prelim2_fa09

# prelim2_fa09 - x is y increasing For which values of x is y...

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Math 1910, Prelim 2 October 29, 2009, 7:30 PM to 9:00 PM You are NOT allowed calculators, the text or any other book or notes. SHOW ALL WORK! Writing clearly and legibly will improve your chances of receiving the maximum credit that your solution deserves. Please label the questions in you answer booklet clearly. Write your name and Lecture/Section number on each booklet you use. You may leave when you have ﬁnished, but if you have not handed in your exam booklet and left the room by 8:45, please remain in your seat so as not to disturb others who are still working. 1) Evaluate the following integrals. a) (6 points) Z 2 1 log 2 x x dx b) (6 points) Z ln2 0 e x (3 - e x ) 2 dx c) (6 points) Z e 2 x 1 + e 4 x dx d) (6 points) Z ln 2 ( x ) dx e) (6 points) Z x + 1 x 3 + x dx 2) For each of the following determine if f ( x ) is O ( e x ) ,O (1), neither, or both. Please give a short reason for each answer. a) (5 points) f ( x ) = sinh x . b) (5 points) f ( x ) = tanh x . c) (5 points) f ( x ) = tan - 1 x . d) (5 points) f ( x ) = e x (1 + sin x ). 3) Consider the function y = x x for x > 0. a) (6 points) Calculate dy dx . b) (6 points) For which values of

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Unformatted text preview: x is y increasing? For which values of x is y decreasing? c) (6 points) Calculate lim x → + x x . (over) c ± by Cornell University. All rights reserved 4) The temperature H of a ﬂuid in degrees Centigrade at time t minutes is given by the function H ( t ) = ± 1 π tan-1 ( t ) + t 1 + t ² 100 ◦ C on the domain t ≥ 0. So H (1) = 75 ◦ C . a) (7 points) Show that H ( t ) is a one-to-one function. b) (7 points) Calculate ( H-1 ) (75 ◦ C ). 5) The size of a population y ( t ) evolves according to the diﬀerential equation dy dt =-2 y, y (0) = y . a) (6 points) Write an expression for y ( t ). b) (6 points) What happens to the size of the population as t → ∞ ? Does your answer depend on the initial size y ? c) (6 points) How long does it take for the size of the population to become a quarter of its starting size? Does your answer depend on the initial size y ?...
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prelim2_fa09 - x is y increasing For which values of x is y...

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