Marh%20119%202006-2007%20Fall%20MidTerm2

Marh%20119%202006-2007%20Fall%20MidTerm2 - Give graph of...

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Group List No. Last Name Name Student No. Department Section Signature : : : : : : : : : : : : : Code Acad. Year Semester Coordinator Date Time Duration M E T U Department of Mathematics CALCULUS I Mid Term 2 Math 119 2006-2007 Fall Muhiddin Uguz December.4.2006 09:30 136 minutes 9 QUESTIONS ON 6 PAGES TOTAL 60 POINTS 1 2 3 4 5 6 7 8 9 Show Your Work Question 1 (9 points) Let f ( x ) be the function f ( x ) = 1 5 x 5 + 1 3 x 3 + 2 x - 8 15 . a) Prove that f ( x ) is invertable. b) Find the ±rst derivative of the inverse of f at x = 2 , i.e. ±nd d dx ( f - 1 ( x ))(2). c) Find the second derivative of the inverse of f at x = 2 , i.e. ±nd d 2 dx 2 ( f - 1 ( x ))(2). Muhiddin UĞUZ
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Question 2 (6 points) Find the area of the region bounded by the curves y = x 2 - 1 , y = 0 , x = - 1 and x = 2. Question 3 (6 points) Find the volume of the solid obtained by rotating the region bounded by the curves y = 2 x - x 2 and y = 0 about y -axis. Muhiddin UĞUZ
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Question 4 (6 points) Let f ( x ) be differentiable function such that Z 0 g ( x ) f ( t ) dt = sin( g ( x )). It is also given that g (1) = π and g 0 (1) 6 = 0. Find f ( π ). Question 5 (6 points) Find all asymptotes of f ( x ) = x 3 + 2 x 2 + 2 x 2 - 1 . Muhiddin UĞUZ
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Question 6 (6 points)
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Unformatted text preview: Give graph of the derivative of f ( x ), that is y = f ( x ), answer the following questions about the function itself : a) Find the intervals on which f ( x ) is decreasing. b) Find the intervals on which f ( x ) is concave up. c) If f (0) = 0 compute f (119). Question 7 (6 points) Find lim n →∞ n X i =1 i 4 n 5 . Muhiddin UĞUZ Surname: . ................ Name: . ................. Id: . ................. Section: . ............... Question 8 (9 points) a) Evaluate Z 4 1 x 2 + 6 √ x dx b) Evaluate Z 6 e dx x ln( x ) c) If y = ( x 3 4 ) √ x 2 + 1 (3 x + 2) 5 , using logarithmic differentiation , Fnd dy dx . Muhiddin UĞUZ Question 9 (6 points) Find an equation of the line through the point (1 , 2) that cuts off the least area from the ±rst quadrant(i.e. where x ≥ , y ≥ 0). Prove your answer. Muhiddin UĞUZ...
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Marh%20119%202006-2007%20Fall%20MidTerm2 - Give graph of...

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