prac2Csol

prac2Csol - 18.02 Practice Exam 2 B Solutions Problem 1. a)...

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18.02 Practice Exam 2 B – Solutions Problem 1. a) f = h 2 xy 2 - 1 , 2 x 2 y i = h 3 , 8 i = 3 ˆ i - 8 ˆ j . b) z - 2 = 3( x - 2) + 8( y - 1) or z = 3 x + 8 y - 12. c) Δ x = 1 . 9 - 2 = - 1 / 10 and Δ y = 1 . 1 - 1 = 1 / 10. So z 2+3Δ x +8Δ y = 2 - 3 / 10+8 / 10 = 2 . 5 d) df ds f f f f ˆ u = f · ˆ u = h 3 , 8 h- 1 , 1 i 2 = - 3 + 8 2 = 5 2 Problem 2. Problem 3. a) w x = - 6 x - 4 y + 16 = 0 ⇒ - 3 x - 2 y + 8 = 0 w y = - 4 x - 2 y - 12 = 0 4 x + 2 y + 12 = 0 ¾ x = - 20 y = 34 Therefore there is just one critical point at ( - 20 , 34). Since w xx w yy - w 2 xy = ( - 6) ( - 2) - ( - 4) 2 = 12 - 16 = - 4 < 0 , the critical point is a saddle point. b) There is no critical point in the Frst quadrant, hence the maximum must be at inFnity or on the boundary of the Frst quadrant. The boundary is composed of two half-lines: x = 0 and y 0 on which w = - y 2 - 12 y . It has a maximum ( w = 0) at y = 0. y
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This note was uploaded on 04/18/2008 for the course 18 18.02 taught by Professor Auroux during the Fall '08 term at MIT.

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prac2Csol - 18.02 Practice Exam 2 B Solutions Problem 1. a)...

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