Unformatted text preview: ) has a vertical tangent line at ( x ,y ). (b) Give an example of a function with a vertical tangent at some regular point such that L ( f,c ) near this point can be expressed as the graph of y as a function of x . (c) Show that in this situation, y cannot be di±erentiable (as a function of x ) at this regular point. Challenge problem: 5. The following example (based on [ 32 , pp. 589] or [ 49 , p. 201]) shows that the hypothesis that f be continuously di±erentiable cannot be ignored in Theorem 3.4.2 . De²ne f ( x,y ) by f ( x,y ) = b xy + y 2 sin 1 y if y n = 0 , if y = 0 ....
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 Fall '08
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 Calculus, Level set, Hyperboloid, vertical tangent line, regular point

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