Math 191 Practice Prelim 2 Fall 2005

# Math 191 Practice Prelim 2 Fall 2005 - Practice Prelim 2...

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Unformatted text preview: Practice Prelim 2, Math 191, Fall 2005 No calculators. Show your work. Clearly mark each answer. 1. Considerthecurvey— 2332—11113: (3) Find the length of this curve between :1:— H 1 and :e— A e"l (b) Consider the segment of the curve between a 2 I and a: = 3 Verify that this piece of curve lies above the :c-axis and ﬁnd the area of the surface obtained by rotating it around the wares 2. (a) Forxin [0, 2], whatissin 1(cosx) ? (b) Evaluate cos(tan1x) . -._ (c) Determine the equation of the line tangent to the graph of y = tan“1 (1112) at a: = 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? (Simply? your answer but do not evaluate it numerically) 4. (a) Prove or disprove: . (i) tan”1 a: = 0(1) (ii) 2:“? 3"” grows slower than a: 23 I . l 1 (111) Iog2 3\$2 grows at the same rate as (a: + '?)2 (1v) ; = o (In—x) (b) If f = 0(9) and g = 0(h), isit true that f : 0(h) :2 Explain. Evaluate the following integrals. :13: (a) f "”5 ‘h (b) / m a 251: A 1 (c) [:23 lnatda: (agé—l) (d) [3:2—+22:+2d\$ 1r/4 6. For each integer n 2 0, let In 2 ] tann :5 dx. 0 (3.) Find In and 11. (b) Find a formula expressing In+2 in terms of In. (G) Deduce a formula expressing In.“ in terms of In. Hence (or otherwise) ﬁnd I4 and 15. ...
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