The following data represent the amount of money and invenstor has in an investment account each year for 10 years.

a. Let x=number of years since 1994 and find an exponential regression model of the form y=ab* for this data set, where y is the amount in the account x years since 1994.

_________________________

b. If the investor plans on retiring in 2021, what will be the predicted value of this accoutn at that time?

________________________________

c. When will the account be worth $50,000?

d. Make a graph of the scatterplot and exponential model below.

Year value of account

1994 $10,000

1995 $10,573

1996 $ 11,260

1997 $11,733

1998 $12,424

1999 $13,269

2000 $13,698

2001 $14,823

2002 $15,297

2003 $16,539

On Z we consider the family of sets τ = {Z, ∅, {−1, 0, 1}, {−2, −1, 0, 1, 2}, . . . }

where the dots mean all sets like the two before that.

a) Prove that τ is a topology.

b) Is {−4, −3, −2, −1, 0, 1, 2, 3, 4} compact in this topology?

c) Is it connected?

d) Is Z compact in this topology?

e) Is it connected?

Let M/F and K/F be Galois extensions with Galois groups G = Gal(M/F) and H = Gal(K/F). Since M/F is Galois, and K/F is a field extension, we have the composite extension field K M.

Show that σ → (σ|_M , σ|_K) is a homomorphism from Gal(K M/F) to G × H, and that it is one-to-one. [As in the notes, σ|_X means the restriction of the map σ to the subset X of its domain.]

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = 2 sin(x) sin(y),     −π < x < π,     −π < y < π

Local minimum:

Local maximum :

Saddle points:

On January 1, 20X8, Parent Company purchased 75% of the common stock of Subsidiary Company for $360,000. On this date, Subsidiary had common stock, other paid in capital, and retained earnings of $20,000, $130,000, and $200,000, respectively. Any excess of cost over book value is due to goodwill. Parent accounts for the Investment in Subsidiary using cost method. On January 1, 20X8, Subsidiary sold $100,000 par value of 6%, ten-year bonds for $97,000. The bonds pay interest semi-annually on January 1 and July 1 of each year. On January 1, 20X9, Parent repurchased all of Subsidiary's bonds for $99,100. The bonds are still held on December 31, 20X9. Both companies have correctly recorded all entries relative to bonds and interest, using straight-line amortization for premium or discount. Calculate NCI's portion of consolidated net income for the year ended of December 31, 20X9. Round all computations to the nearest dollar

0 mod 35 = 〈0 mod 5, 0 mod 7〉 12 mod 35 = 〈2 mod 5, 5 mod 7〉 24 mod 35 = 〈4 mod 5, 3 mod 7〉
1 mod 35 = 〈1 mod 5, 1 mod 7〉 13 mod 35 = 〈3 mod 5, 6 mod 7〉 25 mod 35 = 〈0 mod 5, 4 mod 7〉
2 mod 35 = 〈2 mod 5, 2 mod 7〉 14 mod 35 = 〈4 mod 5, 0 mod 7〉 26 mod 35 = 〈1 mod 5, 5 mod 7〉
3 mod 35 = 〈3 mod 5, 3 mod 7〉 15 mod 35 = 〈0 mod 5, 1 mod 7〉 27 mod 35 = 〈2 mod 5, 6 mod 7〉
4 mod 35 = 〈4 mod 5, 4 mod 7〉 16 mod 35 = 〈1 mod 5, 2 mod 7〉 28 mod 35 = 〈3 mod 5, 0 mod 7〉
5 mod 35 = 〈0 mod 5, 5 mod 7〉 17 mod 35 = 〈2 mod 5, 3 mod 7〉 29 mod 35 = 〈4 mod 5, 1 mod 7〉
6 mod 35 = 〈1 mod 5, 6 mod 7〉 18 mod 35 = 〈3 mod 5, 4 mod 7〉 30 mod 35 = 〈0 mod 5, 2 mod 7〉
7 mod 35 = 〈2 mod 5, 0 mod 7〉 19 mod 35 = 〈4 mod 5, 5 mod 7〉 31 mod 35 = 〈1 mod 5, 3 mod 7〉
8 mod 35 = 〈3 mod 5, 1 mod 7〉 20 mod 35 = 〈0 mod 5, 6 mod 7〉 32 mod 35 = 〈2 mod 5, 4 mod 7〉
9 mod 35 = 〈4 mod 5, 2 mod 7〉 21 mod 35 = 〈1 mod 5, 0 mod 7〉 33 mod 35 = 〈3 mod 5, 5 mod 7〉
10 mod 35 = 〈0 mod 5, 3 mod 7〉 22 mod 35 = 〈2 mod 5, 1 mod 7〉 34 mod 35 = 〈4 mod 5, 6 mod 7〉
11 mod 35 = 〈1 mod 5, 4 mod 7〉 23 mod 35 = 〈3 mod 5, 2 mod 7〉

2.2 Which of the numbers (mod 35) are relatively prime to 35? List them in CRT (Chinese Remainder Theorem) notation.

2.3. For each number x in the answer to #2.2, compute x 2 (mod 35).

2.4 Verify that each square has four square roots (mod 35).

2.5 1 is a square (mod 35). Two of its square roots are 1 and (‐1 ≡ 34 (mod 35)). What are the other two?

Find the optimum solution to the following LP using the Simplex Algorithm. Use Two-Phase method.

??? ?=3?2+2?3 ??
−2?1 + ?2 − ?3 ≤ −3

−?1 + 2?2 + ?3 = 6

?1,?2,?3 ≥0

2. Suppose you have a collection of n items i1, i2, ..., in with weights w1, w2, ..., wn and a bag with capacity W

(a) Describe a simple, efficient algorithm to select as many items as possible to fit inside the bag e.g. the maximum cardinality set of items that have weights that sum to at most W.

(b) Prove your answer.

State and prove Lebesgue Dominated Convergence theorem.

For sets A and B we may define the set difference measure as |A_B| (the cardinality of the set A-B), explain why this is never negative. We know this is not a distance, explain why and modify it so that it is a distance. Prove your claim (in particular be careful to show the triangle inequality holds)

Based off previous events, a demand of 260 banquet attendees is expected at a dinner plate price of $55.00 each. A demand of 340 banquet attendees is expected at a dinner plate price of $35.00 each.
 Your facility can hold up to 500 attendees. There is no rental fee for use of the facility.
 You have two catering companies to choose from:
o Catering Service 1 has a fixed cost of $2,250 and a variable cost of $20 per plate.
o Catering Service 2 has a fixed cost of $2,800 and a variable cost of $14 per plate.
o Costs for both caterers include the food, drinks, plates, utensils, tablecloths, glasses, crew, and cleanup. (That is, there are no other costs to consider.)
 Dinner plates will only be sold as an entire unit (i.e., you can’t sell half of a plate).

A. The price-demand function in terms of x is.
?=?(?)= __________________________________________ Domain: ______ ≤ ? ≤ _________.
B. The revenue function in terms of x is
?(?)= _______________________________________________ Domain: ______ ≤ ? ≤ _________.
C. The cost functions for each of the two possible catering services in terms of ? are
?1(?)= _____________________________________________
?2(?)= ______________________________________________
D. The profit functions for each of the possible catering services in terms of x are
?1(?)= _____________________________________________
?2(?)= _____________________________________________
E. Catering service 1 will break even at a minimum of ___________ banquet attendees.
Catering service 2 will break even at a minimum of ___________ banquet attendees.
F. If the projected sales are 275 banquet tickets,
a. The price per ticket is $__________________.
b. The profit using catering service 1 is $__________________.
c. The profit using catering service 2 is $____________________.
d. Your company should choose catering service # ______ to maximize the banquet’s profit.

Find the matrix A representing the follow transformations T. In each case, check that Av = T(v)

T(T(x,y,z))

where T(x,y,z)=(x-3y+4z, 6x-2z, 8x-y-4z)

A symmetrical 2D plate is bounded by the curves ? = 2?^2 , ? = 6? • Make a clear sketch of the plate. • Choose the appropriate element, on the sketch show its dimensions and the arm length for the element • Write the differentials of the Area • Find the Area of the plate • Find Moment of Inertia and Radius of Gyration of the plate with respect to y axis.

Consider the following initial value problem to be solved by undetermined coefficients. y″ − 16y = 6, y(0) = 1, y′(0) = 0

Write the given differential equation in the form L(y) = g(x) where L is a linear operator with constant coefficients. If possible, factor L. (Use D for the differential operator.)

( )y = 16

Let F be a finite field. Prove that there exists an integer n≥1, such that n.1_F = 0_F .

Show further that the smallest positive integer with this property is a prime number.