Show that the following two vector fields are conservative and find associated scalar potentials.
(a) F = 2ρsin(2φ) ρ + 2ρcos(2φ) φ + k
(b) F = (2r cos2 φ + sinθ cosφ) r + cosθ cosφ θ − [r(sin(2φ)/sinθ) + sinφ] φ

Question 3(a):
When customers arrive at Cool's Ice Cream Shop, they take a number and wait to be called to purchase ice cream from one of the counter servers. From experience in past summers, the store's staff knows that customers arrive at a rate of 150 per hour on summer days between 3:00 p.m. and 10:00 p.m., and a server can serve 1 customer in 1 minute on average. Cool's wants to make sure that customers wait no longer than 10 minutes for service. Cool's is contemplating keeping three servers behind the ice cream counter during the peak summer hours.
(i) Will this number be adequate to meet the waiting time policy?
(ii) What will be the probability that 3 to 4 customers in Shop?
(iii) In winter season, arrival rate of customer is reduced to half from 3:00 p.m. and 10:00 p.m. What decision should be taken by the owner according to cost cutting point of view?
Question 3(b):
Analysis of arrivals at a PSO gas station with a single pump (filler) has shown the time between arrivals with a mean of 10 minutes. Service times were observed with a mean time of 6 minutes.
(i) What is the probability that a car will have to wait?
(ii) What is the mean number of customers at the station?
(iii) What is the mean number of customers waiting to be served?
(iv) PSO is willing to install a second pump when convinced that an arrival would expect to wait at least twelve minutes for the gas. By how much the flow of arrivals is increased in order to justify a second booth?

Explain Fishers Exact Test ?

Difference between BLUE and BLUP ? Different properties of BLUP

Do I need to know where on the dart board the 3, 1, 5 cm darts landed to obtain precision and accuracy?

For the data and calculation for this lab report assume that a dart was thrown at bullseye 3 times with the aim of hitting the middle, the 3 distance from the middle obtain where 3cm, 1cm, and 5cm. Calculate the average, the percent error and discuss in your lab report and emphasize on the accuracy vs precision part

1. Evaluate the double integral for the function f(x,y) and the given region R.

R is the rectangle defined by -2  x  3 and 1   y  e⁴

2. Evaluate the double integral

for the function f(x, y) and the region R.

f(x, y) =

; R is bounded by the lines

x = 1, y = 0, and y = x.

3. Find the average value of the function f(x,y) over the plane region R.

f(x, y) = xy; R is the triangle bounded by y = x, y = 2 - x, and y = 0

4. Verify that y is a general solution of the differential equation and find a particular solution of the differential equation satisfying the initial condition.

y =

;  

= −2xy²;  y(0) = 7

A volume is described as follows:

1. the base is the region bounded by x = − y 2 + 16 y + 5 and x = y 2 − 30 y + 245 ;

2. every cross section perpendicular to the y-axis is a semi-circle. Find the volume of this object.

Prove: Let A be an mxm nonnegative definite matrix with rank(A)=r Then there exists an mxr matrix B having rank of r, such that A=BB^T

1. (15 pts) Is the matrix A =   1 0 1 0 1 1 1 1 2   diagonalizable? If yes, find an invertible matrix P and a diagonal matrix Λ such that P −1AP = Λ.

Verify that V in the previous example satisfies the axioms for a vector space.

14. Extra Credit: Cayley’s Theorem is an important one in advanced algebra. It says that “Every algebraic group is isomorphic to some permutation group.” Demonstrate this to be true by finding a permutation group (Sn, ∘ ) that is isomorphic to (ℤ3, +) for some n.

(a)Show that S = {a+b √ 5 | a, b ∈ Q} is a subring of the real numbers (with the usual + and × of real numbers). Explain why S is a field.

(b) Prove that if r is an element of a ring R and r 3 = 0, then 1 − r is a unit in R.

(c) Write down all the nilpotent elements of Z24, stating the index of nilpotence in each case. Verify the statement in part (b) holds in Z24.

(ii) The market consists of the following stocks. Their prices and number of shares are as follows: Stock Price Number of Shares Outstanding A $10 100,000 B 20 10,000 C 30 200,000 D 40 50,000 (a) What is the percentage increase in the market if a S&P 500 type of measure of the market (value-weighted average) is used? b. The price of Stock C doubles to $60. What is the percentage increase in the market if a S&P 500 type of measure of the market (value-weighted average) is used? c. Repeat question (b) but use a Value Line type of measure of the market (i.e., a geometric average) to determine the percentage increase. d. Suppose the price of stock B doubled instead of stock C. How would the market have fared using the aggregate measures employed in (b) and (c)? Why are your answers different?

Solve the Cauchy-Euler equation

x⁴y'''' - 4x²y'' + 8xy' - 8y = 12xlnx

x > 0