Math%20119%202006-2007%20MidTerm2

# Math%20119%202006-2007%20MidTerm2 - METU Department of...

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Unformatted text preview: METU Department of Mathematics CALCULUS I Mid Term 2 Last Name : : Name Department : : Signature : Math 119 Code Acad. Year : 2006-2007 : Fall Semester Coordinator : Muhiddin Uguz Date Time Duration 1 2 : December.4.2006 : 09:30 : 136 minutes 3 4 5 List No. Student No. : : Section Z Group 9 QUESTIONS ON 6 PAGES TOTAL 60 POINTS 6 7 8 9 UĞU Show Your Work 1 5 1 3 Question 1 (9 points) Let f (x) be the function f (x) = x5 + x3 + 2x − d −1 (f (x))(2). dx iddin a) Prove that f (x) is invertable. 8 . 15 Muh b) Find the ﬁrst derivative of the inverse of f at x = 2, i.e. ﬁnd c) Find the second derivative of the inverse of f at x = 2, i.e. ﬁnd d2 −1 (f (x))(2). dx2 Question 2 (6 points) Find the area of the region bounded by the curves UĞU Z y = x2 − 1, y = 0, x = −1 and x = 2. Question 3 (6 points) Find the volume of the solid obtained by rotating the region Muh iddin bounded by the curves y = 2x − x2 and y = 0 about y -axis. Question 4 (6 points) Let f (x) be diﬀerentiable function such that 0 f (t) dt = sin(g (x)). It is also given that g (1) = π and g (1) = 0. Find f (π ). UĞU Z g (x) x3 + 2x2 + 2 . x2 − 1 iddin Muh Question 5 (6 points) Find all asymptotes of f (x) = Question 6 (6 points) Give graph of the derivative of f (x), that is y = f (x), answer UĞU b)Find the intervals on which f (x) is concave up. Z the following questions about the function itself : a)Find the intervals on which f (x) is decreasing. iddin c)If f (0) = 0 compute f (119). n Question 7 (6 points) Find lim i=1 Muh n→∞ i4 . n5 Surname: ................. Name: .................. Id: .................. Section: ................ Question 8 (9 points) 4 x2 + 6 √ dx x 6 dx x ln(x) a)Evaluate b)Evaluate c)If y = Muh iddin e UĞU Z 1 √ 3 (x 4 ) x2 + 1 dy , using logarithmic diﬀerentiation, ﬁnd . (3x + 2)5 dx Question 9 (6 points) Find an equation of the line through the point (1, 2) that cuts Muh iddin UĞU Z oﬀ the least area from the ﬁrst quadrant(i.e. where x ≥ 0, y ≥ 0). Prove your answer. ...
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Math%20119%202006-2007%20MidTerm2 - METU Department of...

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