Show that the two definitions of continuity in section 2.1 are equivalent. Consider separately

the cases where z0 is an accumulation point of G and where z0 is an isolated point of G.

2.1 :

Definition1. Suppose f : G → C. If z0 ∈ G and either z0 is an isolated point of G or lim f(z) = f(z0) (z→z0)
then f is continuous at z0. More generally, f is continuous on E ⊆ G if f is continuous at every z ∈ E.

Definition 2.

Suppose f : G → C and z0 ∈ G. Then f is continuous at z0 if, for every positive real

number ε there is a positive real number δ so that
|f(z)−f(z0)|<ε for all z∈G satisfying |z−z0|<δ.

Thanks.

a) Find the approximations T₁₀, M₁₀, and S₁₀ for

from pi to 0 , 38sin(x)dx

(Round your answers to six decimal places.)

Find the corresponding errors E_T, E_M, and E_S. (Round your answers to six decimal places.)

(b) Compare the actual errors in part (a) with the error estimates given by the Theorem about Error Bounds for Trapezoidal and Midpoint Rules and the Theorem about Error Bound for Simpson's Rule. (Round your answers to six decimal places.)

(c) How large do we have to choose n so that the approximations T_n, M_n, and S_n to the integral in part (a) are accurate to within 0.00001?

The monthly cost of owning a car depends on the number of kilometers it is driver. Taylor found that in May it cost her $500 to drive 800 km and it June it cost her $650 to drive 1400 km.

1. Express the monthly Cost C as a function of the distance driven d, assuming that a linear function is a suitable model.
2. Use this function to predict the cost of driving 2000 km per month.
3. Find the derivative of the function with appropriate units. What does the derivative of the function represent?
4. Does it cost more to drive the car in December than it does in August? Explain.

Consider the the differential equation

2xy''+ 5y'+xy= 0

1) determine the indicial equation and its roots

2) For each root of the indicial equation, determine the recurrence relation

3) Do the indicial roots differ by an integer? If yes, find the general solution on (0,∞). If not, find the series solution corresponding to the larger root on (0,∞)

Suppose that A equals 10. What are the values for (d1+) and (d1-) in the following constraint? A + (d1-) - (d1+) = 7

Group of answer choices

(d1-) =0, (d1+) =3

(d1-) =3, (d1+) =0

(d1-) =7, (d1+) =0

(d1-) =0, (d1+) =7

(d1-) =10, (d1+) =3

2.

The optimal solution is to select only two of the alternatives. Suppose you wished to add a constraint that stipulated that alternative 2 could only be selected if alternative 1 is also selected (i.e., if alternative 1 is not selected, you may not select alternative 2; however, you may select #1 and not select #2). How would this constraint be written?

Group of answer choices

A - B = 0

A - B <= 0

A -B >= 0

A + B = 2

none of the above

A goal programming problem had two goals (assume equal weights of 1). Goal number 1 was to achieve a cost of $2,400 and goal number 2 was to have no idle time for workers in the factory. The optimal solution to this problem resulted in a cost of $2,400 and no idle time. What was the value for the objective function for this goal programming problem?

Group of answer choices

2300

100

-100

0

none of the above

3.

In a basic transportation problem with 3 factories and 5 warehouses, there would be ___ decision variables.

Group of answer choices

3

5

8

15

None of the above

4.

In a basic transportation problem with 3 factories and 5 warehouses, there would be ___ constraints.

Group of answer choices

3

5

8

15

None of the above

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)