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MultiFinalExam01

# MultiFinalExam01 - Math 2224 Practice Questions...

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Math 2224 Practice Questions Instructions: The Final Exam in this course is scheduled for Tuesday, December 11, from 1:05–3:05 pm in McB 129 (not our usual room!). There will be two review session: one in class on Tuesday, and one on Sunday, December 9, from 7:30–10:00 pm in McB 129 (the same room as the ﬁnal). Here are some practice questions for the ﬁnal. An answer key is included as the last page (don’t peek!) As a whole these questions emphasize computation over theory (though there are a few exceptions!) I believe with only a few days left, the beneﬁt to be gained in terms of speed and accuracy from practicing computation will be worthwhile. You should expect the actual departmental ﬁnal to be more balanced. 1. Find an equation for the plane consist- ing of all points that are equidistant from the two points ( 1, 1, 0 ) and ( 0, 1, 1 ) . (a) x - z = 0 (b) x - z = 1 (c) x - y = 0 (d) x - y = 1 (e) x + z = 1 2. Find the point at which the two lines ~ r = h 1, 1, 0 i + t h 1, - 1, 2 i and ~ r = h 2, 0, 2 i + s h- 1, 1, 0 i intersect. (a) ( 1, 0, 1 ) (b) ( 2, 0, 2 ) (c) ( 1, 1, 1 ) (d) ( 2, 2, 2 ) (e) ( 1, 1, 0 ) 3. Identify the surface 2 = y 2 + z 2 . (a) ellipsoid (b) hyperboloid of one sheet (c) hyperboloid of two sheets (d) cylinder (e) elliptic paraboloid 4. Identify the trace of the surface x = y 2 + z 2 in the plane z = 1. (a) ellipse (but not a circle) (b) parabola (c) hyperbola (d) two intersecting lines (e) circle 5. The point ( 1, 3, 2 ) is expressed in spherical coordinates as: (a) ( 2, π 6 , π 4 ) (b) ( 4, π 3 , π 4 ) (c) ( 2, π 6 , π 4 ) (d) ( 8, π 6 , π 4 ) (e) ( 8, π 3 , π 4 ) 6. Let f ( x , y ) = ( x 2 + y ) 3 . If x = 1, ﬁnd f ( x , 2 x ) . (a) 3 (b) 8 (c) 9 (d) 16 (e) 27 1

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7. Find the range of the function f ( x , y ) = e x - y 2 . (a) ( 0, ) (b) [ 0, ) (c) ( 1, ) (d) [ 1, ) (e) ( - , ) 8. Evaluate lim ( x , y ) ( 1, - 1 ) x 2 - y 2 x + y (a) 0 (b) 1 (c) 2 (d) 3 (e) does not exist 9. Let f ( x , y , z ) = x yz , x > 0. Find the value of f z ( 2, 3, 0 ) . (a) 1
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MultiFinalExam01 - Math 2224 Practice Questions...

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