Question 1 36 points, 3 points each PART A True or False?...

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Northwestern University NetID: Math 230 Final Exam Solutions Winter Quarter 2014 March 17, 2014 Instructions: Read each problem carefully. Write legibly. Show all your work on these sheets. Make sure that your final answer is clearly indicated. If two answers are presented, the average of the points for each answer will be given. This exam has 16 pages and 9 problems. Before starting the exam, please check that your copy contains all of them and obtain a new copy of the exam immediately if it does not. You may not use books, notes, or calculators. Good luck! (2 points) Mark your section and write your NetID in the upper right corner of this page. Do NOT write your name on this exam. Sec. # Time Instructor 21 8:00 Zhu 31 9:00 Zhu 41 10:00 Broderick 51 11:00 Xia 61 12:00 Chau 63 12:00 Yang 71 1:00 Yang 81 2:00 Kahouadji Prob. Points Score possible 0 2 1 36 2 15 3 15 4 12 5 27 6 15 7 18 8 35 9 25 TOTAL 200
Math 230 Final Exam Solutions Winter Quarter 2014 Page 2 of 16 Question 1 (36 points, 3 points each) . PART A True or False? Circle the correct answer. (a) The line x = 3 + 2 t, y = 4 - t, z = 1 + 3 t intersects the y -axis.
(b) The function f ( x, y ) = p x 2 + y 2 is continuous on the entire xy -plane.
(c) Let g ( x, y, z ) be a continuous function of three variables. The level surface g = 1 must not intersect the level surface g = 2 .
(d) If z = h ( x + y ) for some differentiable function h ( u ) , then ∂z ∂x and ∂z ∂y must be equal for all values of x and y .
Math 230 Final Exam Solutions Winter Quarter 2014 Page 3 of 16 (e) Every ellipse has constant curvature.
(f) The point ( x, y, z ) = ( 3 , 3 , 6) lies on the surface described in spherical coordinates by φ = π/ 3 .
(g) Let i , j , and k be the unit vectors along the three-dimensional rectangu- lar coordinate axes. The vectors i and k satisfy ( i × i ) × k = i × ( i × k ) .
(h) A curve represented by a vector-valued function r ( t ) lies entirely on a surface. If r (0) = h 1 , 2 , 3 i , then the tangent vector r 0 (0) can be taken as a normal vector for the tangent plane to the surface at (1 , 2 , 3) .
Math 230 Final Exam Solutions Winter Quarter 2014 Page 4 of 16 PART B Suppose that f : R 2 R has continuous second partial derivatives. A table of values at four points is given. ( x, y ) (5 , 8) ( - 3 , 1) (3 , - 9) ( - 2 , 4) f h 0 , - 1 i h 0 , 0 i h 0 , 0 i h 0 , 0 i f xx - 2 - 4 2 3 f xy - 3 5 - 4 5 f yy 5 - 7 8 4 What does f have at each of these four points? Mark your answers.
Math 230 Final Exam Solutions Winter Quarter 2014 Page 5 of 16

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