Question

find an equation for the plane tangent to the
surface defined by the equation 3xy + z2 − 4 = 0 at the point (1, 1, 1).

Answer 1

Do you know how to find gradient vector?

Answer 2

yeah, gradient provides an equation to determine the "normal" to the plane; then attribute the point to it

Answer 3

df/dx = a

df/dy = b

df/dz = c

a(x-1)+b(y-1)+c(z-1)=0 then right?

Answer 4

for example:

df/dx = 3y ; at (1,1,1). y = 1

df/dx = a = 3(1) = 3

Answer 5

so the equation would be 3(x-1)+3(y-1)+2(Z-1) right?

Answer 6

df/dx = 3y; y=1; a=3

df/dy = 3x; x=1; b=3

df/dz = 2z; z=1; c = 2

yes, that is correct. sometimes they want you to write it all out so there is no (...)s tho