Final answer:
To find the equation of the tangent plane to a parametric surface at a specific point, calculate and evaluate the partial derivatives of the parametric equations, get the normal vector via cross product, and apply the point-normal form of a plane.
Step-by-step explanation:
The question asks to find an equation of the tangent plane to a parametric surface at a given point. The given parametric equations for the surface are x=3u+5v, y=2u^2, and z=5u-3v. To find the equation of the tangent plane, one must first determine the partial derivatives of the parametric equations with respect to u and v, then evaluate these derivatives at the point corresponding to the parameters that yield the given point on the surface.
The normal vector to the surface at the given point is obtained by taking the cross product of these partial derivatives. Finally, the equation of the tangent plane is found by using the point-normal form of a plane.Calculate the partial derivatives of the parametric equations with respect to u and v.Substitute the values of u and v at the given point into the derivative equations to find the slope vectors.Use the slope vectors along with the given point to construct the equation of the tangent plane.