101f03mt2 - MATH 101 Second Midterm Examination 1 2 3 4...

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December 8, 2003 17:00-18:00 Student number: Name: Signature: 1 /40 points 2 /15 points 3 /15 points 4 /30 points Total /100 points In order to get full credit, you have to show your work and explain what you are doing. Calculators are not allowed, nor needed. Good luck. (1) a) Evaluate lim x 0 F 0 ( x ) if F = Z x 1 sin 2 t t dt . (15 points) By the fundamental theorem of calculus, we have d dx F ( x ) = d dx Z x 1 sin 2 t t dt = sin 2 x x and lim x 0 F 0 ( x ) = lim x 0 sin 2 x x = lim x 0 sin x x · sin x = 1 · 0 = 0 . b) Find f 0 ( x ) if f ( x ) = x ln x ( x > 0) . (15 points) We have ln f ( x ) = (ln x )(ln x ) = (ln x ) 2 . DiFerentiating we get 1 f ( x ) f 0 ( x ) = 2(ln x ) 1 · 1 x and so f 0 ( x ) = f ( x )2 ln x x = 2 x ln x ln x x . c) Evaluate lim x 0 (1 + 2 x ) 1 3 x . (10 points) Let y = (1 + 2 x ) 1 3 x . Then ln y = 1 3 x ln(1 + 2 x ) and lim x 0 ln y = lim x 0 ln(1 + 2 x ) 3 x = lim x 0 1 1+2 x · 2 3 = 2 3 and ln lim x 0 y
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This note was uploaded on 11/16/2011 for the course MATH 102 taught by Professor Soysal during the Winter '09 term at Boğaziçi University.

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101f03mt2 - MATH 101 Second Midterm Examination 1 2 3 4...

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