Winter 2008 - Cioaba's Class - Exam 2 (Version 1)

Winter 2008 - Cioaba's Class - Exam 2 (Version 1) - Name...

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Name: PID: Discussion Section - No: Time: TA’s name: Midterm 2, Math 20C - Winter 2008 Duration: 50 minutes Please close your books, turn off your calculators and phones and put them away. To get full credit you should explain your answers. 1. Let f ( x,y ) = x ln( x 2 + y 2 ). a.(2 points) Calculate all first order partial derivatives. Solution. f x = 1 ln( x 2 + y 2 ) + x 2 x x 2 + y 2 = ln( x 2 + y 2 ) + 2 x 2 x 2 + y 2 . f y = x 2 y x 2 + y 2 = 2 xy x 2 + y 2 . b.(3 points) Calculate all second order partial derivatives. Solution. f xx = 2 x x 2 + y 2 + 4 x ( x 2 + y 2 ) - 2 x 2 (2 x + 0) ( x 2 + y 2 ) 2 = 2 x x 2 + y 2 + 4 x 3 + 4 xy 2 - 4 x 3 ( x 2 + y 2 ) 2 = 2 x ( x 2 + y 2 ) + 4 x 3 ( x 2 + y 2 ) 2 = 6 x 3 + 2 xy 2 ( x 2 + y 2 ) 2 . f yy = 2 x ( x 2 + y 2 ) - 2 xy (2 y + 0) ( x 2 + y 2 ) 2 = 2 x 3 - 2 xy 2 ( x 2 + y 2 ) 2 . f xy = f yx = 2 y ( x 2 + y 2 ) - 2 xy (2 x + 0) ( x 2 + y 2 ) 2 = 2 y 3 - 2 xy 2 ( x 2 + y 2 ) 2 .
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2.a. (2 points) Let f ( x,y ) = radicalbig 9 - x 2 - y 2 . Find the domain and the range of the function f . Solution. The domain is formed by the points ( x,y ) such that 9 - x 2 - y 2 0 which is the same as x 2 + y 2 9. This is the disk of center (0 , 0) and radius 3.
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