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solex1sample - Math 114 Fall 2009 Practice Exam 1 with...

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Math 114 Fall 2009 Practice Exam 1 with Solutions Contents 1 Problems 2 2 Solution key 8 3 Solutions 9 1
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1 Problems Question 1: Let L be the line tangent to the curve −→ r ( t ) = (big t 2 + 3 t + 2 , e t cos t, ln( t + 1) )big at t = 0. Find the coordinates of the point of intersection of L and the plane x + y + z = 8. (A) (2 , 1 , 0) (B) (6 , e 2 cos 2 , ln(2)) (C) (2 , 0 , 1) (D) (5 , 2 , 1) (E) (8 , 3 , 2) (F) (0 , 4 , 4) Solution Key: 2.1 Solution: 3.1 Question 2: Find the distance from the point (1 , 1 , 3) to the the plane 3 x + 2 y + 6 z = 6. Solution Key: 2.2 Solution: 3.2 2
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Question 3: Which of these expressions is always equal to the box (triple) product of A , B and C ? Solution Key: 2.3 Solution: 3.3 Question 4: Which of the following points lies on the same plane with (1 , 2 , 0), (2 , 2 , 1), (0 , 1 , 1)? Solution Key: 2.4 Solution: 3.4 3
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Question 5: A particle starts at the origin with initial velocity hatwide ı + hatwide hatwide k . Its acceleration is −→ a ( t ) = t hatwide ı + hatwide + t hatwide k . Find the position of the particle at t = 1. (A) 1 6 hatwide ı + 1 2 hatwide + 1 3 hatwide k (B) 7 6 hatwide ı + 1 2 hatwide 5 6 hatwide k (C) hatwide ı + hatwide + hatwide k (D) 7 6 hatwide ı + 3 2 hatwide 5 6 hatwide k (E) 5 6 hatwide ı + 2 3 hatwide 5 6 hatwide k (F) hatwide ı + 2 hatwide hatwide k Solution Key: 2.5 Solution: 3.5 Question 6: Find the equation of the surface consisting of all points P whose distance to the z -axis is equal to three times the distance from P to the xy -plane. Solution Key: 2.6 Solution: 3.6 4
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Question 7: The set of points P such that −→ QP · −→ B = 2 is Solution Key: 2.7 Solution: 3.7 5
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Question 8: Match the equations to their graphs. Enter your answers in the table at the bottom of the page and show your reasoning on the next page. Equations: (1) z x 2 y 2 = 0 (2) z 2 x 2 y 2 = 1 (3) sin( x ) y + sin( y ) x = 1 (4) z 2 x 2 = 2 Graphs: –2 –1 0 1 2 x –2 –1 0 1 2 y –2 –1 0 1 2 –2 –1 0 1 2 x –2 –1 0 1 2 y –2 –1 0 1 2 –2 –1 0 1 2 x –2 –1 0 1 2 y –2 –1 0 1 2 –2 –1 0 1 2 x –2 –1 0 1 2 y –2 –1 0 1 2 Graphs (a) (b) (c) (d) Equations 6
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Solution Key: 2.8 Solution: 3.8 Question 9: True or false. Explain your reasoning.
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