235-08y-t1-SOLNS - UNIVERSITY OF TORONTO DEPARTMENT OF...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT 235 Y - CALCULUS II TEST #1. OCTOBER 30, 2008. PROBLEMS AND SOLUTIONS. 1. Let C be the curve defined by the parametric equations x = 4 t t 2 and y = 2 t 3 . a) (10 marks) Find the coordinates of each of the points on the curve C where the tangent line has slope 3 . Solution: We just have to find the points on C where 3 dy dx = . Computing the derivates, we obtain 2 /6 /4 2 dy dy dt t dx dx dt t == . So, we need 6 t 2 = 12 – 6 t or t 2 + t – 2 = 0 . That is: t = – 2 or t = 1 . The points on the curve C where the tangent has slope 3 are ( – 12 , – 16 ) and ( 3 , 2 ) . b) (10 marks) Find the values of t for which the curve C is concave upward. Solution: We just have to find the values of t for which 2 2 0 dy dx > . Again, computing the corresponding derivatives, we obtain 2 23 3(4 ) 2(2 ) dd y t t dt dx dx t dt ⎛⎞ ⎜⎟ ⎝⎠ . So, we need 3 0 ) tt t > and the values of t for which this condition holds are 0 < t < 2 or t > 4 .
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Page 2 2. (15 marks) Find the area of the region that lies inside the polar curve r = 3 cos θ but outside the polar curve r = 1 + cos . (Recommended textbook exercise 10.4 #27.) Solution: Notice that, for each of the two given curves, r (– ) = r ( ) because cos (– ) = cos . Therefore, both curves are symmetric with respect to the polar axis. A rough sketch of the graphs of the two curves, a circle and a cardiod, shows that they have exactly three intersection points and that one of these intersection points is the pole.
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This note was uploaded on 10/16/2009 for the course MATH MAT235 taught by Professor Recio during the Fall '08 term at University of Toronto.

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235-08y-t1-SOLNS - UNIVERSITY OF TORONTO DEPARTMENT OF...

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