Consider the surface f (x comma y comma z )f(x, y,z)equals=negative 2 x squared plus 2 y squared minus 3 z squared plus 3 equals 0−2x2+2y2−3z2+3=0, which may be regarded as a level surface of the function wequals=f (x comma y comma z )f(x, y,z). the point p(negative 2 comma 2 comma 1 )(−2,2,1) is on the surface. a. find the (three-dimensional) gradient of f and evaluate it at p. b. the heads of all the vectors orthogonal to the gradient with their tails at p form a plane. find an equation of that plane.

a) (8,8,-6)

b) 4x+4y+3z = -3

Step-by-step explanation:

a)

The surface is given by the equation  

f(x,y,z) = 0 where

The gradient of this function is the vector

If we evaluate it in the point P = (-2,2,1) we obtain the point

(8,8,-6)

b)

The vectors with their tails at P are of the form  

(-2,2,1)-(x,y,z) = (-2-x, 2-y, 1-z)

as they must be orthogonal to the gradient, they must be orthogonal to the vector (8,8,6) so their inner product is 0

and the equation of the desired plane is

4x+4y+3z = -3

c-   36

taxicab circle radius 9 has 4(9) = 36 points