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Unformatted text preview: UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT 235 Y CALCULUS II FALL-WINTER 2008-2009 SAMPLE QUESTIONS FROM PREVIOUS YEARS TERM TEST #1 1. Let C be the curve defined by the parametric equations x = 5 + t 2 and y = 8 t 3 t 4 . Find the values of t for which the curve C is concave upward. 2. Find the length of the loop of the curve x = 3 t t 3 , y = 3 t 2 . 3. Find the area of the region that lies inside the polar curve r = 1 + sin but outside the polar curve r = 2 sin . 4. Let A ( 1 , 2 , 3 ) , B ( 2 , 2 , 2 ) and C ( 0 , 4 , 5 ) . a) Find the angle between the vectors AB JJJG and AC JJJG . b) Find the area of the triangle ABC . 5. Let P ( 4 , 2 , 3 ) , Q ( 5 , 3 , 1 ) , R ( 3 , 5 , 5 ) and let L be the line that passes through P and Q . a) Find parametric equations for the line that passes through the point R and is parallel to the line L . b) Find an equation of the plane that passes through the midpoint of the segment QR and is perpendicular to the line L . c) Find the coordinates of the point of the line L which is closest to the point R . 6. Find an equation of the plane that contains the line of intersection of the planes x z = 1 and y + 2 z = 3 and is perpendicular to the plane x + y 2...
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