Sketch the region of continuity for f (x; y) on a set of axes and sketch the region of

continuity for df/dy (x. y) on a separate set of axes. Apply Picard’s Theorem to determine whether the

solution exists and whether it is unique.

a) y'  = 2x²y + 3xy² ; y(1) = 2

b) y' = sqrt(2x - 3y) ; y(3) = 2

Let S be a subset of a vector space V . Show that span(S) = span(span(S)). Show that span(S) is the unique smallest linear subspace of V containing S as a subset, and that it is the intersection of all subspaces of V that contain S as a subset.

1.let {v=(1,2,3,5,9),v2=(3,1,2,8,9),v3=(2,-5,5,9,4)} and {u1=(0,1,1,1,2),u2=(0,2,-2,-2,0)} be basis of subspaces V and U of R5 respectively.find a basis and the dimension of V+U and V intersection U.

2.does a matrix have a right inverse ?if so find one A=[2,-3,-7,11;3,-1,-7,13;1,2,0,2]

3.find the interpolating polynomial that passes through the point (1,2),)(-1,-8) and (2,1)

Consider the equation uux + uy = 0 with the initial condition

u(x, 0) = h(x) = ⇢ 0 for x > 0

u_o for x < 0,   with u_o< 0.

Show that there is a second weak solution with a shock along the line x = u_o y / 2

The solution in both mathematical and graphical presentation before and after the shock.

Find the adjoint of matrix A, the determinant of matrix A, and the determinant of the adjoint A.

A= 1 1 0 2

2 1 1 0

0 2 1 1

1 0 2 1

A SEIRS model with stochastic transmission :project proposal

National governments issue debt securities known as sovereign bonds, which can be denominated in either local currency or global reserve currencies, like the U.S. dollar or euro. First define what these bonds are. Why are these issued? Then discuss the issues that can arise when investors invest in these types of bonds. What are the advantages and disadvantages of these bonds? Are there unique issues that can arise only with this type of bond? Would you invest in sovereign bonds?

Show that radical 3, radical 5, radical 7, radical 24, and radical 31 are not rational numbers

Consider the initial value problem given below.

y'=x+4cos(xy), Y(0)=0

Use the improved​ Euler's method subroutine with step size h=0.3 to approximate the solution to the initial value problem at points x= 0.0,0.3,0.6.....3.0

4. (Applying LU and LUP decompositions) In this problem, we'll use the LU/LUP decomposition to solve a linear system of equations.

a) For A = [12 -8 13 -1 13;14 11 -5 -5 -7;1 -8 -9 10 8;-11 10 -8 3 8;-11 -8 4 2 -4] find matrices P, L, and U so that PA = LU using Matlab's lu function. Based on your results: did Matlab use pivoting during the lu-computation?

b) For b = [4;-4;-5;3;7] solve Ax = b using the LU decomposition as follows. Solving Ax = b is the same as solving PAx = Pb. (With P from a). Since PA = LU, we need to solve LUx = Pb, and we can split that into two triangular systems as follows: Ly = Pb, and Ux = y. Solve both of these systems using Matlab's linsolve, state x and y explicitly.

c) Compare the quality of the x you found in b to the solution of Ax = b you get from using linsolve. (As in 2d, work with the differences Ax - b).

d) You want so solve Ax = b for various vectors b, so you collect them into a single matrix B. So your goal is to find a matrix X so that AX = B (one column in X for each column in B). Working with P, L, and U from parts a/b we see that this amounts to solving two systems: LY = PB and UX = Y. For B = [18 -7 -14 10 -14 -2 13 12 -15 -15;-15 -14 4 -2 13 -16 15 -3 -15 14;3 12 -10 -17 2 19 -17 17 15 5;-1 -8 6 -11 20 -20 -4 -13 3 -6;-20 1 8 17 -17 11 -10 -10 2 1] solve these two equations using Matlab's linsolve.

First find Y in LY = PB, and then use that to find X in UX = Y. Check that AX - B is close to the zero matrix.

For this last problem, work with format short (or even format compact) so that the matrices don't use up too much screenspace.

2. (Solving linear systems)

Consider the linear system Ax = b

with A =[14 9 14 6 -10;-11 -11 5 8 6;15 -2 -14 8 -15;14 13 11 -3 -7;0 9 13 5 -14], and . b = [-4;8;6;0;10].

a) Verify that the linear system has a unique solution. Hint: use rref, rank, det, or any other Matlab method. Briefly explain the answer please.

You'll now solve Ax = b in three different ways. Store the three different solutions in four different variables (x1,x2,x3,x4, say), we need to compare them in parts c and d.

b1) Using rref, determine the solution x of the system Ax = b, and store it in variable x1. Hint: x1 is the final column of the reduced row echelon form of M = (A|b). Remember horzcat.

b2) Using Matlab's linsolve method, find a solution to the system Ax = b. Store the solution in x2.

b3) Using Matlab's inverse method (^(-1) or inv), solve the system as x = A^(-1)b, store the result in x3.

b4) Use the rref method from class to find an inverse matrix B of A. Use that to calculate x = Bb. Store the result in x4.

c) Compare the solutions x1, x2, x3,x4. Do any two of them agree? Does that contradict a) or not?

d) Try ranking the solutions by quality. To do so, compute Ax-b, for the various values of x, and see which x produces the smallest difference between Ax and b.

Q: (LU decomposition) Find the LU decomposition of A = [-3 2 5 1; 12 -4 -20 -2; -6 0 15 1; -9 6 35 4]. You can use the compact method which works within a single matrix or you can build L and U separately. State L and U explicitly, and verify (in Matlab) that A = L*U. Hint: Matlab's built-in lu function isn't useful, since it pivots.

Q- A manufacturer of outdoor clothing makes wax jackets and trousers. Each jacket requires 1 hour to make, whereas each pair of trousers takes 40 minutes. The materials for a jacket cost $32 and those for a pair of trousers cost $40. The company can devote only 34 hours per week to the production of jackets and trousers, and the firm’s total weekly cost for materials must not exceed $1200. The company sells the jackets at a profit of $12 each and the trousers at a profit of $14 per pair. Market research indicates that the firm can sell all of the jackets that are produced, but that it can sell at most half as many pairs of trousers as jackets.

1. A- How many jackets and trousers should the firm produce each week to maximize profit?

b- Due to the changes in demand, the company has to change its profit margin on a pair of trousers. Assuming that the profit margin on a jacket remains at $12 and the manufacturing constraints are unchanged, find the minimum and maximum profit margins on a pair of trousers which the company can allow before it should change its strategy for optimum output.

I am asked to find the square roots using the bisection method for x * x - a = 0.

I was wondering how the bisection method is performed.

Let's suppose a = 9, so I would need to find the roots of x * x - 9 = 0.

Also, from the 1st equation, when would the bisection method NOT output a root?

Solve the initial value problem dy/dx = −(2x cos(x^2))y + 6(x^2)e^(− sin(x^2)) , y(0) = −5

Solve the initial value problem dy/dt = (6t^5/(1 + t^6))y + 7(1 + t^6)^2 , y(1) = 8.

Find the general solution of dy/dt = (2/t)*y + 3t^2* cos3t