Question 1. The math department needs a printer in the computer room of the second floor of MSB, and the department is planning the purchase and maintenance schedule of the printer for the next six years. The cost of a new printer is $80, and it can be used for at most five years. The maintenance cost each year of operation is as follows: year 1, $30; year 2, $40; year 3, $60, year 4, $90; year 5, $100. The department just purchased a new printer, and assume a printer has no salvage value. Formulate a shortest path problem to minimize the total cost of purchasing and operating a printer for the next six years. (Note: you need to draw a network, clearly explain the meaning of the nodes, arcs and clearly give the cost on each arc. You don’t need to solve it).

^{2. Recall that the set Q of rational numbers consists of equivalence classes of elements of Z × Z\{0} under the equivalence relation R defined by: (a, b)R(c, d) ⇐⇒ ad = bc. We write [a, b] for the equivalence class of the element (a, b). Using this setup, do the following problems: 2A. Show that the following definition of multiplication of elements of Q makes sense (i.e. is “well-defined”): [a, b] · [r, s] = [ar, bs]. (Recall this means that we must check that the definition gives the same answer no matter which representative of the equivalence class we use to compute the product.) [This is the same as problem 19 of section 4.2.]}

4. Suppose that the price per unit of input K is 1 euro, the price per unit of input C is 12 euros and the price per unit of input L is 3 euros a) What is the minimum cost of producing 40 units of output y for the firm if the firm’s production function is Y =min {K;4C}+ L/2 ?

The Government is planning a policy, which envisages simultaneously the following two policies: 1)

introduction of a tax of 1 euro per unit of input L, and 2) introduction of a subsidy of 2 euros per unit of

input C. Find and explain briefly, how would the planned policy affect this firm. Would it affect the

minimum cost of production of 40 units of output? Would it affect the technology used? Provide

calculations for proof.

Carlos takes a glass of cold water from the refrigerator and puts it on a table. The day is sunny
and the temperature is 30 °C. Once outside the refrigerator, the water temperature was
0 °C and after 10 minutes it rose to 18 °C. Determine a differential equation that models the
temperature change over time, assuming the reason at which the temperature changes
of the drink is proportional:

a) The difference between its own temperature and that of the surrounding environment.

b) Determine what happens with temperature when time tends to infinity.

c) the square of the difference between its own temperature and that of the surrounding environment.

consider the vectors:
v1=(1,1,1)
v2=(2,-1,1)
v3=(3,0,2)
v4=(6,0,4)

a)find the dimension and a basis W=Span(v1,v2,v3,v4)
b) Does the vector v=(3,3,1) belong to W. Justify your answer
c) Is it true that W=Span(v3,v4)? Justify your answer

1. (15pts)
A standard 52-card deck consists of 13 cards from each of 4 suits (spades, hearts, diamonds, clubs). What is the probability that a 5-card poker hand drawn from a standard 52-card deck has (please give necessary explanation as the solution)

(a) (5pts) 2 clubs, 1 hearts and 2 diamonds?
(b) (5pts) 1 ace and 2 face cards (face cards refer to king, queen, or jack)?

(c) (5pts) at least one ace?
You may leave your expressions without evaluating the numerical value.

6) a) Find the general solution to the 2^nd order differential equation y''+6y'+8y=0          [8 pts]

b) Find the general solution to y''+6y'+8y=2e-x. Use the method of undetermined coefficients.   [8 pts]

c) Solve the IVP y''+6y'+8y=2e-x, y0=0, y'0=0           [5 pts]

Type a proposition involving p, q, r and s that is true just when at least two of the propositional variables are true. For example, your proposition would be true in any case for which p and r are both true but false when, say, p is true while q, r and s are all false.

2. In the proof of Gauss’ Lemma, I stated that each of the terms |a|, |2a|, …, |(p-1)a/2| are distinct modulo p. Prove this by contradiction. [Hint: As in college algebra, there are two cases.]

Advanced Calculus 1

Problem 1 If the function f : D → R is uniformly continuous and α is any number, show that the function αf : D → R also is uniformly continuous.

Problem2 Provethatiff:D→Randg:D→Rareuniformlycontinuousthensois the sum f + g : D → R.

Problem 3 Define f (x) = 2x + 1 for all x ∈ R. Prove that f is uniformly continuous.

Problem 4 Define f (x) = x3 + 1 for all x ∈ R. Prove that f is not uniformly continuous.

Show that two m×n matrices are equivalent if and only if they have the same invariant factors, i.e. (by Problem 4), if and only if they have the same Smith normal form.

Consider the second-order boundary value problem

y′′ +(2x^2 +3)y′ −y =6x, 0≤x ≤1, (4)

y(0) = 1, y(1) = 0.
(a) Rewrite the second-order equation (4) as a system of two first-order equations

involving variables y and z. [2]

(b) Suppose that yn and zn are approximations to y(xn) and z(xn), respectively, where xn = nh, n = 0,...,N and h = 1/N for some positive integer N. Find the iterative formula when using the modified Euler method to approximate (4) with the modified boundary conditions:

y(0) = 1, y′(0) = z0.
(c) Hence, employ the shooting method, with underlying modified Euler method, to

find approximations yn, n = 1, . . . , N to problem (4)-(5), when N = 5. [Hint: Notice that differential equation (4) is linear.]

Boris and Natasha agree to play the following game. They will flip a (fair) coin 5 times in a row. They will compute S = (number of heads H – number of tails T).

a) Boris will pay Natasha S. Graph Natasha’s payoff as a function of S. What is the expected value of S?

b) How much should Natasha be willing to pay Boris to play this game? After paying this amount, what is her best case and worst case outcome?

This time, after 5 flips of the coin, if there are more heads H than tails T, Boris will pay Natasha H – T. If there are more tails T than heads H, Boris will pay Natasha nothing.

c) Graph Natasha’s payoff as a function of S = H – T. What does this graph remind you of?

d) What is the expected value of Natasha’s payoff? How much should she be willing to pay to play this game? After paying this amount, what is her best case and worst case outcome?

3. For the inhomogeneous differential equation x ′′ + 2x ′ + 10x = 100 cos(4t),

(a) Describe a system for which this differential equation would be an appropriate model.

(b) Find the general solution, x(t), to the equation.

(c) Does the general solution have the expected terms? What behavior do the terms describe?

(d) Find the specific solution that fits the initial conditions x(0) = 0 and x ′ (0) = 0.

(e) Plot the solution and discuss how you see the expected behaviors

This is a combinatorics problem

Suppose we wish to find the number of integer solutions to the equation below, where 3 ≤ x1 ≤ 9, 0 ≤ x2 ≤ 8, and
7 ≤ x3 ≤ 17.

x1 + x2 + x3 = r

Write a generating function for this problem, and use it to solve this problem for r = 20.