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Test2Sol(1804)(08-092nd)

Test2Sol(1804)(08-092nd) - T2/MATH1804/YMC/2009 THE...

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Unformatted text preview: T2/MATH1804/YMC/2009 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1804 University Mathematics A Test 2 Name : U. No. : Group : Answer ALL 4 Questions • Write your answer in the space provided. • For full credits, show your steps clearly and give details to your solution. • No books and notes are allowed to use. 1. ( 8 points ) (a) Compute lim x → Z x 2 t sin tdt x 6 . [ Hint : Use L’Hˆ opital’s rule.] (b) Use logarithmic differentiation to find the derivative of y = x x (2 x + 1) 2 √ x + 3 . (c) Find the linear approximation P 1 ( x ) of log 8 x at x = 8 . Solution : (a) lim x → Z x 2 t sin tdt x 6 = lim x → d dx Z x 2 t sin tdt 6 x 5 = lim x → 2 x ( x 2 sin x 2 ) 6 x 5 = lim x → sin x 2 3 x 2 = lim x → 2 x cos x 2 6 x = lim x → cos x 2 3 = 1 3 (b) Taking the natural logarithm of both sides gives ln y = x ln x- 2 ln(2 x + 1)- 1 2 ln( x + 3) Differentiating both sides then yields d dx ln y = d dx ( x ln x- 2 ln(2 x + 1)- 1 2 ln( x + 3)) 1 y · dy dx = ln x + 1- 4 2 x + 1- 1 2 · 1 x + 3 Thus, dy dx = x x (2 x + 1) 2 √...
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Test2Sol(1804)(08-092nd) - T2/MATH1804/YMC/2009 THE...

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