Math241_Fall08_9AM_Exam3 - Math 241 – Exam 3 – 9AM V1...

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Unformatted text preview: Math 241 – Exam 3 – 9AM V1 Name: November 16, 2008 Section Registered In: 55 points possible 1. Let f : R n → R . (a) (3pts) State the formula for the second-order Taylor polynomial of f at the point ~a in R n . (b) (2pts) In order for this expansion to be valid, we require that f be C k . Briefly explain what the value of k must be. (c) (3pts) How did we use the second-order Taylor polynomial to derive the second-derivative test for local extrema? Briefly explain. 2. (7pts) Let f ( x,y,z ) = x 3 + x 2 y- yz 2 + 2 z 3 . Find the second-order Taylor polynomial of f at the point (1 , , 1). (You do not need to simplify your answer.) 3. Let f ( x,y,z ) = x 3 + xy 2 + x 2 + y 2 + 3 z 2 . (a) (3pts) Show that the point- 2 3 , , ¶ is a critical point of f . (b) (2pts) If H f (- 2 3 , , ) = - 2 0 0 2 3 0 6 , classify the critical point. 4. (8pts) Find the maximum and minimum values of f ( x,y ) = y 2- x + 200 subject to the constraint x 2 + y 2 = 1.= 1....
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This note was uploaded on 02/15/2011 for the course MATH 241 taught by Professor Kim during the Fall '08 term at University of Illinois, Urbana Champaign.

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Math241_Fall08_9AM_Exam3 - Math 241 – Exam 3 – 9AM V1...

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