101s06mt2

101s06mt2 - B U Department of Mathematics Math 101 Calculus...

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B U Department of Mathematics Math 101 Calculus I Spring 2006 Second Midterm This archive is a property of Bo˘gazi¸ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1. Evaluate (a) lim x 0 + sin( x 2 ) 1 / ln x Solution: y = ( sin x 2 ) 1 / ln x ln y = 1 ln x ln sin x 2 lim x 0 + ln y = = lim x 0 + 2 x cos x 2 sin x 2 1 /x ln y = lim x 0 + 2 x 2 cos x 2 sin x 2 = lim x 0 + x 2 sin x 2 · lim x 0 + 2 cos x 2 = 2 lim x 0 + ln y = 2 ln lim x 0 + y = 2 lim x 0 + y = e 2 lim x 0 + ( sin x 2 ) 1 / ln x = e 2 (b) lim n →∞ n j =1 1 n + jc for c > 1 Solution: 1 nj + c = 1 /n 1 + jc n lim n →∞ n j =1 1 n · 1 1 + jc n = 1 0 dx 1 + xc 1 0 dx 1 + xc = 1 c 1 0 cdx 1 + xc

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