that AB = BA = I. Then show that B = A−1 .

Briefly compare and contrast Trapezoid Rule and Simpson’s Rule. Talk about the ways in which they are conceptually similar, and important ways in which they differ. Use the error bound formulas (found in the notes, and on the practice final exam) to show that the error in using these formulas must approach zero as h (the distance between adjacent nodes) approaches zero.

Find a series solution ofy′′−xy′+ 2y= 0.

Consider the following model of interacting species:

R' = R(2 + 3R − S)

S' = S(1 − S + 4R)

(a) Find all the equilibrium points, and determine the type of those points which are in the first quadrant (including those on the axes)

(b) Plot the phase portrait of the system.

(c) If the initial conditions are R(0) = 1 and S(0) = 1, what will be the population size of each species when t → ∞?

Haskell

Map and Filter

6. Let f1 = filter (\ x -> x > 0) and f2 = filter (\x -> x < 10), and let nbrFilter g x = length (filter g x).

a. Rewrite f1(f2[-5..15]) so that it uses function composition to apply just one function to the list.

b. Rewrite the nbrFilter function definition to have the form

nbrFilter g = function composition involving length and filter … and leaving out x.

Use Stoke's Theorem to find the circulation of F⃗ =7yi⃗ +3zj⃗ +2xk⃗ around the triangle obtained by tracing out the path (5,0,0) to (5,0,3) to (5,5,3) back to (5,0,0)

3. A machine is used to fill containers with a liquid product. Fill volume can be assumed to be
normally distributed. A random sample of ten containers is selected, and the net contents
(oz) are as follows: 12.03, 12.01, 12.04, 12.02, 12.05, 11.98, 11.96, 12.02, 12.05, 11.99.
a. Suppose that the manufacturer wants to be sure       that the mean net contents
exceeds 12 oz. What conclusions can be drawn from the data. Use a = 0.01.
b. Construct a 95% two-sided confidence interval on the mean fill volume.
c. Does the assumption of normality seem appropriate for the fill volume data?
d. Repeat (a,b,c) above for a = 0.05. Compare with results for a = 0.01.

A particular county in a certain state of a far away country has 17 cities and wants to build roads between them to connect them all, that is, there should be a paved path between any two cities. What is the minimum number of roads, each connecting two cities, to guarantee that all cities are connected? Be careful, wrong answers have negative weights.

Denition:
An orthogonal array OA(k, n) on n symbols is an n² x k array such that, in any two columns, each ordered pair of symbols occurs exactly once.
Prove that there exists an OA(k, n) if and only if there exist (k - 2) mutually orthogonal Latin squares of order n.

(combinatorics and design)

Carlo and Anita make mailboxes and toys in their craft shop near Lincoln. Each mailbox requires 1 hour of work from Carlo and 2 hours from Anita. Each toy requires 1 hour of work from Carlo and 3 hours from Anita. Carlo cannot work more than 55 hours per week and Anita cannot work more than 12 hours per week. If each mailbox sells for $ 15 and each toy sells for $ 24, then how many of each should they make to maximize their​ revenue? What is their maximum​ revenue?

Carlo and Anita should make __ mailboxes and __ toys. Their maximum revenue is $__.

Use the Fourier-Motzkin Elimination method to solve problem 4.1-5 from the textbook (10th edition).

Maximize Z=x₁ + 2x₂,

subject to

x₁ + 3x₂ <=8

x₁ + x₂ <=4

and

x₁ >=0, x₂>=0.