Math 237 - Final Exam Winter 2008 NOTE: The questions on this exam does not exactly reﬂect which questions will be on this terms exam. That is, some questions asked on this exam may not be asked on our exam and there may be some questions on our exam not asked here. 1. Short Answer Problems  a) Let f : R 2 → R . What is the greatest rate of change of f from a point ( a, b ) and in which direction does it lie?  b) Let G : R 2 → R 2 and F : R 2 → R 2 . State the Chain Rule in matrix form.  c) Does the function f ( x, y ) = y cos( e xy ) + ln( | y | +2 x ) x 2 +1 have an minimum on the unit disc x 2 + y 2 ≤ 1? 2. Let f ( x, y ) = p 1-x 2-y 2  a) Sketch the domain of f .  b) Draw the level curves C = f ( x, y ) for C = 0 and C = 1.  c) Determine if f is continuous at the point (0 , 1).  d) Determine where f is diﬀerentiable on its domain. 3. Let f ( x, y ) = e x 2 + y 2 .  a) Find the linear approximation
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