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...
DUE DATE: 23 MARCH 2020 1 1. Let f(x,y) = (x, y) + (0,0) 0. (x, y) = (0,0) evaluate lim(x,y)=(4,3) [5] 2r + 8y 2. Show that lim does not exist. [10] (*.w)-(2,-1) 2.ry + 2 3. Find the first and second partial derivatives of f(x,y) = tan-'(x + 2y). [16] 4. If z is implicitly defined as a function of x and y by I?+y2 + 2 = 1, show az Əz that +y=z [14] ar ду 5....
1 1 Consider the function f(x.y,z) 2x y 2 the point P(3,0,1), and the unit vector u 0 Compute the gradient of f and evaluate it at P b. Find the unit vector in the direction of maximum increase of f at P c. Find the rate of change of the function in the direction of maximum increase at P d. Find the directional derivative at P in the direction of the given vector. a. 1 1 Consider the function...
Please do the parts in the given order tyā (x,y)メ(0,0) (x,y)= (0,0). if if 1 (d) Given the unit vector u-( find the directional derivative of f(x, y) at the 리지, ,- point (to,m) = (0,0), in the direction of the vector a. (e) Find the gradient of f(x, y) at the point (zo,o) (0,0) (c) Find the equation of the tangent plane to the graph of the function z -f(x, y) at the point (x,y,z) (1,0,0). tyā (x,y)メ(0,0) (x,y)=...
(;) (1 point) Let f = ye-2x, a = and y = Calculate the directional derivative of f at a in the direction of and in the direction of v/llv
cense. This license can only be 5. Given f (x,y) = 12 + x2 + 2y2 the point P(2,1)and vector u =< a. Find the directional derivative at the point in the direction of u b. Find the Gradiant
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).
Exercise 2. Directional derivative (6 pts + 9 pts) Let f(x,y,z) = xy + y2 – 23 – 105. 1) vf = a. yi + (x + 2y); – 3z2K b. (y + y2)i + (x + 2y); - 3z2k c. xi + (y + 2y); - 3z2K d. None of the above a. b. C. d. 2) The direction in which ſ decreases most rapidly at A(0,1,1) is: a. 14 14 c. wait tasta d tai-haiti e. None of...
4. Let f(x, y) = 2 - 2x – y + xy. (a) Find the directional derivative of f at the point (2,1) in the direction (-1,1). [2] (b) Find all the critical points of the function f and classify them as local extrema, saddle points, etc. [2]
true or false is zero. F 9. The plane tangent to the surface za the point (0,0, 3) is given by the equation 2x - 12y -z+3-0. 10. If f is a differentiable function and zf(x -y), then z +. T 11. If a unit vector u makes the angle of π/4 with the gradient ▽f(P), the directional derivative Duf(P) is equal to |Vf(P)I/2. F 12. There is a point on the hyperboloid 2 -y is parallel to the plane...