Tanks A and Bare filled with 100 gal and 200 gal of brine respectively. Initially, 40 lb of salt dissolved in the solution in tank A and 20 lb of salt dissolved in the solution in tankB. The system is closed in that the well-mixed liquid pumped from A into B and from B into A at a rate of 25gal/min each (so, the liquid volumes do not change over the time in both tanks).

(a) Find the general equation which represents the salt quantity. (Don’t forget that this is an IVP)

(b) Find the salt concentration of tank A after 1hr (60min). Explain the long term behavior briefly.

Suppose A is the set of positive real numbers, and suppose u and v are two strictly increasing functions.1 It is intuitive that u and v are ordinally equivalent, since both rank larger numbers higher, and therefore generate the same ranking of numbers. Write this intuition as a proof.

ALGORITHMS AND ANALYSIS:

Show with a counterexample that the greedy approach does not always yield an optimal solution for the Change problem when the coins are U.S. coins and we do not have at least one of each type of coin.

if y(t) is the solution of y′′+2y′+y=δ(t−3),y(0)=0,y′(0)=0 the find y(4)

introduction and background on the design of water supply system. Present the concepts of water flow in piping system focusing on energy equation (Bernoulli equation), head losses, and momentum. Support all the information with references. Provide the objectives at the end of the section.

Briefly present the main objective of this report. State the approach used to achieve the tasks focusing into the equations used in the calculation. You can evaluate your design through the results you achieved such as velocities, flowrate, head losses, pressure, etc. it is recommended to discuss the challenges and design limitation

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)