Question

Find an equation of the plane tangent to the following surface at the given point.z equals tangent Superscript negative 1 Baseline (xy )​;(0 comma 3 comma 0 )

Answer 1

Let f(x, y, z) = z - arctan(x y). Compute the gradient of f at the point (0, 3, 0):

∇ f(x, y, z) = (-y / (1 + x²y²), -x / (1 + x²y²), 1)

∇ f (0, 3, 0) = (-3, 0, 1)

This vector is orthogonal to the surface z = f(x, y). Then the equation of the tangent plane is

∇ f (0, 3, 0) • (x, y - 3, z) = 0

(-3, 0, 1) • (x, y - 3, z) = 0

-3x + z = 0

z = 3x