(10 pts) 4/ Let f(x,y)= xet-3 and P = (1,1). a) Find the rate of change...
(1 point) Letf(x,y) = xer'-y and P = (8, 64). (a) Calculate lIVf(P)lI. (b) Find the rate of change of f in the direction Vf(P) (e) Find the rate of change of f in the direction of a vector making an angle of 4 5 with Vf(P). Answers (a) 129.25
rty. I 5. [16 pointsj Consider the function f(x, y,z) Let S denote the level surface consisting of all points in space such that f(,y,z)-4, and let P- (2,-2,1), which is on S. a) Calculate Vf. b) Determine the maximum value of Daf(P), where u is any unit vector at P c) Find the angle between Vfp and PO, where O denotes the origin. d) Find an equation for the tangent plane to S at P rty. I 5. [16...
Find the directional derivative of the function f(x,y,z) = z4−x3y2 at P(1,-1,1) in the direction of the vector from P(1,-1,1) to the point Q(2,1,0). What is the maximum rate of change of f at the point P(1,-1,1) and in which direction
17. Given f(x, y, z) = x^yz -- xyz', P(2,-1,1) and vector v =<1,0,1 >. Find i. the directional derivative of the function at the point P in the direction of v. ii. the maximum rate of change of f.
please circle the answer! (1 point) Suppose f (x, y) = , P = (1, 3) and v 3i - 2j A. Find the gradient of f Vf = i+ j Note: Your answers should be expressions of x and y; e.g. "3x - 4y" B. Find the gradient of f at the point P. (Vf) (P) it j Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of v. Duf...
Consider the following. f(x, y, z) = xe3yz, P(2,0,1), u } (a) Find the gradient of f. Vf(x, y, z) = (b) Evaluate the gradient at the point P. VF(2,0, 1) = (c) Find the rate of change of fat P in the direction of the vector u. Duf(2, 0, 1) =
4. Let 3 f(x, y, z) = x’yz-xyz3, 4 P(2, -1, 1), u =< 0, > 5 a). Find the gradient of f. b). Evaluate the gradient at the point P. c). Find the rate of change of f at the point of P in the direction of the vector u.
3) Suppose that w = f(x, y, z) = ln(x y2z3). a) (20 pts.) Find the unit vector in the direction of most rapid increase in w at the point (x,y,z) = (1,-2,-3) b) (15 pts.) Find the rate of change in w in this direction at (1,-2,-3).
please solve now (a) 3 marks The directional derivative of f(x,y) at a point P in the direction of the vector <2,3 > equals 7, and the directional derivative of f(x,y) at a point P in the direction of the vector < 1,-2 > equals 5. Find Vf at P. (b) 4 marks (c) 4 marks Find Zxy if z3 = xz+y. (d) 4 marks Find and classify all local extreme points of f(x,y) = x3 + y3 - 3x...
1. Let X and Y have a discrete joint distribution with ( P(X = x, Y = y) = {1, 10, if (x, y) = (-1,1) if x = y = 0 elsewhere Show that X and Y are uncorrelated but not independent. [5 points] 2. Let X and Y have a discrete joint distribution with f(-1,0) = 0, f(-1,1) = 1/4, f(0,0) = 1/6, f(0, 1) = 0, $(1,0) = 1/12, f(1,1) = 1/2. Show that (a) the two...