True or False: If and are defined for a function and a point , then defines the tangent line to the curve at .
We want to see what a typical tangent plane to the graph of looks like. Fix the point where . This is a point on the graph .
Every plane in that lies on has equation for numbers . For which does this equation not give a plane?
Under what conditions on is the plane parallel to the -plane?
Under what conditions on does the plane contain the line parallel to the -axis through ?
It is not hard to see that if the plane contains any parallel to the -axis, then it must contain the parallel through . Given that this cannot happen for the graph of a differentiable function in the form , we can express the tangent plane at in the form . By considering the section of the graph in the -plane when , find .