final solution4 - FORTH PRACTICE FINAL Name Math 21a Fall...

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1/17/2007, FORTH 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 projection vector proj vectorv ( vectorw ) is parallel to vectorw . Solution: It is parallel to vectorv . 2) T F Any parametrized surface S is a graph of a function f ( x, y ). Solution: A counter example is the sphere. 3) T F If the directional derivatives D vectorv ( f )(1 , 1) and D vectorw ( f )(1 , 1) are both 0 for vectorv = ( 1 , 1 ) / 2 and vectorw = ( 1 , 1 ) / 2, then (1 , 1) is a critical point. Solution: Indeed f (1 , 1) must be perpendicular to vectorv and vectorw and so be the zero vector. 4) T F The linearization L ( x, y ) of f ( x, y ) = x + y + 4 at (0 , 0) satisfies L ( x, y ) = f ( x, y ). Solution: The linearization of any linear function at (0 , 0) is the function itself. 5) T F For any function f ( x, y ) of two variables, the line integral of the vector field vector F = f on a level curve { f = c } is always zero. Solution: The gradient is perpendicular to the velocity vector. 6) T F If vector F is a vector field of unit vectors defined in 1 / 2 x 2 + y 2 2 and vector F is tangent to the unit circle C , then integraltext C vector F · dvector r is either equal to 2 π or 2 π . Solution: vector r is parallel to vector F so that vector F · vector r is equal to 1 or 1.
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7) T F If a curve C intersects a surface S at a right angle, then at the point of intersection, the tangent vector to the curve is parallel to the normal vector of the surface. Solution: This is clear once you know what the question means. 8) T F The curvature of the curve vector r ( t ) = ( cos(3 t ) , sin(6 t ) ) at the point vector r (0) is smaller than the curvature of the curve vector r ( t ) = ( cos(30 t ) , sin(60 t ) ) at the point vector r (0). Solution: The curvature is independent of the parametrization of the curve. 9) T F At every point ( x, y, z ) on the hyperboloid x 2 + y 2 z 2 = 1, the vector ( x, y, z ) is tangent to the hyperboloid. Solution: It is normal to the hyperboloid. 10) T F The set { φ = π/ 2 , θ = π } in spherical coordinates is the negative x axis.
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