3. Let f(,y) = cos(xy) and a =(,1). (a) Find f(a). (b) Find a unit vector which is normal to the level set {(x,y): f(x,y) = 0} at the point a. (c) For the unit vector ū= (-3), find the directional derivative Daf(a). (d) What is the largest possible value for Duf(a) among all unit vectors ü? What is the least possible value? (e) Consider the path elt) = (1,7)+(-), and the composition g(t) = f oct). Find g(0).
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 : z? + y2 < 4} of radius 2. (a) Find the maximu and minimu of Duf at (0,0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk(,y) E R2 r2 + y2 < 4} and all unit vectors u. (llint. Think of IvJF as a function ofェand y in the disk.)
Let f(x,y) = x2 - xy + y2 - y. Find the directions u and the values of Duf(1, -1) for which the following is true. a. Duf(1, - 1) is largest c. Duf(1, - 1) = 0 e. Duf(1, -1)= -3 b. Duf(1, -1) is smallest d. Duf(1, -1) = 4 a. Find the direction u and the value of Duf(1, - 1) for which Duf(1 - 1) is largest u=[i+Oj and D,f(1, - 1) = 0 b. Find...
1. (1.5 points) Sketch the following vector fields: (B) B(x,y)=(z-y,2). (C) Vf where f(x,y) = xy 1. (1.5 points) Sketch the following vector fields: (B) B(x,y)=(z-y,2). (C) Vf where f(x,y) = xy
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...
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) =
2. Let if r and y are not both 0 f(x, y) = 0 if (x, y) = (0,0) (a) Show that and we both exist at the origin are are zero (b) Let v = (v1, v2) be a unit vector with vị and v2 both not zero. Prove that V (f) at the origin exists, and compute it directly from the definition. Does the formula Vu(f) = (Vf). ✓ hold at the origin? (c) Is f differentiable at...
Please describe the contour map and list important aspects of it, thanks! Let f(x,y) -2(xy 1) be a scalar function in R2. a) Find the vector field F(x, y) for which f(x, y) is a potential function, b) c) sketch a contour map of f (x, y) and, on the same figure, sketch F(x,y) (on R2). Comment on any important aspects of your sketch. Let f(x,y) -2(xy 1) be a scalar function in R2. a) Find the vector field F(x,...
2. Let f(x,y) == xy + sin(x). Find a unit vector ū such that for the directional derivative Daf(7,0) one has Daf(1,0) = -_. 27+127. b. None of the other alternatives is correct. Ocū7-7
5) The level curves of a function f(x,y) are given in the graph below. 2 X -1 -2 i Estimate f(3,3) ii Estimate Vf(-3, 1) Let u be a unit vector parallel to (1,4). Calculate Daf using your answer from i iv) Find the location of all critical points of the function f, on the set -5 <r< of these is a saddle point) iii) Let D be the domain bounded between the curves y = x and y= 2...