math318.final Spring 2013 - Math 318 Krantz Spring 2013 May...

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Math 318 Spring, 2013 Krantz May 7, 2013 Final Exam General Instructions: Read the statement of each problem carefully. Do only what is requested—nothing more and nothing less. Provide a complete solution to each problem. If you only write the answer then you will not get full credit. If you need extra room for your work then use the backs of the pages. Be sure to ask questions if anything is unclear. (10 points) 1. If f : R 2 R is differentiable at a point a , then the entries of the matrix df ( a ) are the partial derivatives of f . Explain why this is true. 1
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(10 points) 2. If f and g are real-valued functions on an open set U R n and if D v f ( a ) = D v g ( a ) for every point a U and every unit vector v , then show that f and g differ by a constant. (10 points) 3. If f ( x, y ) = parenleftBigg x 2 - y 2 x 2 + y 2 parenrightBigg and g ( x, y ) = x 3 - 4 y 2 , then use the Chain Rule to calculate the derivative at the origin of g f . 2
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(10 points) 4. If T : R n R m is a linear mapping then define the quantity bardbl T bardbl .
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