Problem #1: Mary found the optimum value of x2 + 2x + 3y2 + 3y + 3 subject to x + y = 7 to be at the [2 marks] point (43, 1) What was the value of the Lagrange multiplier, 22
2x+5xy* 1) Let f(x,y) = *3+x3y2 Which among the following is true about limf(x,y)? (x,y)--(0,0) a. By using the two path test we can deduce that the limit does not exist b. By using the two path test we can deduce that the limit exists c. The limit is 2 d. None of the above O a. O b. O c. O d. 2) Let f(x,y) Vx+1-y+1 xy Then lim f(x,y) (xy)+(0,0) a. is 0 b.is c. is 1 d....
Determine if the following pairs of lines are parallel, perpendicular or none. a) 2x + 3y = 6 3x - 2y = 6 b) y = 2x + 3 x = 2y + 3 c) x = -2 - 3y 2x + 6y = 5
5. Evaluate the integral c (2x -y)dx + (x + 3y)dy along the path C: line segment from (0,0) to (3,0) and (3,0) to (3,3) 5. Evaluate the integral c (2x -y)dx + (x + 3y)dy along the path C: line segment from (0,0) to (3,0) and (3,0) to (3,3)
D.E (1) y"+2y'+y=x2-1-3, y(0)=-2, y'(0)=1 (2) + y'= -8cos2x+6sin 2 x (3) y*- 3y + 2 y =e" (x2 + 2x - 1)
3. Suppose x,y,z satisfy the competing species equations <(6 - 2x – 3y - 2) y(7 - 2x - 3y - 22) z(5 - 2x - y -22) (a) (6 points) Find the critical point (0,Ye, ze) where ye, we >0, and sketch the nullclines and direction arrows in the yz-plane. (b) (6 points) Determine if (0, yc, ze) is stable. (c) (8 points) Determine if the critical point (2,0,0) is stable, where I > 0.
The area of the region enclosed by the curves y = 2x, 3y = X and 2y = -x+5 is none of these o o 3,5 0 2.5 0 o o
3. Consider the function f(x,y) = 4 + 2x - 3y - x2 + 2y2 - 3xy. a) (5 pts.) Calculate the partial derivative functions, and use them to calculate the gradient vector evaluated at c = b) (5 pts.) Write down the affine approximation to at the e given in a) /(x) = f(c)+ Vf(e)'(x - c) . Use it to calculate (1.1, 1.1). (Hint: it should be close to f(1.1, 1.1))
1. xy' = 3y + 3 3. (x + 1)y' - (2x + 3) y = 0
4. Consider the set S = {(x, y) | x ∈ [0, 1], x2 ≤ y ≤ x}. Prove that S is a Jordan region and integrate the function 2x 2 + 3y 2 on S.