A function of two variables being continuous means

Select all that apply

The function is built from elementary functions and algebraic operations.

We can evaluate limits of the function by simply plugging in values.

Its graph can be drawn without lifting up the pencil

The function's value at each point of the domain is equal to its limit there.

All the partial derivatives exist.

Suppose h, k, r, s, t∈Z.Set a=3rt(2s+t) and b=3rs(s+2t).Prove the cubic polynomial

f (x) = (x − h)(x − a − h)(x − b − h) + k
passes through the point (h,k), has integer roots, has local extrema with integer coordinates, and has an inflection

point with integer coordinates.

How to do a value stream map (VSM) of the customer ordering process for the X-opoly scenario?

X-Opoly, Inc., was founded by two first-year college students to produce a knockoff real estate board game similar to the popular Parker Brothers; game Monopoly®. Initially, the partners started the company just to produce a board game based on popular local landmarks in their small college town, as a way to help pay for their college expenses. However, the game was a big success and because they enjoyed running their own business, they decided to pursue the business full-time after graduation.

X-Opoly has grown rapidly over the last couple of years, designing and producing custom real estate trading games for universities, municipalities, chambers of commerce, and lately even some businesses. Orders range from a couple of hundred games to an occasional order for several thousand. This year X-Opoly expects to sell 50,000 units and projects that its sales will grow 25 percent annually for the next five years.

X-Opoly’s orders are either for a new game board that has not been produced before, or repeat orders for a game that was previously produced. If the order is for a new game, the client first meets with a graphic designer from X-Opoly’s art department and the actual game board is designed. The design of the board can take anywhere from a few hours to several weeks, depending on how much the client has thought about the game before the meeting. All design work is done on personal computers.

After the client approves the design, a copy of the computer file containing the design is transferred electronically to the printing department. Workers in the printing department load the file onto their own personal computers and print out the board design on special decals, 19.25 inches by 19.25 inches, using high-quality color inkjet printers. The side of the decal that is printed on is usually light gray, and the other side contains an adhesive that is covered by a removable backing.

The printing department is also responsible for printing the property cards, game cards, and money. The money is printed on colored-paper using standard laser printers. Ten copies of a particular denomination are printed on each 8.5-inch by 11-inch piece of paper. The money is then moved to the cutting department, where it is cut into individual bills. The property cards and game cards are produced similarly, the major difference being that they are printed on material resembling posterboard.

In addition to cutting the money, game cards, and property cards, the cutting department also cuts the cardboard that serves as the substrate for the actual game board. The game board consists of two boards created by cutting a single 19-inch by 19.25-inch piece of cardboard in half, yielding two boards each measuring 19.25 inches by 9.5 inches. After being cut, game boards, money, and cards are stored in totes in a work-in-process area and delivered to the appropriate station on the assembly line as needed.

Because of its explosive growth, X-Opoly’s assembly line was never formally planned. It simply evolved into the 19 stations shown in the following table.

A polynomial in Z[x] is said to be primitive if the greatest common divisor of its coefficients is 1. Prove the product of two primitive polynomials is primitive. [Hint: Use proof by contradiction.]

Let ? and W be finite dimensional vector spaces and let ?:?→? be a linear transformation. We say a linear transformation ?:?→? is a left inverse of ? if ST=I_v, where ?_v denotes the identity transformation on ?. We say a linear transformation ?:?→? is a right inverse of ? if ??=?_w, where ?_w denotes the identity transformation on ?. Finally, we say a linear transformation ?:?→? is an inverse of ? if it is both a left and right inverse of ? . When ? has an inverse, we say ? is invertible.

Show that

(a)  ? has a left inverse iff ? is injective.

(b) If ? is a basis for ?V and ? is a basis for ?, then [?]^?_?(Transformation from basis ? to ?) has a left inverse iff its columns are linearly independent.

Use Matlab to solve the system x²+xy³=9 , 3x²y-y³ =4 using Newton's method for nonlinear system. use each of initial guesses (x₀,y₀)=(1.2,2.5), (-2,2.5), (-1.2,-2.5), (2,-2.5) observe which root to which the method converges, the number of iterates required and the speed of convergence.

Write the system in the form f(u) = 0, and report for each case the number of iterations needed for ||f(u)||²≤ 10^-12−.

Show that there is a continuous, strictly increasing function on the interval [0, 1] that maps a set of positive measure onto a set of measure zero.

(Use the Cantor set and the Cantor-Lebesgue Function)

Are the following functions satisfiable?
If the function is satisfiable, with a single line containing 4 comma-separated values, each of

which is either True or False, for x, y, z, v in this order. For example, you would submit: True,False,True,False.

If the function is not satisfiable, use the laws of propositional logic to prove that the function is a contradiction.

a) xy ̄+zv
b) (x+y)(x ̄+z)(y ̄+z ̄)(x+v)
c) xx ̄+yy ̄+zz ̄+vv ̄
d) (x+y)(x+y+z)+not(x+y+z)not(x+y+z+v)

Write a Matlab m-script to compute the backward difference approximation A = f(a)−f(a−h) h of the derivative T = f0(a) for f(x) = sin(x) and a = π/3 using each value of h in the sequence 2−n (n = 1,2,3,···,52).

1)

Solve the Laplace equation ∇^2(u)=0 (two dimensions so ∂^2/∂a^2 + ∂^2/∂b^2) where the boundaries of the rectangle are 0 < a < m, 0 < b < n with the boundary conditions:

u(a,0) = 0

u(a,n) = 0

u(0,b) = 0

u(m,b)= b^2

There is a hill 600 feet tall the slope of the road originally has a 27% grade(slope) and it is decreased to 14% to make it a legal road in the town. legal roads have a 12-15% grade maximum. Find the length of the new road and the percentage increase between the old and new road

A company operates a solar installation in the desert in Western Australia. It is reviewing its operating practices with a view to making them more efficient

. a) The solar installation generates electric power from sunlight and incurs operating costs for cleaning the solar modules (sometimes called solar panels) and replacing solar modules that have failed. The annual revenue from the electric power is variable due to variable cloudiness and solar module failure and has a mean of $2.78m and a standard deviation of $0.32m. The annual operating costs have a mean of $0.51m and a standard deviation of $0.12m. Calculate the mean and standard deviation of the annual profit = annual revenue – annual operating costs.

b) Expected revenue varies systematically from one month to another, being higher in the summer when there is more sunshine. Monthly operating costs follow the same probability model regardless of the month (same mean and standard deviation apply to all months). Calculate, if possible, the mean and standard deviation of (i) monthly operating costs (ii) monthly profits. If a calculation is not possible, give the reason.

c) The solar installation is located in the desert 100 km from the nearest office of the company that operates it and the company sends a maintenance crew out quarterly (once every 3 months) to clean dust and sand off the solar modules and check for mechanical or electrical problems. Each solar module is also monitored electronically over the Internet so that the operating company is alerted immediately when a solar module fails. On average 1.3 modules fail per month and the maintenance crew replaces any failed modules on their quarterly visits. Module failures are independent of each other and occur at random. The loss of a few solar modules does not impact revenue enough to justify the cost of sending the maintenance crew before the next quarterly visit. However the operating company decides that if more than 7 modules have failed they should send the maintenance crew out immediately to replace the failed modules. What is the probability of the maintenance crew having to go to the solar installation before the end of the regular 3-month period?

d) If 8 modules fail, the maintenance crew loads 9 replacement modules into their truck in case one is smashed during the 100 km drive, much of which is over uneven dirt tracks through the desert. Past experience shows that the probability of any individual module being smashed on this journey is 0.043. The operations manager wants the probability that the crew arrives with less than 8 working modules to be < 0.05. How many replacement modules should the maintenance crew load into their truck so as to achieve this objective? Answer this question, stating your assumptions clearly, and comment on whether the assumptions are likely to be true.

e) The solar modules are covered by a 25-year warranty which covers the cost of the replacement module itself but not the cost of driving 100 km and installing it. The operating company plans on visiting the site only once every 3 months and is therefore considering purchasing “business continuity insurance” which would cover the loss of revenue from failed solar modules for an annual premium of $5000. In order to decide whether it is worth paying this premium the company needs to calculate its expected revenue loss from failed modules. The average loss of revenue from one failed module is $200 per month. If one module fails during a 3-month period, we assume it fails in the middle of that period so that it has failed for a total of 1.5 months and the loss of revenue is 1.5*200 = $300. We make similar assumptions if 2,3,4, … modules fail during the 3 month period. Considering the probabilities of 0,1,2, …,10 modules failing during a 3-month period, what is the expected revenue loss during a 3-month period? Based on this expected loss, should the company purchase business continuity insurance?

3) Determine the longest interval in which the given initial value problem is certain to have a unique twice-differentiable solution. Do not attempt to find the solution t(t − 4)y" + 3ty' + 4y = 2 = 0, y(3) = 0, y'(3) = −1.

4) Consider the ODE: y" + y' − 2y = 0. Find the fundamental set of solutions y1, y2 satisfying y1(0) = 1, y'1 (0) = 0, y2(0) = 0, y'2 (0) = 1.