review_01 - Exam Course Due Date Instructor 16s MATH 2451...

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Exam : 16s MATH 2451 review 01 Course : Multivariable Calculus Due Date : 2099-02-11 (Wed) Instructor : McCary T.C. Use (First Name) (Last Name) 2016-02-15 02:58
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1. (10 points) You’ll have to prove Fermat’s theorem in the R R case in the oral exam. You’ll have to prove Fermat’s theorem in the R n R case on the written exam (which is a straightforward consequence of the R R case).
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2. (10 points) The proof of Clairaut’s theorem will be a bonus question on the exam.
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3. (10 points) Suppose G x y = 0 and that y is an implicit function of x : y = f ( x ). Show dy dx = - ∂G/∂x ∂G/∂y where ∂G ∂y 6 = 0. Hint : consider the function p ( x ) = x f ( x ) , the composition ( G p )( x ) = 0, and the chain rule. This is the reason implicit differentiation from scalar calculus works.
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4. (10 points) In this exercise, we’ll arrange a situation where ∂y ∂x ∂z ∂y ∂x ∂z = - 1. (a) Assume a surface S given as the zero level set of a function F : F x y z = 0.
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