final solution1

# Final solution1 - FIRST PRACTICE FINAL Math 21a Fall 2007 Name MWF 9 Chen-Yu Chi MWF 10 Oliver Knill MWF 10 Corina Tarnita MWF 11 Veronique Godin

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Unformatted text preview: 1/17/2007, FIRST PRACTICE FINAL Math 21a, Fall 2007 Name: MWF 9 Chen-Yu Chi MWF 10 Oliver Knill MWF 10 Corina Tarnita MWF 11 Veronique Godin MWF 11 Stefan Hornet MWF 11 Jay Pottharst MWF 12 Chen-Yu Chi MWF 12 Ming-Tao Chuan TTH 10 Thomas Barnet-Lamb TTH 10 Rehana Patel TTH 11:30 Thomas Barnet-Lamb TTH 11:30 Thomas Lam • Please mark the box to the left which lists your section. • Do not detach pages from this exam packet or unstaple the packet. • Show your work. Answers without reason- ing can not be given credit except for the True/False and multiple choice problems. • Please write neatly. • Do not use notes, books, calculators, comput- ers, or other electronic aids. • Unspecified functions are assumed to be smooth and defined everywhere unless stated otherwise. • You have 180 minutes time to complete your work. 1 20 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 10 13 10 14 10 Total: 150 Problem 1) True/False questions (20 points) Mark for each of the 20 questions the correct letter. No justifications are needed. 1) T F The length of the curve vector r ( t ) = ( sin( t ) , t 4 + t, cos( t ) ) on t ∈ [0 , 1] is the same as the length of the curve vector r ( t ) = ( sin( t 2 ) , t 8 + t 2 , cos( t 2 ) ) on [0 , 1]. Solution: This is a consequence of the chain rule: integraltext b a | r ( s ( t )) ′ | dt = integraltext b a | r ′ ( s ( t )) || s ′ ( t ) | dt = integraltext s ( b ) s ( a ) | r ′ ( s ) | ds . 2) T F The parametric surface vector r ( u, v ) = (5 u − 3 v, u − v − 1 , 5 u − v − 7) is a plane. Solution: Yes, the coordinate functions r ( u, v ) = ( x ( u, v ) , y ( u, v ) , z ( u, v )) are all linear. 3) T F Any function u ( x, y ) that obeys the differential equation u xx + u x − u y = 1 has no local maxima. Solution: If ∇ u = ( u x , u y ) = (0 , 0), then u xx = 1 which is incompatible with a local maximum, where u xx > 0 by the second derivative test. 4) T F The length of the vector vector b onto a vector vectora is smaller or equal than the length of the vector vector b . Solution: The length is vector b times the absolute value of the cos of the angle between the vectors. 5) T F If f ( x, y ) is a function such that f x − f y = 0 then f is called conservative. Solution: The notion of conservative applies to vector fields and not to functions. 6) T F ( vectoru × vectorv ) · vectorw = ( vectoru × vectorw ) · vectorv for all vectors vectoru,vectorv, vectorw . Solution: While | ( vectoru × vectorw ) · vectorv | and | ( vectoru × vectorw ) · vectorv | are both equal to the volume of the parallelepiped determined by vectoru,vectorv and vectorw , the sign is different. An example: for vectoru = ( 1 , , ) ,vectorv = ( , 1 , ) , vectorw = ( , , 1 ) , we have ( vectoru × vectorv ) · vectorw = 1 and ( vectoru × vectorw ) · vectorv = − 1....
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## This note was uploaded on 04/06/2008 for the course MATH 21a taught by Professor Knill during the Fall '07 term at Harvard.

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Final solution1 - FIRST PRACTICE FINAL Math 21a Fall 2007 Name MWF 9 Chen-Yu Chi MWF 10 Oliver Knill MWF 10 Corina Tarnita MWF 11 Veronique Godin

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