1. First-order PDE examples. For each of the following problems find the char- acteristic lines, sketch some of them in the xy-plane, state the general solution to the PDE, and find the particular solution satisfying the boundary data. (a) 3ux +4uy = 0, u(x, 0) = sin(2x). (b) Ux+2uy = 0, u(0, y) = y. (c) √1 − x² Ux + Uy = 0, u(0, y) = y. Consider only the restricted domain |x| ≤ 1. (d) Ux+ Uy = 1, u = x on y = x/2.

2. Sketch out the possible solution area (shade the area). Label all corner points with an ordered pair (x,y). x+y >1 4x + 3y 0 3. The solution area in problem number 2 represents how to maximize profits for your company. Your boss wants to know ALL the possible ways to maximize profits. a. Tell me one function in the form z= Ax + By, that will produce an infinite number of ways to maximize profits. b. Write a 2 or 3 sentence summary to your boss on why the function will work. Explain it from a mathematical perspective, because your boss also took Math 12.

Oo.116. Subject  :- Advance Mathematic

Find the minimum value of x +y +z² givenx+y+z=3a.,

this is my last attempt and I really need help

Find the minimum and maximum values of the function ƒ(x, y) = x²y + x + y subject to the constraint xy 4. (Give exact answers. Use symbolic notation and fractions where needed. Enter DNE if the extreme value does not exist.) minimum: maximum:

3. Suppose that a Pizza Store revenues about 20,000r pesos when employing y clerks. Let P be the weekly profit function (in pesos) of the store, which is given by P(r, y) = 3,000 + 240y + 20y(x – 2y) – 10(r – 12)*. What should be the number of clerks and their collected revenue to obtain the greatest profit? If the profit function is constrained by the equation 61x – y = 6133, is there still any profit? Justify your answer.

2. Find the equation f(x) = ax+b of the least square line for the points (1,0), (−1, 2), (2, 1).

Sketch the following problem graphically: max f = x2 X2 subject to g1 = X1 – (1 – x2)³ < 0 92 = x1 2 0 Find the solution graphically. Apply the optimality conditions and monotonicity rules. Discuss. (From Kuhn & Tucker, 1951.)

3. The quadratic functions f and g are defined by f(x) = 4x + p, xeR %3D g(x) = x +qx+ p, x€R where p. q and r are non-zero constants such that p=-2r and q=-3r. Given that (x+r)is a factor of f and g. Determine the value of p. q and r. Hence, factorise / and g completely. [10 marks] C3, CLO 3, PLO4, PEO 3, MQF LOD 6