. Solve the Initial value problem by using Laplace transforms: ? ′′ + 3? ′ + 2? = 6? −? , ?(0) = 2 ? ′ (0) = 8

Find a basis and the dimension of W. Show algebraically how you found your answer.

a. W = {(x1, x2, x3, x4) ∈ R^4 | x2 = x3 and x1 + x4 = 0}

b. W = {( A ∈ M 3x3 (R) | A is an upper triangular matrix}

c. W = { f ∈ P3 (R) | f(0) = 0.

Show that if Y is a subspace of X, and A is a subset of Y, then the subspace topology on A as a subspace of Y is the same as the subspace topology on A as a subspace of X.

Prove or disprove. (If a proof seems difficult to finish, at least tell what you tried, or how far you got, or what seems to make the proof difficult.) (a) If a | b, then a 2 | b 2 . (b) If a 2 | b 2 , then a | b. (c) If a | b 2 , then a | b. (d) If (a, c) = 1 and (b, c) = 1, then (ab, c) = 1. (e) If (a, c) = 1 and (b, c) = 1, then (a, b) = 1.

Given X′=AX with X(t)=[x(t)y(t)], A=[23−4012−21] and X(0)=[3−4]. (a) Write the eigenvalues and eigenvectors of A λ1= , V1= ⎡⎣⎢⎢ ⎤⎦⎥⎥ , and λ2= , V2= ⎡⎣⎢⎢ ⎤⎦⎥⎥ (b) Write the solution of the initial-value problem in terms of x(t),y(t) x(t)= y(t)=

Show that the product of two Hausdorff spaces is Hausdorff.

(A) Prove division with remainder makes sense for integers as well as natural numbers. In other words prove the following.

Proposition: Let d be a nonzero integer. For any integer n, there exist unique integers q and r such that n = dq + r and 0 ≤ r < |d|.

Show that every order topology is Hausdorff.

For any two real sequences {an} and {bn}, prove that Rudin’s Ex. 5 We assume that the right hand side is defined, that is, not of the form ∞ − ∞ or −∞ + ∞.

lim sup (an + bn) ≤ lim sup an + lim sup bn.

Proof If lim sup an = ∞ or lim sup bn = ∞, there is nothing to prove

How do the three types of integer programming problems differ? Which do you think is most common, and why? Be detailed.

Prove that each element in Pentagon D5 has a unique inverse under the binary operation.

D5={AF, BF, CF, DF, EF,0,72,144,216,288}

Juan is a salesman for L. L. Bowers Corp. He has a choice of three compensation plans. Plan 1 pays $2500 per month. Plan 2 pays $2000 per month plus 15% commission. Plan 3 pays $1700 per month plus 30% commission. Graph the three plans and determine which is best. I need this graphed in excel but i am not sure how to do it.

The cost of controlling emissions at a firm goes up rapidly as the amount of emissions reduced goes up. Here is a possible model: C(x, y) = 1,000 + 200x2 + 200y2 where x is the reduction in sulfur emissions, y is the reduction in lead emissions (in pounds of pollutant per day), and C is the daily cost to the firm (in dollars) of this reduction. Government clean-air subsidies amount to $100 per pound of sulfur and $200 per pound of lead removed. How many pounds of pollutant should the firm remove each day to minimize net cost (cost minus subsidy)?

1. Find a possible formula for the trigonometric function whose values are in the following table.

X 0 2 4 6 8 10 12

Y 5 1 -3 1 5 1 -3

y=?

2. A population of rabbits oscillates 15 above and below an average of 128 during the year, hitting the lowest value in January (t = 0). Find an equation for the population, P, in terms of the months since January, t.

P(t) =   

What if the lowest value of the rabbit population occurred in April instead?

P(t)) =   

3. Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the high temperature of 59 degrees occurs at 4 PM and the average temperature for the day is 50 degrees. Find the temperature, to the nearest degree, at 8 AM

Degrees:

4. Outside temperature over a day can be modeled as a sinusoidal function. Suppose you know the temperature varies between 32 and 68 degrees during the day and the average daily temperature first occurs at 10 AM. How many hours after midnight, to two decimal places, does the temperature first reach 45 degrees?

Hours:

The usual ε − δ definition of limits, Definition. limx→a f(x) = L exactly when for every ε > 0 there is a δ > 0 such that for any x with |x − a| < δ we are guaranteed to have |f(x) − L| < ε as well.

1. Use the ε − δ definition of limits to verify that limx→1 (−2x + 1) = −1. [2]

2. Use the definition of limits that you didn’t use in answering question 1 to verify that limx→2 (−x + 2) does not =1. [2]

3. Use either definition of limits above to verify that limx→3 (x^2− 5)= 2. [3] Hint: The choice of δ in 3 will probably require some slightly indirect reasoning. Pick some arbitrary smallish positive number for δ as a first cut. If it doesn’t do the job, but x is at least that close, you’ll have more information to help pin down the δ you really need. Note: The problems above are probably easiest done by hand, though Maple and its competitors do have tools for solving inequalities which could be useful.

5. Compute limx→0 sin (x + π)/x by hand. [1