For each family of vectors, determine wether the vectors are linearly independent or not, and in case they are linearly dependent, find a linear relation between them.

a) x1 = (2, 2, 0), x2 = (0, 2, 2), x3 = (1, 0, 1)

b) x1 = (2, 1, 0), x2 = (0, 1, 0), x3 = (1, 2, 0)

c) x1 = (1, 1, 0, 0), x2 = (0, 1, 1, 0), x3 = (0, 0, 1, 1), x4 = (0, 0, 0, 1)

(1 point) In general for a non-homogeneous problem y′′+p(x)y′+q(x)y=f(x) assume that y1,y2 is a fundamental set of solutions for the homogeneous problem y′′+p(x)y′+q(x)y=0. Then the formula for the particular solution using the method of variation of parameters is yp=y1u1+y2u2 where u′1=−y2(x)f(x)W(x) and u′2=y1(x)f(x)W(x) where W(x) is the Wronskian given by the determinant W(x)=∣∣∣y1(x)y′1(x)y2(x)y′2(x)∣∣∣ So we have u1=∫−y2(x)f(x)W(x)dx and u2=∫y1(x)f(x)W(x)dx. NOTE When evaluating these indefinite integrals we take the arbitrary constant of integration to be zero. In other words we have the single integral formula yp(x)=y1(x)∫−y2(x)f(x)W(x)dx+y2(x)∫y1(x)f(x)W(x)dx As a specific example we consider the non-homogeneous problem y′′+9y=sec2(3x) (1) The general solution of the homogeneous problem (called the complementary solution, yc=ay1+by2 ) is given in terms of a pair of linearly independent solutions, y1, y2. Here a and b are arbitrary constants. Find a fundamental set for y′′+9y=0 and enter your results as a comma separated list ∗ BEWARE Notice that the above set does not require you to decide which function is to be called y1 or y2 and normally the order you name them is irrelevant. But for the method of variation of parameters an order must be chosen and you need to stick to that order. In order to more easily allow WeBWorK to grade your work I have selected a particular order for y1 and y2. In order to ascertain the order you need to use please enter a choice for y1= and if your answer is marked as incorrect simply enter the other function from the complementary set. Once you get this box marked as correct then y2= . With this appropriate order we are now ready to apply the method of variation of parameters. (2) For our particular problem we have W(x)= u1=∫−y2(x)f(x)W(x)dx=∫ dx= u2=∫y1(x)f(x)W(x)dx=∫ dx= And combining these results we arrive at yp= (3) Finally, using a and b for the arbitrary constants in yc, the general solution can then be written as y=yc+yp=

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit in R[x] iff f(x)=a is of degree 0 and is a unit in R.

Find a constant coefficient linear second-order differential equation whose general solution is:

(c1 + c2(x))e^(-2x) + 1

Prove that the Ramsey number R(3,4) = 9 by showing that both the lower bound and the upper bound is 9.

For this problem, let ke = max{2,the largest even number in your student number}, ko = max{3,the largest odd number in your student number} So, e.g., if your student number is 5135731, then ke = 2 and ko = 7. The data in Table 2 represents points (xi,yi) sampled from an experimentally generated triangular wave function with period 2π.
(a) Use the least-squares technique we developed in class to estimate the coefficients a, b and c for the optimal least-squares fit of the data points to the function yf(x) = acos(x) + bcos(kox) + ccos(kex)
(b) The repeating triangular wave function can be expressed as y(x) =(2x−π π 0 ≤ x ≤ π3 π−2x π π ≤ x ≤ 2π Using the inner product hf(x),g(x)i = 1 πR2π 0 f(x)g(x) dx, compare the least-squares coefficients you have obtained with the inner productshy(x),cos(nx)i, for suitable choices of n. How do they compare? What can you conclude?
x y 0.00000 -0.96468 0.50000 -0.64637 1.00000 -0.32806 1.50000 -0.00975 2.00000 0.30856 2.50000 0.62687 3.00000 0.94518 3.50000 0.80715 4.00000 0.48884 4.50000 0.17053 5.00000 -0.14778 5.50000 -0.46609 6.00000 -0.78440

1) A farmer finds that if she plants 60 trees per acre, each tree will yield 40 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 2 bushels. How many trees should she plant per acre to maximize her harvest?

2)A baseball team plays in a stadium that holds 46000 spectators. With the ticket price at $10 the average attendence has been 19000. When the price dropped to $7, the average attendence rose to 23000. Assume that attendence is linearly related to ticket price. What ticket price would maximize revenue?

The largest Ferris Wheel in the world is the London Eye in England. The height (in meters) of a rider on the London Eye after t minutes can be described by the function h(t) = 65 sin[12(t − 7.5)] + 70. (15)

1. How long does it take for the Ferris wheel to go through one rotation?
2. What is the minimum value of this function? Explain the significance of this value.

Make a numerical and graphical summary of the data, commenting on any features that you find interesting. Limit the output you present to a quality that a busy reader would find sufficient to get a basic understanding of the data. PLEASE SHOW ME HOW TO USE R TO SOLVE THIS.

race - racial composition in percent minority

fire - fires per 100 housing units

theft - theft per 1000 population

age - percent of housing units built before 1939

volact - new homeowner policies plus renewals minus cancellations and non renewals per 100 housing units

involact - new FAIR plan policies and renewals per 100 housing units

income - median family income

Note that first column of the data set is ZIP code in Chicago

race fire theft age volact involact income
60626 10.0 6.2 29 60.4 5.3 0.0 11744
60640 22.2 9.5 44 76.5 3.1 0.1 9323
60613 19.6 10.5 36 73.5 4.8 1.2 9948
60657 17.3 7.7 37 66.9 5.7 0.5 10656
60614 24.5 8.6 53 81.4 5.9 0.7 9730
60610 54.0 34.1 68 52.6 4.0 0.3 8231
60611 4.9 11.0 75 42.6 7.9 0.0 21480
60625 7.1 6.9 18 78.5 6.9 0.0 11104
60618 5.3 7.3 31 90.1 7.6 0.4 10694
60647 21.5 15.1 25 89.8 3.1 1.1 9631
60622 43.1 29.1 34 82.7 1.3 1.9 7995
60631 1.1 2.2 14 40.2 14.3 0.0 13722
60646 1.0 5.7 11 27.9 12.1 0.0 16250
60656 1.7 2.0 11 7.7 10.9 0.0 13686
60630 1.6 2.5 22 63.8 10.7 0.0 12405
60634 1.5 3.0 17 51.2 13.8 0.0 12198
60641 1.8 5.4 27 85.1 8.9 0.0 11600
60635 1.0 2.2 9 44.4 11.5 0.0 12765
60639 2.5 7.2 29 84.2 8.5 0.2 11084
60651 13.4 15.1 30 89.8 5.2 0.8 10510
60644 59.8 16.5 40 72.7 2.7 0.8 9784
60624 94.4 18.4 32 72.9 1.2 1.8 7342
60612 86.2 36.2 41 63.1 0.8 1.8 6565
60607 50.2 39.7 147 83.0 5.2 0.9 7459
60623 74.2 18.5 22 78.3 1.8 1.9 8014
60608 55.5 23.3 29 79.0 2.1 1.5 8177
60616 62.3 12.2 46 48.0 3.4 0.6 8212
60632 4.4 5.6 23 71.5 8.0 0.3 11230
60609 46.2 21.8 4 73.1 2.6 1.3 8330
60653 99.7 21.6 31 65.0 0.5 0.9 5583
60615 73.5 9.0 39 75.4 2.7 0.4 8564
60638 10.7 3.6 15 20.8 9.1 0.0 12102
60629 1.5 5.0 32 61.8 11.6 0.0 11876
60636 48.8 28.6 27 78.1 4.0 1.4 9742
60621 98.9 17.4 32 68.6 1.7 2.2 7520
60637 90.6 11.3 34 73.4 1.9 0.8 7388
60652 1.4 3.4 17 2.0 12.9 0.0 13842
60620 71.2 11.9 46 57.0 4.8 0.9 11040
60619 94.1 10.5 42 55.9 6.6 0.9 10332
60649 66.1 10.7 43 67.5 3.1 0.4 10908
60617 36.4 10.8 34 58.0 7.8 0.9 11156
60655 1.0 4.8 19 15.2 13.0 0.0 13323
60643 42.5 10.4 25 40.8 10.2 0.5 12960
60628 35.1 15.6 28 57.8 7.5 1.0 11260
60627 47.4 7.0 3 11.4 7.7 0.2 10080
60633 34.0 7.1 23 49.2 11.6 0.3 11428
60645 3.1 4.9 27 46.6 10.9 0.0 13731

Required information

[The following information applies to the questions displayed below.]

Professor John Morton has just been appointed chairperson of the Finance Department at Westland University. In reviewing the department’s cost records, Professor Morton has found the following total cost associated with Finance 101 over the last five terms:

Professor Morton knows that there are some variable costs, such as amounts paid to graduate assistants, associated with the course. He would like to have the variable and fixed costs separated for planning purposes.

3-a. Assume that because of the small number of sections offered during the Winter Term this year, Professor Morton will have to offer eight sections of Finance 101 during the Fall Term. Compute the expected total cost for Finance 101.

3-b. Can you see any problem with using the cost formula from (2) above to derive this total cost figure?

In how many ways can an 8×8 chessboard be tiled with dominoes? Ex-plain your answer.

Solve the given nonhomogeneous ODE by variation of parameters or undetermined coefficients. Give a general solution and show all details of your work.

a) y''-4y'+4y=x^2e^x

b) (D^2-2D+I)y=x^2+x^(-2)e^x

Solve the following initial value problem

d2y/dx2−6dy/dx+25y=0,y(0)=0, dy/dx(0)=1.

Let A = {1,2,3}. In each part, give an example of the requested function and justify that your example has the required properties, or explain why no such function exists.

(a) A function f : A → A that does not have an inverse.
(b) A function f : A → A×A that is surjective (i.e., onto).
(c) A function f : P(A) → A such that for all X ∈P(A), f(X) ∈ X.
(d) A function f : A → A that is also a transitive relation.

A) Use Jacobi or Gauss-Seidel iteration and perform three iterations by hand.

B) Use Jacobi or Gauss-Siedel iteration for ten iterations with a MAT-LAB function.

* A= [10 -2 1;-2 10 -2;-2 -5 10] , B=[9;12;18]