Exercises 40–49 concern the implicit function theorems and the inverse function theorem (Theorems 6.5, 6.6, and 6.7).
Let S be the set of points described by the equation sin xy + exz + x3 y = 1.
(a) Near which points can we describe S as the graph of a C1 function z = f (x, y)? What is f (x, y) in this case?
(b) Describe the set of “bad” points of S, that is, the points (x0, y0, z0) ∈ S where we cannot describe S as the graph of a function z = f (x, y). T
(c) Use a computer to help give a complete picture of S.
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