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 d y d x > . Again, computing the corresponding derivatives, we obtain 2 2 3 3 (4 ) 2(2 ) d dy d y t t dt dx dx d x t dt = = . So, we need 3 3 (4 ) 0 2(2 ) t t t > and the values of t for which this condition holds are 0 < t < 2 or t > 4 .
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