Final answer:
The equation of the tangent plane to the surface at the given point z=ex-3y (3,1,1) is z = 20.0855x - 3y - 39.0855.
Step-by-step explanation:
The equation of the tangent plane to the surface at the given point z=ex-3y (3,1,1) is:
First, we need to find the partial derivatives of z with respect to x and y.
∂z/∂x = e^x, ∂z/∂y = -3
Using the point (3,1,1), we can find the values of the partial derivatives: ∂z/∂x = e^3 = 20.0855, ∂z/∂y = -3
The equation of the tangent plane is given by: z - z0 = (∂z/∂x)(x-x0) + (∂z/∂y)(y-y0), where (x0,y0,z0) is the given point.
Substituting the values: z - 1 = 20.0855(x-3) - 3(y-1)
The equation of the tangent plane to the surface at the given point is: z = 20.0855x - 3y - 39.0855