5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x,...
5 and 6 please 5) Given the surface f(x, y, z) = 0 or z = f(x,y), find the tangent plane at P. a) z2 – 2x2 – 2y2 = 12 @ P=(1,-1,4) b) f(x,y) = 2x - 3xy3 @ 12,-1) c) f(x,y) = sin(x) @ (3,5) 6) Find an equation of the tangent plane and the equation of the normal line to surface f(x..zb=0 @P x2 + y2 + z2 = 9 P = (2,2,1)
Consider the surface given as a graph of the function g(x, y) = x∗y 2 ∗cos(y). The gradient of g represents the direction in which g increases the fastest. Notice that this is the direction in the xy plane corresponding to the steepest slope up the surface, with magnitude equal to the slope in that direction. 1. At the point (2, π), find the gradient, and explain what it means. 2. Use it to construct a vector in the tangent...
1. Find the first and second partial derivatives: A. z=f(x,y) = x2y3 - 4x2 + x2y-20 B. z=f(x,y) = x+ y - 4x2 + x2y-20 2. Find w w w x2 - 4x-z-5xw + 6xyz2 + wx - wz+4 = 0 Given the surface F(x,y) = 3x2 - y2 + z2 = 0 3. Find an equation of the plane tangent to the surface at the point (-1,2,1) a. Find the gradient VF(x,y) b. Find an equation of the plane...
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
Let F(x,y,z) = 4i – 3j + 5k and S be the surface defined by z= x2 + y2 and 22 + y2 < 4. Evaluate SJ, F. nds, where n is the upward unit normal vector.
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...
12. (5 points) (a): Find the directional derivative of f(x, y) = y² In r at P(1,4) in the direction of u = -3i + 3j. (b): Find the equation for the tangent plane and normal line to the surface cos(70) – z’y+e*2 + y2 = 4 at P(0,1,2).
7. (20 pts) Consider the surface given by z wy - 4xy + 3x - 2y. (a) Find the equation of the tangent plane to this surface at the point where (x,y) - (1,3). (b) Find the gradient f at the point where (x,y) = (1,3). (c) Find the directional derivative DaS(1,3) where it is the unit vector in the direction of (1, -2). such that the directional derivative Daf(1.3) is a maximum (d) Find a unit vector in the...
6 (20 pts). Let F(x, y, z) = x2 + y2 + x2 - 6xyz. (1) Find the gradient vector of F(x, y, z); (2) Find the tangent plane of the level surface F(x, y, z) = x2 + y2 + x2 - 6xyz = 4 at (0, 0, 2); (3) The level surface F(x, y, z) = 4 defines a function z = f(x,y). Use linear approxi- mation to approximate z = = f(-0.002,0.003).
b) Consider the surface in R3 described by f(x,y,z) = 2x²y3 + z + ye*2 = 9 (i) Find Vf(x,y,z). [3 marks] (ii) Verify that (2,1,0) is a point on this surface. Find the cartesian equation of the tangent plane to this surface at the point (2.1,0). [5 marks]