This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nonzero, 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....
View Full
Document
 Spring '08
 UPPAL

Click to edit the document details