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

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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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