[llamaz]

Which is what differential geometry is based on - you take a surface (for example - the one dimensional functions you mentioned work too, but those are too trivial). Then you associate to each point a plane (re-centered at to the origin). The collection of all vectors that fit into the plane is a vector space called a tangent space, and the collection of all tangent spaces is a tangent bundle. And now you've set up differential geometry and can study it.