Question

If f (x, y) = x y, find the gradient vector ∇f (3, 2) and use it to find the tangent line to the level curve f (x, y) = 6 at the point (3, 2). Sketch the level curve, the tangent line, and the gradient vector

Answer 1

Even though this function depends only on two variables, instead of three, we can use the same methods to compute the gradient and tangent line. The gradient vector is

∇f (x, y) = yi + xj

∇f (3, 2) = 2i + 3j

The tangent line is the line through (3, 2) that is orthogonal to ∇f (3, 2). It has the equation

fx (3, 2)(x − 3) + fy (3, 2)( y − 2) = 0

2x + 3y = 12