Tangent Planes

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Before we began discussing tangent planes, we reviewed the five properties of the gradient:


We utilized the fifth property to find the plane tangent to a curve at a certain point. In order to use it, however, the gradient had to be normal to a "level curve"--not the surface itself. To do this, we had to "dimension-up" our current equation to an equation of three variables by moving all variables and constants including z (which may be written as f(x, y)) to one side of the equation. The side containing 0 is rewritten as F(x, y, z).

We proceeded to solving for the gradient of F(x, y, z) at various points and chose to find the tangent plane at one of those points by using the formula for a plane:


So in general, to find the tangent plane to an equation of two variables at a certain point that lies on the graph of that equation:
1. Rewrite it as a function of three variables
2. Find the gradient of that function
3. Plug in the values of x, y, and z into the gradient to find the normal vector
4. Use the vector and the point and plug it into the formula for a plane to get
your tangent plane.

Happy 4-D differentiating!