Find an equation of the tangent plane to the given surface at the specified point. z = 4(x - 1)2 + 3(y + 3)2 + 6, (2, -1, 22)?

1.
f(x,y) = 4(x-1)^2+3(y+3)^2+6

f(2,-1) = 22

fx(x,y) = 8(x-1)
fx(2,-1) = 8(2-1) = 8

fy(x,y) = 6(y+3)
fy(2,-1) = 12

z = f(2,-1) + (x-2) fx(2,-1) + (y+1) fy(2,-1)
z = 22 + (x-2) (8) + (y+1) (12)
z = 22 + 8x-16 +12y+12

z = 8x+12y+18

2.
f(x,y) = y cos(x-y)

f(-2,-2) = -2 cos(-2+2) = -2 cos(0) = -2

fx (x,y) = -y sin(x-y)
fx (-2,-2) = - (-2) sin (-2-(-2))
fx (-2,-2) = 2 sin(0) = 0

fy (x,y) = cos(x-y) + y (-sin(x-y) ) (-1)
fy (x,y) = cos(x-y) + y sin (x-y)
fy(-2,-2) = cos(0) - 2 sin(0) = 1

z = f(-2,-2) + (x+2) fx(-2,-2) + (y+2) fy(-2,-2)
z = -2 + (x+2) (0) + (y+2) (1)
z = -2 + y + 2
z = y