Final answer:
To find the equation of the tangent plane to a surface at a given point, determine the gradient vector and use it to write the equation of a plane.
Step-by-step explanation:
To find the equation of the tangent plane to a surface at a given point, we need to determine the gradient vector of the surface at that point. The gradient vector represents the direction of the steepest increase of the surface at the point.
Once we have the gradient vector, we can use it to write the equation of a plane in the form: Ax + By + Cz = D, where A, B, C are the components of the gradient vector and x, y, z are the variables.
Let's say the gradient vector is σ = (a, b, c) and the given point is P(x0, y0, z0). The equation of the tangent plane is: a(x - x0) + b(y - y0) + c(z - z0) = 0.