please show the steps and clear hand writing

please show how to Derive the Newton-Raphson formula. Diagrams and explanations will be needed.

In all parts of this problem, assume that we are using fair, regular dice (six-sided with values 1, 2, 3, 4, 5, 6 appearing equally likely). Furthermore, assume that all dice rolls are mutually independent events.

(a) [4 pts] You roll two dice and look at the sum of the faces that come up. What is the expected value of this sum? Express your answer as a real number.

(b) [7 pts] Assuming that the two dice are independent, calculate the variance of their sum. Express your answer as a real number.

(c) [7 pts] You repeatedly roll two fair dice and look at the sum. What is the probability that you will roll a sum of 4 before you roll a sum of 7? Express your answer as a real number.

(d) [7 pts] What is the expected number of rolls until you get a sum of 4 or a sum of 7? (For example, if you get 7 on the first roll, the number of rolls is 1.) Express your answer as a real number.

(e) [7 pts] You roll 10 dice. Using the Chernoff Bound, give an upper bound for the probability that 8 or more of them rolled a 1 or a 2? You don’t need to calculate the value with a calculator (since you do not have one), but please write it in simplest terms.

A division of the Winston Furniture Company manufactures dining tables and chairs. Each table requires 40 board feet of wood and 3 labor-hours. Each chair requires 16 board feet of wood and 4 labor-hours. The profit for each table is $50, and the profit for each chair is $40. In a certain week, the company has 3200 board feet of wood available and 520 labor-hours available. How many tables and chairs should Winston manufacture to maximize its profit? (Let x represent the number of tables Winton manufactures and let y represent the number of chairs they manufacture.) (x, y) = What is the maximum profit? $

A committee of eight people, labelled a, b,...,h, has been split into the following eight

subcommittees:
{a,b,c,h},{c,d,e},{a,b,d,g},{c,d,e,f},{c,d,f},{b,d,g,h},{d,e,f},{c,e,f}

Is it possible for each subcommittee to choose from amongst its members a chairperson, so that nobody chairs more than one subcommittee? Justify your answer.

2 parts:

Solve the PDE using LaPlace Transforme: (A and B are constants)

(Part 1) u_t = u_xx, 0<x<l, t>0

u(x,0) = A,

u_x(0, t) = 0,

u(l, t)= B

(Part 2) u_tt = u_xx,  0< x < l , t >0

u(x,0) = u_t(x,0) = 0

u(0,t) =0,

u(l, t)=A

Suppose we replace Incidence Axiom 4 with the following: Given any line, there are at least three distinct points that lie on it.

What is the smallest number of points in a model for this geometry? More precisely, find a number n such that every model has at least n points and there is at least one model that has only n points, and explain why your answer is correct.

Consider n numbers x1, x2, . . . , xn laid out on a circle and some value α. Consider the requirement that every number equals α times the sum of its two neighbors. For example, if α were zero, this would force all the numbers to be zero. (a) Show that, no matter what α is, the system has a solution. (b) Show that if α = 1 2 , then the system has a nontrivial solution. (c) Show that if α = − 1 2 , then there is a nontrivial solution if and only if n is even.

Blue Company sold 30,000 units of its only product and incurred a $85,000 loss (ignoring taxes) for the current year as shown here. During a planning session for year 2020’s activities, the production manager notes that variable costs can be reduced 25% by installing a machine that automates several operations. To obtain these savings, the company must increase its annual fixed costs by $175,000. The maximum output capacity of the company is 55,000 units per year. Blue Company Contribution Margin Income Statement For Year Ending December 31, 2019 Sales $900,000 Variable $680,000 Contribution Margin $220,000 Fixed Cost $305,000 Net Loss $(85,000) Compute the break-even point in dollar sales for year 2019. (Round your answers to 2 decimal places.) Compute the predicted break-even point in dollar sales for year 2020 assuming the machine is installed and there is no change in the unit selling price. (Round your answers to 2 decimal places.)

Let V be the set of positive reals, V = {x ∈ R : x > 0}. Define “addition” on V by x“ + ”y = xy, and for α ∈ R, define “scalar multiplication” on V by “αx” = x^α . Is V a vector space with these unusual operations of addition and scalar multiplication? Prove your answer.

1. Represent the following in propositional logic. Label the component propositions. Note you may have to reword some propositions.
   1. Bread contains carbs and so does rice.
   2. Either John or Chris was the first student in class this morning.
   3. I have a bad shoulder and whenever it is raining, my shoulder aches.
   4. If the weather is nice, then if Dr Marlowe is lecturing, we will sleep in and then go for a hike.
   5. Henry will stay in his job only if he likes his teammates or he is given a rais
   6. I don’t like grapefruit and I don’t like asparagus unless they are grilled.
   7. Wildwood never has good weather and small crowds.

As a farmer, suppose you want to fence off a rectangular field that borders a river. You wish to find out the dimensions of the field that occupies the largest area, if you have 1200 feet of fencing. Remember that a square maximizes area, so use a square in your work.

• Draw several diagrams to express the situation and calculate the area for each configuration, then estimate the dimension of largest possible field.
• Find the function that models the area in terms of one of its sides.
• Find the point that maximizes the function you found in the second bulleted item.
• Calculate the area of the field at the point you found in the third bulleted item, and then compare your results with the results in the first bulleted item.

Determine the x- and y-intercepts (if any), and vertical and horizontal a symptotes of the rational function r, given by

r(x) =3x^2+ 18x+ 24 / x^2−3x+ 2

and then use this information to sketch a graph ofr. As part of your analysis, you should explicitly examine the behaviour of the function on both sides of each vertical asymptote, and evaluate the function at appropriate test points.

You are reading a research study where the researchers are investigating osteoarthritis pain. In the study, subjects are given either an anti-inflammatory drug injection or a saline injection to reduce osteoarthritis joint pain. A study subject reported that since she received her study injection, her arthritis pain does not seem to have improved. In fact, her pain is getting worse. What is a possible reason for her continued osteoarthritis pain? The research subject likely received the anti-inflammatory study drug, but it just has not yet taken effect. The research subject is resistant to anti-inflammatory agents. The research subject likely received a placebo injection, such as saline. The research subject has an allergy to anti-inflammatory agents.

Please provide a detail explanation and a reference if possible.

What do you think about qualitative research and how it fits in within scientific method research?