Let (Xn) be a monotone sequence. Suppose that (Xn) has a Cauchy subsequence. Prove that (Xn) converges.

Differential equations-Cengel 2.98
A spherical tank of radius R is initially filled with water. Now a hole of radius r is opened at the bottom of the tank and the water is let out. According to Torricelli's law, water comes out of the opening at a velocity v = (2gy) ^ 1/2, where y is the height of the water over the hole at a given moment, and g is the acceleration of gravity. Obtain a relation for the depth of the water with function of time t, and determine how long it takes to empty.
Explain your procedure

Give an example of a set A subset the real numbers for which both A and the complement of A are unbounded.

The phone company calculates a customer’s monthly cell phone bill as follows: monthly service charge: $35 voice charges:

First 100 minutes: $0.018 per minute

Second 100 minutes: $0.034 per minute

Additional minutes: $0.055 per minute

a. What is the charge for a month in which 158 minutes were used?

b. What is the charge for a month in which 311 minutes were used?

c. Construct a piecewise function that gives the monthly charge for x minutes of usage.

why do the scale factors h_1, h_2, and h_3 show up for curvilinear coordinate systems?

A certain region currently has wind farms capable of generating a total of 2100 megawatts ​(2.1 ​gigawatts) of power. Complete parts​ (a) and​ (b) below.

a. Assuming wind farms typically generate 25​% of their​ capacity, how much​ energy, in​ kilowatt-hours, can the​ region's wind farms generate in one​ year? Given that the average household in the region uses about​ 10,000 kilowatt-hours of energy each​ year, how many households can be powered by these wind​ farms? The wind farms can generate nothing ​kilowatt-hours in one year. ​(Simplify your​ answer.) The wind farms can power nothing households. ​(Simplify your​ answer.)

b. One of the great advantages of wind power is that it does not produce the carbon dioxide emissions that contribute to global warming. On​ average, energy produced from fossil fuels generates about 1.5 pounds of carbon dioxide for every​ kilowatt-hour of energy. Suppose the region did not have its wind farms and the energy were instead produced from fossil fuels. How much more carbon dioxide would be entering the atmosphere each​ year? nothing lbs ​(Simplify your​ answer.)

Consider P|r_j, prec|C_max. Show that the greedy algorithm is a 2-approximation.

(It pertains to Scheduling Theory, Algorithms, and Systems.)

Sketch the following regions:                                                                    

1. | z + 3e^πi/4| = 3                          2. { e^z | Im z = 1}

1. In Excel, create a 2x30 data table with the left column representing a population of prey and the right column representing a population of predators. Use the population model presented below.

Let the starting values of the model parameters be: r = 1.3, k = 1, s = .5, v = 1.6, and u = .7

Let the starting population of P = 1.1 and Q = .4

Difference equations: P[t + 1] = P[t](1 + r(1 – P[t]/K)) - sP[t]Q[t]

Qt + 1 = (1-u)Q[t] + vP[t]Q[t]

a. What does sP[t]Q[t] and vP[t]Q[t] represent?

b. Plot the values of P[t] and Q[t] in a graph.

c. Describe in words the changes in P[t] and Q[t] through time.

d. Build table in excel and describe in words what happens if you increase the growth rate of prey (r)? What about if we decrease the growth rate?

e. What does u represent? Why should u be less than 1? What happens if we make u = 1? Can you think of any biological systems in which u = 1 is a realistic assumption?

f. Create a second Excel worksheet representing another population model. Use the instructions from question 1, except that your model should now include a term to represent the amount of prey which cannot be eaten because they are hiding in refuges (just like in question 2) represented by the term: w. Also, for the predators, include a term f representing what happens if a constant, external food source contributes to the predator population.

Let w = 0.3 and let f = 0.25

g. Graph the new population levels.

h. Explore different values of w and f. Try setting w = 0, or f = 0 to see what effect each of these has individually. Describe your results.

How do I check something is analytic.
C-R is necessary but not sufficient. i was told that I must also check if the partial derivatives are continuous. How would I do that?

Solve the given differential equation by undetermined coefficients. y'' − y' + (1/4) y = 6 + e^(x/2)

A lottery offers the chance to win a prize of receiving payments forever starting with $200 for the first payment followed by each consecutive payment increasing by $250 until the payment size reaches $700. If you receive a payment every quarter, with the first in one quarter and interest is earned at j₄ = 5%, what amount must the lottery have in the account today to fund the prize?

8. Which FPU register number (R0 through R7) is identified as ST(0)?

9. Which FPU register determines the precision and rounding methods for calculations?

10. If the divide by zero exception flag in the FPU is set, what happens when the FPU tries to execute a division by zero?