Final answer:
To find the equation of the tangent plane to the surface z = x²eˣˈ, calculate the partial derivatives of z with respect to x and y, then substitute these along with a point on the surface into the point-slope form of the plane's equation.
Step-by-step explanation:
The student asks for the equation of the tangent plane to a surface defined by z = x²eˣˈ. To find this, we need to calculate the partial derivatives of z with respect to x and y, which are the slopes of the tangent plane in the x and z directions, respectively.
The partial derivative of z with respect to x would be (∂z/∂x) and the partial derivative with respect to y would be (∂z/∂y). Once these derivatives are calculated, we use the point-slope form of the equation for the plane z - z0 = (∂z/∂x)(x - x0) + (∂z/∂y)(y - y0), where (x0, y0, z0) is a specific point on the surface. By substituting the partial derivatives and the point's coordinates into this equation, we obtain the equation of the tangent plane.