practice_prelim2_fa05 - (Simplify your answer but do not...

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Practice Prelim 2, Math 191, Fall 2005 No calculators. Show your work. Clearly mark each answer. 1. Consider the curve y = 1 2 x 2 - 1 4 ln x . (a) Find the length of this curve between x = 1 and x = e 4 . (b) Consider the segment of the curve between x = 1 and x = e . Verify that this piece of curve lies above the x -axis and find the area of the surface obtained by rotating it around the x -axis. 2. (a) For x in £ 0 , π 2 / , what is sin - 1 (cos x ) ? (b) Evaluate cos(tan - 1 x ) . (c) Determine the equation of the line tangent to the graph of y = tan - 1 (ln x ) at x = e . 3. A colony of bacteria is grown under ideal conditions in a laboratory so that the population increases exponentially with time. At the end of 2 hours there are 10,000 bacteria. At the end of 5 hours there are 70,000. How many bacteria were present initially?
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Unformatted text preview: (Simplify your answer but do not evaluate it numerically.) 4. (a) Prove or disprove: (i) tan-1 x = O (1) (ii) x-2 3 x grows slower than x 2 x (iii) log 2 3 x 2 grows at the same rate as ( x + 7) 2 (iv) 1 x = o ± 1 ln x ¶ (b) If f = O ( g ) and g = O ( h ) , is it true that f = O ( h ) ? Explain. 5. Evaluate the following integrals: (a) Z x e-x 2 dx (b) Z dx √ x-x 2 (c) Z x a ln x dx ( a ± =-1) (d) Z 2 x-1 x 2 + 2 x + 2 dx 6. For each integer n ≥ , let I n = Z π/ 4 tan n x dx . (a) Find I and I 1 . (b) Find a formula expressing I n +2 in terms of I n . (c) Deduce a formula expressing I n +4 in terms of I n . Hence (or otherwise) find I 4 and I 5 ....
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