# pm1 - Let f R 2 R be dened by f u,v = uv and let g R 2 R 2...

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Practice Midterm Examination Instructor J. Verstraete Time: 40 minutes No notes allowed All questions carry equal weight Question 1. (a) Show that xy 1 2 ( x 2 + y 2 ). (b) Use part (a) and the ² - δ deﬁnition of limits to show lim ( x,y ) (0 , 0) xy p x 2 + y 2 = 0 . 1

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Question 2. Deﬁne what it means for a function f ( x ) : R n R to be diﬀerentiable at a point a . Then prove that the function f ( x,y ) = | xy | 1 / 2 is not diﬀerentiable at ( x,y ) = (0 , 0). 2
Question 3.

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Unformatted text preview: Let f : R 2 R be dened by f ( u,v ) = uv and let g : R 2 R 2 be dene by g ( x,y ) = ( y,x ). If ( x,y ) = f ( g ( x,y )), use the chain rule to nd x . 3 Question 4. Compute the second order Taylor formula for f ( x,y ) = log(1 + x + y ) about the point ( x,y ) = (0 , 0). Taylors Theorem Not Examinable 4...
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## This note was uploaded on 12/25/2008 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.

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pm1 - Let f R 2 R be dened by f u,v = uv and let g R 2 R 2...

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