a) Your initial belief about stock A is that its future price cannot be predicted on the basis of existing public information. An insider comes forward claiming that the price will fall. You know the insider is not totally reliable and tells the truth with probability p=0.3. Use Bayes’ theorem to calculate the posterior probability that the stock price will fall, based on the insider’s evidence.A second insider, equally unreliable, comes forward and also claims that the price will fall. Assuming that the insiders are not colluding, what is your posterior probability of a price fall?  Based on your above answers, does the probability of future stock price depend on unreliable insiders? Would you expect this outcome? Explain your argument.

(9) (a) If f(t) = x₁e ^−kt cos(ωt) + x₂e ^−kt sin(ωt), then f'(t) has the same form with coefficients y₁, y₂.

(i) Find the matrix A such that y = Ax.

(ii) Find the characteristic polynomial of A.

(iii) What can you say about eigenvalues of A? (iv) Interpret your answer to

(iii) as a calculus statement. That is, explain how your answer to (iii) could have been predicted from a basic fact of calculus.

(b) If f(t) = x₁ sin(t) + x₂ cos(t) + x₃tsin(t) + x₄t cos(t), then f'(t) has the same form with coefficients y1, ..., y4. Same questions (a)-(d) as previously.

(i) Find the rate of change of the function f(x) = (x + 2)/( 1 − 8x) with respect to x when x = 1. (ii) The number of units Q of a particular commodity that will be produced with K thousand dollars of capital expenditure is modeled by Q(K) = 500 K ^(2/3). Suppose that capital expenditure varies with time in such a way that t months from now there will be K(t) thousand dollars of capital expenditure,where K(t) = (2t^4 + 3t + 149)/( t + 2) (a) What will be the capital expenditure 3 months from now? How many units will be produced at this time? (b) At what rate will production be changing with respect to time 5 months from now? Will production be increasing or decreasing at this time?

On the island of Knights and Knaves we have three people A, B and C. (The island must be known for its inhabitants’ very short names.) A says: We are all knaves. B says: Only one of us is a knave. Using an approach similar to the one in the notes, determine if A, B and C are each a knight or a knave. (The problem might have no solutions, one solution, or many solutions.)

The number of commercial airline boardings on domestic flights increased steadily during the 1990s as shown in the table below. Let​ f(t) be the number of commercial airline boardings on domestic flights​ (in millions) for the year that is t years since 1990.

Numbers of Commercial Airline Boardings on Domestic Flights

f f(x)

Year Number of Boardings​ (millions)

1991 452

1995 547

1997 599

1999 635

2000 666

Find an equation of f.

Let (X,d) be the Cartesian product of the two metric spaces (X_1,d₁) and (X_2,d₂).

a) show that a sequence {(x_n¹,x_n²)} in X is Cauchy sequence in X if and only if {x_n¹} is a Cauchy sequence in X₁ and {x_n²} is a Cauchy in X_2.

b) show that X is complete if and only if both X₁ and X₂​​​​​​​ are complete.

Now assume that the harvesting is not done at a constant rate, but rather at rates that vary at different times of the year. This can be modeled by ??/?? = .25? (1 − ?/4 ) − ?(1 + sin(?)). This equation cannot be solved by any technique we have learned. In fact, it cannot be solved analytically, but it can still be analyzed graphically.

8. Let c=0.16. Use MATLAB to graph a slopefield and approximate solutions for several different values of p(0), and interpret what you see. Turn in the graphs together with your analysis. Note: 0<p<5, and 0<t<50 is a reasonable viewing window to start with, though you may want to change it as you proceed.

a) Verify that the indicial equation of Bessel's equation of order p is (r-p)(r+p)=0

b) Suppose that p is not an integer. Carry out the computation to obtain the solutions y1 and y2 above.

y"+y=cos(9t/10)

1. general solution of corresponding homongenous equation

2. particular solution

3.solution of initial value problem with initial conditions y(0)=y'(0)=0

4. sketch solution in part 3

1. (a) Let p be a prime. Prove that in (Z/_pZ)[x], x^p−x= x(x−1)(x−2)···(x−(p−1)).

(b) Use your answer to part (a) to prove that for any prime p, (p−1)!≡−1 (modp).

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.