quiz_07-key - 2 θ 2 ±(c Is f globally invertible State...

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Exam : 16s MATH 2451 quiz 07 Course : Multivariable Calculus Duration : 25 min Due Date : 2016-03-02 (Wed) Instructor : McCary T.C. Use (First Name) (Last Name) 2016-03-02 01:28
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1. (10 points) Determine whether each function has a local inverse at the point in question, or explain why this question does not make any sense. (a) f x y = x + x 3 + y 5 at 2 3 (b) p x y z = x y z x 2 z 2 at 0 0 0
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2. (10 points) Let f t θ = e t cos θ e t sin θ . (a) Show that f is locally invertible at every point. (b) Find two di ff erent points t 1 θ 1 and t 2 θ 2 such that f t 1 θ 1 = f t 2 θ 2 . (c) Is f
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Unformatted text preview: 2 θ 2 ± . (c) Is f globally invertible? State your reasons. 3. (10 points) Let f : R → R be a smooth function, and let g : R 2 → R be given by g x y ± = y − f ( x ), let M = ² x y ± ³ ³ ³ ³ g x y ± = 0 ´ , and let c be a stationary point of f . (a) Compute µ D g x y ±¶ . (b) Is µ D g x y ±¶ onto on M ? (c) Is µ D g c y ±¶ onto at x = c ? (d) Is it possible to locally represent M as a function of y at x = c ?...
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