Question

Please help me check my work. I needed to find the tangent plane and normal line of the function f(x,y) = x^3*y at point (1,2,f(1,2)).

Answer 1

I wrote for the plane

z = f(1,2) + a (x - 1) + b (y - 2)

where a and b are the partial derivatives of f(x,y) at the point (1,2)

f(1,2) = 2
a = 3x^2 y at the point (1,2) = 6
b = x^3 at the point (1,2) = 1

z = 2 + 6 (x - 1) + (y - 2)

Then I chose two points on the plane:

(1,2,2) and (2,-4,2) and subtracted the second from the first getting:

(-1,6,0), so (-1,6) is parallel to the plane, and the normal line goes through (1,2,2) and is normal to it.

r: (x,y,z) = (1,2,2) + lambda (6,1,0)