Find the Legendre’s polynomial ??(?) from the differential equation (1 − ?²) ?²?/??² − 2? ??/?? + 6? = 0 and represent the Legendre’s polynomial with equation.

Show that the given relation R is an equivalence relation on set S. Then describe the equivalence class containing the given element z in S, and determine the number of distinct equivalence classes of R.

Let S be the set of all possible strings of 3 or 4 letters, let z = ABCD and define x R y to mean that x has the same first letter as y and also the same third letter as y.

Use undetermined coefficients to solve the differential equation. Linearly independent solutions to the associated homogeneous ODE are also shown. (please provide a detailed procedure to understand)

?²?′′ − 4??′ + 6? = ?⁴e^{-3x                       y}₁=x²              y₂=x³

Problem 15

How many (a) 1×2 (b) 2×2 (c) 1×4 (d) 2×4 rectangles that form implicants are there in a 4×4 Karnaugh map? Describe their implicants as products of literals, assuming the variables are p, q, r, and s.

Romans Food Market, located in Saratoga, New York, carries a variety of specialty foods from around the world. Two of the store’s leading products use the Romans Food Market name: Romans Regular Coffee and Romans DeCaf Coffee. These coffees are blends of Brazilian Natural and Colombian Mild coffee beans, which are purchased from a distributor located in New York City. Because Romans purchases large quantities, the coffee beans may be purchased on an as-needed basis for a price 12% higher than the market price the distributor pays for the beans. The current market price is $0.45 per pound for Brazilian Natural and $0.67 per pound for Colombian Mild. The compositions of each coffee blend are as follows:

Romans sells the Regular blend for $3.3 per pound and the DeCaf blend for $4.5 per pound. Romans would like to place an order for the Brazilian and Colombian coffee beans that will enable the production of 1100 pounds of Romans Regular coffee and 525 pounds of Romans DeCaf coffee. The production cost is $0.84 per pound for the Regular blend. Because of the extra steps required to produce DeCaf, the production cost for the DeCaf blend is $1.08 per pound. Packaging costs for both products are $0.25 per pound. Formulate a linear programming model that can be used to determine the pounds of Brazilian Natural and Colombian Mild that will maximize the total contribution to profit.

If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)

The complete linear program is

What is the contribution to profit?

Optimal solution:

Let R be a relation on the set of all integers such that aRb if and only if 3a − 5b is even. Tell if R is an equivalence relation. Justify your answer. (Hint: 3b − 5a = 3a − 5b + 8b − 8a)

1. Find integers x and y such that 23x 107y 1
2. Find 17-1 (mod 97) using the Extended Euclidean algorithm. Based on your results,
what is 97-1 (mod 17)?
3. Find gcd(30150,243) using the Euclidian algorithm
4. Find the inverse of 719 (mod 3728). Show all the steps.

AN OBJECT IS THROWN UPWARD WITH A VELOCITY OF 70FT/S. WHEN WILL IT BE 85FT ABOVE ITS INITIAL POSITION . USING "NEWTONS PROJECTILE MOTION" APPLYING QUADRATIC FORMULA a=16.1 b=70/s c=-85ft im showing 70+ = .9892s 70-=-5.33705 i need to see detail im comming up with a reverse for pos and neg im comming up pos 5.33705 and neg .98925 i need to see where ive made a mistake

Provided a standard sequence of n independent Bernoulli trials in which the probability of success is θ and the probability of failure is 1−θ.

If A represents the observed number of success and B represents the observed number of failures, (with A+B = n), then find I(θ), the Fisher information matrix. (Hint: Recall that the sum of n Bernoulli trials is a Binomial random variable. Also assume that n, A and B are fixed and so the only unknown parameter is θ, in the case I(θ) will be a scalar.)

Please show all work/provide explanations as appropriate, including clearly defining any variables that you use in the applied problems.

1) According to United Nations estimates ( http://www.worldometers.info/world-population/ ) , the population of the world at the start of 2020 was about 7.795 billion, and it is growing at about 1.05% per year.

a. Find an exponential growth model (i.e. write down an exponential function) that gives the earth’s population in billions as a function of time as measured by number of years after 2020. Be sure to clearly and carefully define your variables.

b. According to the model, what will the world population be in 2030? Solve algebraically as opposed to using a table of values or a graph.

c. Again, according to the model, how long will it take the population to double? Solve this algebraically; that is, do not simply use a table or graph.

d. In what year will the population reach 20 billion? Again, solve without resorting to a table of values or a graph.

2) Refer to the population model you found in problem 1. a. Sketch a graph of your model, displaying at least the next 200 years. You may carefully sketch the graph by hand on graph paper,

b. What is the average rate of change of the world population as described by your model over the next 100 years? (so, from 2020 to 2120, or t = 0 to t = 100)

c. What is the average rate of change from 2020 to 2050?

d. Describe how you could estimate the rate of change of the world population in 2030, and then carry out and show the steps you describe to give an estimate.

How many irreducible polynomials of the form x³ + ax² + bx + c are in Z_p[x], such that p is a prime?

1. a.) Find the values of x for which the series converges. Express your answer in interval notation. n = 1 ∞ ∑(−3)ⁿ?ⁿ

b.) Find the sum of the series (valid over the values of x found in part a).

(8) Suppose T : R 4 → R 4 with T(x) = Ax is a linear transformation such that • (0, 0, 1, 0) and (0, 0, 0, 1) lie in the kernel of T, and • all vectors of the form (x1, x2, 0, 0) are reflected about the line 2x1 − x2 = 0.

(a) Compute all the eigenvalues of A and a basis of each eigenspace.

(b) Is A invertible? Explain.

(c) Is A diagonalizable? If yes, write down its diagonalization (you can leave it as a product of matrices). If no, why not?

The Simpson numerical method tries to obtain a result closer to the analytic solution with and independent variable x, using the Simpson method (both 1/3 and 3/8 methods). Use a MATLAB script. The program should ask for the user to submit the function he wishes to integrate and the values of the a & b limits. Additionally it should ask the number of elements that would divide the function( sub-intervals).

We are playing with a 52-card standard deck (four ”suits” ♥, ♠, ♦, ♣, with ”faces” A, 2, 3,...,10, J, Q, K). In how many ways can we choose a 5-card hand, such that

(a) four cards have same face?

(b) all five cards of same suit?

(c) the five cards have consecutive faces (where A can be either the lowest or the highest face)?

(d) the five cards form a ”straight flush” (all five cards of same suit, and have consecutive faces)?

(e) the five cards form a ”flush” (all five cards of same suit, but do not form a ”straight flush”)?

(f) the five cards form a ”full house” (three cards have same face, and the other two have same face)?