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

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

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Unformatted text preview: Name: PID: Discussion Section- No: Time: TAs 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 x 2 + y 2 . a.(2 points) Calculate all first order partial derivatives. Solution. f x = 1( x 2 + y 2 )- x (2 x + 0) ( x 2 + y 2 ) 2 = y 2- x 2 ( x 2 + y 2 ) 2 . f y = x- 2 y ( x 2 + y 2 ) 2 =- 2 xy ( x 2 + y 2 ) 2 . b.(3 points) Calculate all second order partial derivatives. Solution. f xx =- 2 x ( x 2 + y 2 ) 2- ( y 2- x 2 )4 x ( x 2 + y 2 ) ( x 2 + y 2 ) 4 = ( x 2 + y 2 )(- 2 x ( x 2 + y 2 )- 4 x ( y 2- x 2 )) ( x 2 + y 2 ) 4 = 2 x (- x 2- y 2- 2 y 2 + 2 x 2 ) ( x 2 + y 2 ) 3 = 2 x ( x 2- 3 y 2 ) ( x 2 + y 2 ) 3 . f yy = 2 x ( x 2 + y 2 )- 2 xy (0 + 2 y ) ( 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 x 2 y ( x 2 + y 2 ) 2 . 2.a. (2 points) Let f ( x, y ) =- radicalbig 25- x 2- y 2 . Find the domain and the range of the function...
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Winter 2008 - Cioaba's Class - Exam 2 (Version 2) - Name:...

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