3. Let f(,y) = cos(xy) and a =(,1). (a) Find f(a). (b) Find a unit vector...
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
Consider the following function 6 f(x, y,z)=z - x? cos(my) + xy? (i) Find the gradient of the function f(x, y, z) at the point P,(2,-1,-7). (ii) Find the directional derivative of f(x, y, z) at P,(2,-1,-7) along the direction of the vector ū = 2î+j+2k. (iii) Find the equation of the tangent plane to the surface given below at the point P,(2,-1, -7). 6 :- xcos(ty) + = 0 xy
2. Let f(x, y) = xy (2] (a) Findäf af and Vf. 5 (b) Find a unit vector u for which Duf(v2, V2) = 0. 2. Let f(x, y) = xy (2] (a) Findäf af and Vf. 5 (b) Find a unit vector u for which Duf(v2, V2) = 0.
6. Let f(x,y) = xy+sin(x). Find all directions (unit vectors) so that the directional derivative off at the point (1,0) equals -
s (ls points) 1/ Given f(x,>)-xy+e" sin y and P(1,0) a) Find the directional derivative of fat P in the direction of Q(2, 5). b) Find the directions in which the function increases and decreases most rapidly atP e) Find the maximum value of the directional derivative of fat P. d) Is there a direction u in which the directional derivative o f fat P equals 1? If there is, find u. If there is no such direction, explain. e)...
Problem 1: Let F(, y,) be a function given by F(, y, z) (r2+y)e. Let S be the surface in R given by the equation Fr, y, 2) 2. (a) Find an equation of the tangent plane to the surface S at the point p(-1,1,0) (b)Find the directional derivative -1,1,0) of F(,y,2) in the direction of the unit vector u = (ui, t», t's) at the point p(-1,1,0) - In what direction is this derivative maximal? In what direction is...
Find the directional derivative of f(x,y,z)=xy+z^3 at the point (2,3,1) in the direction of a vector making an angle of 3π/4 with ∇f(2,3,1).
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...
(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.)
1. Let f(x,y) = (2-7-% and g(x,y) = v f(x,y). J(1)(4 points) Find the maximum value of g(y). |(272 points) At which point(s) (x,y) and in the direction of which unit vector(s) ů does the maximum value for the directional derivative Dif(x,y) occur?