Final answer:
To find the equation for the tangent plane to the surface z² = xy²cos(z), we need to find the partial derivatives, and use the point-normal form to form the equation of the tangent plane.
Step-by-step explanation:
To find the equation for the tangent plane to the surface z² = xy²cos(z), we need to find the partial derivatives of the equation with respect to x and y. These derivatives will give us the slopes of the surface in the x and y directions. Then, we can use the point (x₀, y₀, z₀) on the surface to form the equation of the tangent plane using the point-normal form.
Let's start by finding the partial derivatives:
∂z/∂x = 2xy²cos(z) - y² ∂z/∂y = 2xy²cos(z) - 2xyzsin(z)
Now, let (x₀, y₀, z₀) be a point on the surface. The equation of the tangent plane is:
2xy₀²cos(z₀)(x - x₀) - y₀²(y - y₀) + z₀ = 0