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Problem 1: (Gradient vector) Given f (x, y) = x2 + y2, find Vf. Graph some level curves and vectors Vf at the following points: P1 (1,1), P2 (3,4), P3 (-1, -2). Check out this website for visualization of surface and gradient vector: https://mathinsight.org/directional derivative_gradient introduction Problem 2: (a) Find a normal vector of the surface x2 + 3y2 + z2 = 28 at P(4, 1, 3). (Ans: Vfl(4,1,3) = 81 + 6j + 6k) (b) Find the equation for the tangent plane to the surface z = -6x2 + 10y at point (2,1, -14). (Ans: 24x - 20y + z = 14) Problem 3: Find the directional derivative of the function at Po in the direction of A. (Note that the given direction vector A may not be a unit vector). (a) f(x, y) = 2xy - 3y2, Po (5,5), A = 41 + 3j (Ans: -4) (b) f(x, y) = 2x2 + y2, Po(-1,1), A = 31 - 4j (Ans: -4)Problem 4: The plane x = 1, intersects the paraboloid at z = x2 + y2 in a parabola. Find the slope of the tangent to the parabola at (1, 2, 5). (Ans: 4) i. Problem 5: Find the directions in which the functions increases and decreases most rapidly at P0, respectively. (or nd the directions in which the directional derivative is maximum and minimum, respectively). (a) f(x,y) = x2 + xy + yz, Pu(1,1) (Ans: max 3/5, min x/) (b) f(x, y) = xzy + ex? sin y, at P0 (1,0) (Ans: 2, 2) Problem 6 In what direction is the derivative off (x, y) = xy + y2 at P(3,2) equal to zero. Problem 7: Is there a direction 51 in which the rate of change of f(x, y) = x2 33:}! + 4y2 at P(1,2) equal 14? Give reasons for your answer. (Ans: not possible) (Bonus, 2 points) If such direction exists, how would you obtain the unit vector for this direction based on given conditions? Just give your thought on how to approach this problem. Problem 8: By about how much will g(x, y, z) = x + x cos 2 y sinz + 32* change if he point P(x, y, 2') moves from P.3(Z, 1,0) a distance of (13 = 0.2 unit toward the point P1(0,1,2)? (Ans: 0, no change) Problem 9: Find all the local maxima, local minima, and saddle points of the following function. foxy) = 4X)! - If\" -y'*