Math241_Spring08_Exam2 - Math 241 Exam 2 Version 1 1....

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Math 241 – Exam 2 – Version 1 1. (5pts) Find the domain and range of the function f ( x,y ) = r x y . 2. (a) (4pts) Define what it means for a function f : R n R to be continuous at ~x = ~a . (b) (6pts) Show that f ( x,y ) = 3 x 2 y - y 3 x 2 + y 2 if ( x,y ) 6 = (0 , 0) 0 if ( x,y ) = (0 , 0) is continuous at the origin. 3. (5pts) Determine whether or not f ( x,y ) = xy 2 is differentiable. Briefly justify your answer. 4. (5pts) Find the direction in which the value of z decreases the fastest when z = x 2 y + y at the point (2 , 1). What is the rate of change of z in this direction. 5. Let f ( x,y ) = y 2 + x 2 y + x 2 - 2 y . (a) (4pts) Find all the critical points of f ( x,y ). (b) (6pts) Use the Second Derivative Test to classify the critical points. 6. (5pts) Explain why there is no function with continuous second-order partial derivatives such that f x ( x,y ) = 6 xy 2 and f y ( x,y ) = 8 xy . 7.
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This note was uploaded on 02/15/2011 for the course MATH 241 taught by Professor Kim during the Spring '08 term at University of Illinois, Urbana Champaign.

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Math241_Spring08_Exam2 - Math 241 Exam 2 Version 1 1....

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