Fall 2007 - Cioaba's Class - Exam 2 (Version 1)

Fall 2007 - Cioaba's Class - Exam 2 (Version 1) - Name:...

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Unformatted text preview: Name: PID: Discussion Section- No: Time: TAs name: Midterm 2, Math 20C - Lecture D (Fall 2007) Duration: 50 minutes Please close your books, turn off your phones and put them away. You can use one page of handwritten notes. To get full credit you should explain your answers. Dont forget to write your name, ID number and your TAs name. 1.a.(3 points) Find all second order partial derivatives of the function f ( x, y ) = radicalbig x + y 2 . Solution. We write f ( x, y ) = ( x + y 2 ) 1 2 . The first order partial derivatives are f x = ( x + y 2 ) 1 2 2 f y = y ( x + y 2 ) 1 2 The second order partial derivatives are f xx = 1 2 1 2 ( x + y 2 ) 3 2 = ( x + y 2 ) 3 2 4 f xy = 1 2 1 2 ( x + y 2 ) 3 2 2 y = y ( x + y 2 ) 3 2 2 f yy = ( x + y 2 ) 1 2 + y 1 2 ( x + y 2 ) 3 2 2 y = ( x + y 2 ) 1 2 y 2 ( x + y 2 ) 3 2 b.(2 points) Find the linear approximation of the function g ( x, y ) = ln(2 x + y ) at (0 , 1) and use it to approximate f (0 . 1 , 1 . 1). Solution. The linear approximation at (0 , 1) is f ( x, y ) f (0 , 1) + f x (0 , 1)( x 0) + f y (0 , 1)( y 1) We have f x ( x, y ) = 2 2 x + y and f y ( x, y ) = 1 2 x + y which implies f x (0 , 1) = 2 and f y (0 , 1) = 1. Thus, the linear approximation near (0 , 1) is f ( x, y ) 0 + 2 x + y 1 which implies that f (0 . 1 , 1 . 1) . 2 + 1 . 1 1 = 0 . 3 2. Suppose you are climbing a hill whose shape is given by the equation z = 1000 . 01 x 2 . 02 y 2 where x, y and z are measured in meters, and you are standing at a point with coordinates...
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This note was uploaded on 04/29/2008 for the course MATH 20C taught by Professor Helton during the Fall '08 term at UCSD.

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Fall 2007 - Cioaba's Class - Exam 2 (Version 1) - Name:...

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