235-08-ps1-SOLNS - UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT 235 Y CALCULUS II FALL-WINTER 2008-09 ASSIGNMENT#1 DUE ON OCTOBER 3 PROBLEMS

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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 xx + ++ + + x , if the limit exists. Solution: Let 2 91 7 l n 9 l n A =+ + and 2 31l n l n B + . Now, 22 AB −= + , where . 2 2 7 l l n 9 ( 1l n ) 8 l n x x x x + + −+ + = x Notice that 7 7 9 17ln ln ( 9) ln 9 ln ln ln ln Ax x x x + = + += + + and similarly, 2 11 3ln 1 ln ln Bx x x + . So, 7 1 1 ln ( 9 3 1 ) ln ln ln ln x + ++ + + . Notice also that and 0 lim ln x x + =−∞ ln ln x x =− when 0 < x < 1 . Therefore, () 00 8ln 8 4 lim lim 33 3 7 1 1 ln ( 9 3 1 ) ln ln ln ln x x →→ = + + . b) Let P ( t , f ( t ) ) be any point on the curve (1 ) x fx 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 tt f t and whose height has a length of t . Therefore, the area of the triangle OPQ is 1 ( () ) 2 At t ft tf t . Now, 2 23 13 ) x x = + , then 24 1( 1 3 ) ( ) 2( 1)( 1) ( t t t = + 2 t . Finally, 32 4( 2 ) ) 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 , 9 ± ± ) .
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Page 2 2. More review on single variable calculus.
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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- Toronto.

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235-08-ps1-SOLNS - UNIVERSITY OF TORONTO DEPARTMENT OF MATHEMATICS MAT 235 Y CALCULUS II FALL-WINTER 2008-09 ASSIGNMENT#1 DUE ON OCTOBER 3 PROBLEMS

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