Let f : R n R be a differentiable map and let F : R n+1 R be given by (x, z) 7 f(x) z, where x R n and z R. Let M R n+1 be the graph of f and let N = F 1 ({0}). (a) Show M = N. (b) Let a R n . Describe the tangent space on M at (a, f(a)) R n+1 using f. (c) Let a R n . Describe the tangent space on N at (a, f(a)) R n+1 using F. (d) Show that the descriptions in (b) and (c) agree.