redo Fundamental Theorem of Arithmetic for F[x], F=field, including whatever preliminary results.

1. Use the ε-δ definition of continuity to prove that (a) f(x) = x 2 is continuous at every x0. (b) f(x) = 1/x is continuous at every x0 not equal to 0.

3. Let f(x) = ( x, x ∈ Q 0, x /∈ Q (a) Prove that f is discontinuous at every x0 not equal to 0. (b) Is f continuous at x0 = 0 ? Give an answer and then prove it.

4. Let f and g be two functions on R. Prove that if f and g are continuous on R and f(q) = g(q) for every q ∈ Q, then f(x) = g(x) for every x ∈ R.

1.   Solve the following system:

2x1- 6x2- x3 = -38
-3x1–x2 +7x3 = -34
-8x1 +x2 – 2x3 = -20

By:
a.   LU Factorization
b.   Gauss-Siedel Method, error less that10-4

Hint (pivoting is needed, switch rows).

for y(t) function ty'' - ty' + ty = 0, y(0)= 0 , y'(0)= 1

solve this initial value problem by using Laplace Transform. (The equation could have been given such as "y'' - y' + y = 0" but it is not. Please, be careful and solve this question step by step.) )

G is solvable if and only if G/Z(G) is solvable

5. Of the 9-letter passwords formed by rearranging the letters AAAABBCCC (4 A’s, 2 B’s, and 3 C’s), I select one at random. Determine the following probabilities.

(a) Prob (my word is a palindrome and has no two C’s next to each other). 4 points

(b) Prob (my word has two C’s next to each other and the other C not next to them).

(c) Prob (my word has the three C’s next to each other and the B’s apart from each other).

Leave your probabilities with factorials in them.

Suppose we define a relation on the set of natural numbers as follows. Two numbers are related iff they leave the same remainder when divided by 5. Is it an equivalence relation? If yes, prove it and write the equivalence classes. If no, give formal justification.

Does there exist a partition of R that is an uncountably infinite partition that consists of uncountably infinite sets? If so, construct such a partition, otherwise prove that such a partition can not exist.

Assume there are 14 homes in the Quail Creek area and 4 of them have a security system. Three homes are selected at random:

1. What is the probability all three of the selected homes have a security system? (Round your answer to 4 decimal places.)

2. What is the probability none of the three selected homes has a security system? (Round your answer to 4 decimal places.)

3. What is the probability at least one of the selected homes has a security system? (Round your answer to 4 decimal places.)

4. Are the events dependent or independent?

• Dependent

• Independent

• Joint

Parametrize the osculating circle to the parabola y = x² at x = -1

Insert K = 3

(kx^2+1)dy/dx + Kxy = Kx

Describe whether the first order type: Linear or exact and solve to find the general Solution.

What is a feasible solution of an LP? how can we define it in terms of Mathematical proof.

We suggest the use of a spreadsheet to create the amortization tables.

You take out a 30-year mortgage for $70,000 at 9.65%, to be paid off monthly. Construct an amortization table showing how much you will pay in interest each year for the first 15 years and how much goes toward paying off the principal. If you sell your house after 15 years, how much will you still owe on the mortgage according to the amortization table? HINT [See Example 8.] (Round your answer to the nearest cent.)

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