Fluidyne Corporation manufactures and sells water filters. The economic forecasting unit of the company has supplied the following demand equation:                                      Points: 5

QB=    2000 -     5PB +   2.5PC   +   0.82Y   + 0.6AB

          (1200)       (1.5)       (1.2)            (0.5)      (0.2)

Where,

QB=quantity sold

PB=price per unit

PC=average unit price of competitor’s product

Y=income per household

AB=advertising expenditure

            R2= 0.86

S.E.E=5

Standard error of coefficients in parentheses (1200) (1.5) (1.2,) (0.5) , ( 0.2)

Given, PB=$50,    PC=$45,      AB=$12,500      Y=$2,000

1. Does each independent variable have a significant effect on the sales of Fluidyne water filters?
2. Interpret the coefficient of variables that are significant?
3. Interpret the coefficient of determination(R²)
4. Determine the monthly quantity demanded sold (QB) for water filter?
5. Is the quantity demanded for water filter (QB) sensitive to its own price?
6. Is water filter a luxury or necessity?

Consider the relation R on N such that xRy if and only if the sum of the digits of x and y coincide.

(i) Prove or disprove R is an equivalence relation. (ii) What are the equivalence classes of R.

Differential Equations. A 10kg weight is attached to a wall horizontally via a spring. The rest (equilibrium) length of the spring is 0.2m. The spring has stiffness coefficient k = 50. The weight drags along the ground, giving an effective damping coefficient of b = 40. We will ignore gravity and all other outside forces on the spring. If the weight is pulled to a position 0.5m away from the wall (thus +0.3m past equilibrium) and then flicked toward the wall at 2m/s. How close is the weight to the wall after 1 second? Bonus question: What is the closest the weight gets to the wall? Must use differential equations formula not physics

Determine if the following is is reflexive, symmetric, antisymmetric and transitive and why?

x relates y <-> x divides y 2 (on all positive numbers)

Compare your sample estimate to the current mean gas in the State of Ohio. Find the mean price of gasoline in the State of Ohio. You might try this GasBuddy website. Set up a null and alternative hypothesis to see if your sample for Dayton is enough to prove that the population mean gasoline price in Dayton is different than the mean price in Ohio. Test the hypotheses. Show your work. Using a significance level of 0.05, what is your conclusion?

Part 3 - Compare your sample estimate to the current mean gas in the U.S. What is the mean price of gasoline in the U.S.? You might try reading the graph on this GasBuddy website. Set up a null and alternative hypothesis to see if your sample for Dayton is enough to prove that the population mean price in Dayton is different than the mean price in the U.S. Test the hypotheses. Show your work. Using a significance level of 0.05, what is your conclusion?

For each of the following problems (even the book ones) do the following:

a. Use the x- and y- nullclines to find all the equilibrium points.

b. Compute the Jacobian matrix of the system.

c. Determine the type of each equilibrium point (if it is a hyperbolic equilibrium).

d. Plot the phase portrait in CoCalc.

1. dx/dt = x-xy-8x^2

dy/dt = -y+xy

2. dx/dt = x-y+x^2

dy/dt = x+y

3. dx/dt = 8x-4x^2-xy

dy/dt = 3y-3xy-y^2

4. dx/dt = y-x^2y

dy/dt = -x+xy^2

According to​ Kepler's first​ law, a comet should have an​ elliptic, parabolic, or hyperbolic orbit​ (with gravitational attractions from the planets​ ignored). In suitable polar​ coordinates, the position left parenthesis r comma script theta right parenthesis of a comet satisfies an equation of the form r = betaplus​e(r*cosine script theta​), where beta is a constant and e is the eccentricity of the​ orbit, with 0less than or equals less than1 for an​ ellipse, e equals1 for a​ parabola, and e greater than1 for a hyperbola. Suppose observations of a newly discovered comet provide the data below. Determine the type of​ orbit, and predict where the comet will be when script theta equals 4.1 left parenthesis radians right parenthesis. script theta 0.88 1.14 1.48 1.72 2.17 r 3.68 3.08 2.08 1.06 0.54 The comet has either a. hyperbolic b parabolic c elliptic  orbit. When script thetaequals4.1 ​(radians), the comet will be at requals nothing.

1.Find the derivative of the product between a scalar function and a vector function using the product formula.

2. Find the volume of an irregular solid using triple integration, the first integral should have at least one limit with variables.

3. Determine the moment of inertia of an irregular solid using triple integration. the first integral should have at least one limit with variables.

4. Find the angle between two lines using dot product. the two lines should not pass through zero.

5. Determine the work done (line integral) in a close path using two methods. The path should contain a curve and a line. the line should not pass through (0,0). The first method should be by using directly the formula F∙dr and the second method using Green's Theorem. Give your own vector field function F. F should be of the form <ax^myⁿ,ax^myⁿ>

6. Discuss a practical application of the cross product (vectors)

Sketch the graph of a function f(x) that satisfies all the given conditions. Clearly label any asymptotes, extreme values and points of inflection.

f(x) is only discontinuous at x = −4.

f(x) has a global minimum but no global maximum.

f'(x) > 0 only on the intervals (−∞, −4) and (1, 3).

f(x) only changes concavity at x = −1 and x = 4.

limx→∞ f(x) = 4.

Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing three red marbles, five green ones, two white ones, and two purple ones. She grabs seven of them. Find the probability of the following event, expressing it as a fraction in lowest terms. HINT [See Example 1.]

She has all the red ones.

Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing three red marbles, three green ones, four white ones, and three purple ones. She grabs eight of them. Find the probability of the following event, expressing it as a fraction in lowest terms. HINT [See Example 1.]

She has at least one green one

Recall from Example 1 that whenever Suzan sees a bag of marbles, she grabs a handful at random. She has seen a bag containing four red marbles, three green ones, four white ones, and two purple ones. She grabs five of them. Find the probability of the following event, expressing it as a fraction in lowest terms. HINT [See Example 1.]

She has two red ones and one of each of the other colors.

Question 2: A bipartite graph with 2n vertices (namely |V1| = |V2| = n) is d-regular if and only if the degree of every vertex in V1 ∪ V2 is exactly d. Show that a d-regular bipartite graph always has a perfect matching (a matching of size n that includes all vertices).

***Remarks: All the graphs here are without self loops and parallel or anti-parallel edges. A network is a directed graph with source s and sink t and capacity ce > 0 on every edge e. In all the algorithms, always explain their correctness and analyze their complexity. The complexity should be as small as possible. A correct algorithm with large complexity, may not get full credit. The number of vertices is denoted by n, and the number of edges by m. ***

How would I prove this ?

R > A / R > ( A v W )

I was thinking that this would work but I am not sure.

2. R > (A v W) 1, MP