Final answer:
To find the equation of the tangent plane to the surface at a given point, we need to compute the partial derivatives of the surface equation with respect to x and y. Then, we substitute the values of the point into the partial derivatives and form the equation of the plane.
Step-by-step explanation:
To find the equation of the tangent plane to the surface at the point (x₀, y₀, z₀), we need the partial derivatives of the surface equation with respect to x and y. Let's assume our surface equation is f(x, y, z) = 8x⁴. Taking the partial derivatives, we get:
∂f/∂x = 32x³
∂f/∂y = 0
At the given point (x₀, y₀, z₀), substitute the values into the partial derivatives:
∂f/∂x(x₀, y₀, z₀) = 32x₀³
∂f/∂y(x₀, y₀, z₀) = 0
The equation of the tangent plane at the point (x₀, y₀, z₀) is:
32x₀³(x - x₀) + 0(y - y₀) + (z - z₀) = 0