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235-08-ps1-SOLNS - UNIVERSITY OF TORONTO DEPARTMENT OF...

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UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT 235 Y – CALCULUS II FALL-WINTER 2008-09 ASSIGNMENT #1. DUE ON OCTOBER 3. PROBLEMS AND SOLUTIONS 1. A brief review of single variable calculus. a) Evaluate ( ) 2 0 lim 9 17ln 9ln 3 1 ln ln x 2 x x x + + + + + x , if the limit exists. Solution: Let 2 9 17ln 9ln A x x = + + and 2 3 1 ln ln B x x = + + . Now, 2 2 A B A B A B = + , where . 2 2 2 2 9 17ln 9ln 9(1 ln ln ) 8ln A B x x x x = + + + + = x Notice that 2 2 2 2 9 17 9 17 9 17ln 9ln ln ( 9) ln 9 ln ln ln ln A x x x x x x x x = + + = + + = + + and similarly, 2 1 1 3 ln 1 ln ln B x x x = + + . So, 2 2 9 17 1 1 ln ( 9 3 1 ) ln ln ln ln A B x x x x x + = + + + + + . Notice also that and 0 lim ln x x + = −∞ ln ln x x = − when 0 < x < 1 . Therefore, ( ) 0 0 2 2 8ln 8 4 lim lim 3 3 3 9 17 1 1 ln ( 9 3 1 ) ln ln ln ln x x x A B x x x x x + + = = − = − + + + + + + . b) Let P ( t , f ( t ) ) be any point on the curve 2 2 ( ) (1 ) x f x x = + and let Q be the point at which the y -axis intersects the line that is tangent to the given curve at the point P . Suppose that A ( t ) denotes the area of the triangle OPQ , where O = ( 0 , 0 ) . Is there an absolute maximum value for the area A ( t ) ? If so, find this absolute maximum value and determine the coordinates of all the points P for which that maximum is reached. Solution: An equation of the line that is tangent to the given curve at the given point is y f ( t ) = f ( t ) ( x t ) . If x = 0 then the value of y in this equation is y = f ( t ) – t f ( t ) . So, Q is the point with coordinates ( 0 , f ( t ) – t f ( t ) ) and OPQ is a triangle whose base has a length of ( ) ( ) f t t f t and whose height has a length of t . Therefore, the area of the triangle OPQ is 1 ( ) ( ( ) ( )) 2 A t t f t t f t = . Now, 2 2 3 1 3 ( ) (1 ) x f x x = + , then 2 4 2 2 2 3 2 3 1 (1 3 ) ( ) ( ) 2 (1 ) (1 ) (1 ) t t t t A t t t t = = + + + 2 t . Finally, 3 2 2 4 4 (2 ) ( ) (1 ) t t A t t = + which shows that A ( t ) reaches its absolute maximum at 2 t = ± . The absolute maximum value for the area of the triangle OPQ is 8 ( 2 ) 27 A ± = and this maximum area is reached only when the tangency point P has coordinates 2 ( 2 , 9 ± ± ) .
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