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final_f07_post (2)

final_f07_post (2) - Math 237 Final F07 NOTE This exam does...

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Math 237 Final F07 NOTE: This exam does not necessarily reflect which type of questions will be asked on this semester’s final. i.e. something covered on this test may not be covered on this semester’s final and material not covered on this test may be covered on this year’s final. 1. Short Answer Problems [2] a) State the second derivative test. [3] b) Does the mapping F ( x, y ) = ( y - x 2 , y + x 2 ) have an inverse transformation in a neighbourhood of (1 , 2)? Justify your answer [2] c) Find the derivative matrix of the mapping F ( x, y ) = ( x 2 sin y, xy 2 ). 2. Let f ( x, y ) = x y - y x . [3] a) Find the equation of the tangent plane to the surface z = f ( x, y ) at (1 , 1 , 0). [4] b) Use the 2nd degree Taylor Polynomial to approximate f (1 . 1 , 0 . 9). 3. Consider the function f ( x, y, z ) = ln( x + e yz ). [2] a) Write the definition of the directional derivative. [3] b) In what direction(s) does f have a rate of change of - 1 at the point (0 , 1 , 0)? [2] c) Is there a direction in which f has a rate of change of 2 at the point (0 , 1 , 0)? Justify your answer.
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