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101s05fin - B U Department of Mathematics Math 101 Calculus...

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B U Department of Mathematics Math 101 Calculus I Spring 2005 Final Exam Calculus 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. Question 1 Find the volume of the solid obtained by revolving the region enclosed by the graphs of f ( x ) = tan x , g ( x ) = sin x and x = π 4 about the x - axis. Solution. tan x = sin x holds for x = 0 , kπ, k = ± 1 , ± 2 , . . . but only for x = 0 both graphs enclose a region with x = π 4 . Then V = π π/ 4 Z 0 (tan 2 x - sin 2 x ) dx . Use tan 2 x = sec 2 x - 1 and sin 2 x = 1 - cos 2 x 2 . V = π Z π/ 4 0 (sec 2 x - 1 - 1 - cos 2 x 2 ) dx = π Z π/ 4 0 (sec 2 x - 3 2 + cos 2 x 2 ) dx = π tan x - 3 2 x + sin 2 x 4 π/ 4 0 = π tan π 4 - 3 2 π 4 + sin 2 · π/ 4 4 - (tan 0 - 0 + sin 0 4 ) = π 1 - 3 π 8 + 1 4 = π 5 4 - 3 π 8
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Question 2 Show that the function f ( x ) = x 4 + 2 x 3 - 2 has exactly one zero in [0 , 1]. Solution. First note that f ( x ) is everywhere continuous and differentiable being a polynomial. f (0) = - 2 f (1) = 1 f (0) = 0 and f (1) > 0 hence f , being continuous, has at least one zero in (0 , 1).
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