Before we compute the tangent plane, let’s take a step back and draw a few parallels for tangent lines and planes and how we find them.
  • When we talk about lines in the -plane, we need a point and a slope. For tangent lines, we evaluate the derivative at the point of tangency to find the slope.
  • When we talk about lines in the -plane, “slope” is not well-defined; we need a vector parallel to the line and a point on the line. For tangent lines to a curve with parameterization , we evaluate the derivative at the appropriate -value to find the vector parallel to the line.
  • When we talk about planes in the -plane, “slope” also does not make sense; we need a normal vector and a point. When we want to find the tangent plane to a surface , we use the gradient to construct a normal vector.

In fact, the tangent plane at has normal vector

Now, given that , give the equation of the tangent plane to the surface where .

  • The point on the plane can be found by evaluating at . Doing so, we find .
  • To construct a normal vector, note that
    • so .
    • so .

    Thus, a normal vector to the tangent plane is .

We now compute the tangent plane. Recall that the equation of the plane with normal vector that passes through is

Using