Question

Find an equation of the tangent plane to the given parametric surface at the specified point. The parametric surface is defined by r(u, v) = u² i + 3u sin(v) j + u cos(v) k, and the point is u = 1, v = 0.

Answer 1

To find the equation of the tangent plane to the given parametric surface at the specified point, follow the steps of finding partial derivatives, evaluating them at the given values, and using the point-point form of a plane equation.

Step-by-step explanation:

To find the equation of the tangent plane to the given parametric surface at the specified point, we need to follow these steps:

1. First, find the partial derivatives of the position vector r(u, v) with respect to both u and v.
2. Next, evaluate the partial derivatives at the given values of u = 1 and v = 0 to get the normal vector to the tangent plane.
3. Finally, use the point-point form of a plane equation with the known point and the normal vector to write the equation of the tangent plane.

By following these steps, we can find the equation of the tangent plane to the given parametric surface at the specified point.