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final solution2

# final solution2 - SECOND PRACTICE FINAL Name Math 21a Fall...

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1/17/2007, SECOND 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

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Problem 1) True/False questions (20 points) 1) T F The distance from (1 , 2 , 1) to (3 , 2 , 1) is ( 2 , 4 , 2). Solution: The distance is a number, not a vector. 2) T F The plane y = 3 is perpendicular to the xz plane. Solution: The plane is parallel to the xz plane. 3) T F All functions u ( x, y ) that obey u x = u at all points obey u y = 0 at all points. Solution: The function u ( x, y ) = e x y satisfies u x = u but u y = e x . 4) T F The best linear approximation at (1 , 1 , 1) to the function f ( x, y, z ) = x 3 + y 3 + z 3 is the function L ( x, y, z ) = 3 x 2 + 3 y 2 + 3 z 2 Solution: The linear approximation is a linear function in x, y, z . The correct linear approximation would here be L ( x, y, z ) = 0. 5) T F If f ( x, y ) is any function of two variables, then integraltext 1 0 parenleftBig integraltext 1 x f ( x, y ) dy parenrightBig dx = integraltext 1 0 parenleftBig integraltext 1 y f ( x, y ) dx parenrightBig dy . Solution: The correct identity would be integraltext 1 0 parenleftBig integraltext 1 x f ( x, y ) dy parenrightBig dx = integraltext 1 0 ( integraltext y 0 f ( x, y ) dx ) dy . 6) T F Let C = { ( x, y ) | x 2 + y 2 = 1 } be the unit circle in the plane and vector F ( x, y ) a vector field satisfying | vector F | ≤ 1. Then 2 π integraltext C vector F · dr 2 π .
Solution: By definition: integraltext vector F · dr = integraltext 2 π 0 vector F ( vector r ( t )) · vector r ( t ) dt and so | integraltext vector F · vector dr | = integraltext 2 π 0 | vector F ( vector r ( t )) · vector r ( t ) | dt integraltext 2 π 0 | vector F ( vector r ( t )) || vector r ( t ) | dt integraltext 2 π 0 | vector r ( t ) | dt = 2 π . 7) T F Let vectora and vector b be two nonzero vectors. Then the vectors vectora + vector b and vectora vector b always point in different directions. Solution: Take vectora = ( 4 , 2 ) and vector b = ( 2 , 1 ) . Then vectora + vector b and vectora vector b point in the same direction. 8) T F If all the second-order partial derivatives of f ( x, y ) vanish at ( x 0 , y 0 ) then ( x 0 , y 0 ) is a critical point of f .

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