sol2-05 - MAT 137Y, 20052006 Solutions to Term Test 2 1....

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Unformatted text preview: MAT 137Y, 20052006 Solutions to Term Test 2 1. (10%) (a) Find a constant k for which lim x sin x- x- kx 3 x 5 exists and the limit is also non-zero, and determine the value of the limit. Let L be the limit above. Regardless of the value of k , the limit is of the form ( ) , so by applying LHopitals Rule three times, L = lim x cos x- 1- 3 kx 2 5 x 4 = lim x - sin x- 6 kx 20 x 3 = lim x - cos x- 6 k 60 x 2 . The resulting limit does not exist unless the numerator goes to zero, which occurs when k =- 1 6 . This gives L = lim x - cos x + 1 60 x 2 which by LHopitals Rule gives L = lim x sin x 120 x = 1 120 lim x sin x x = 1 120 . (10%) (b) Find the equation of all tangent lines to the curve y = x 2 that intersect the point ( 1 4 ,- 3 2 ) . Draw a picture. Clearly ( 1 4 ,- 3 2 ) is NOT on the curve. Suppose the tangent line intersects the curve at the point ( a , a 2 ) . Using two different ways to obtain the slope of the tangent line, we have m = a 2 + 3 2 a- 1 4 = 2 a . Solving for a , 2 a ( a- 1 4 ) = a 2 + 3 2 = 2 a 2- a 2 = a 2 + 3 2 = a 2- a 2- 3 2 = = 2 a 2- a- 3 = = ( 2 a- 3 )( a + 1 ) = = a = 3 2 ,- 1 . The corresponding slopes are m = 2 a = 3 ,- 2. Therefore the equations of the tangent lines are y + 3 2 = 3 x- 1 4 = y = 3 x- 9 4 , y + 3 2 =- 2 x- 1 4 = y =- 2 x- 2 . 1 (12%) 2. A poster is to have an area of 180 square inches with 1 inch margins at the bottom and sides and a 2 inch margin at the top. What dimensions will give the largest printed area? Make sure to verify that your answer yields a maximum. Let x be the width of the poster and y be the height of the poster....
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sol2-05 - MAT 137Y, 20052006 Solutions to Term Test 2 1....

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