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practice_prelim2_fa06

practice_prelim2_fa06 - dy dt = ky where t is the time in...

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Practice Prelim 2, Math 191, Fall 2006 1. (a) Show that the function f ( x ) = x 3 + 3 x is one-to-one and hence has an inveerse f 1 ( x ). (b) What are the domain and the range of f 1 ( x )? (c) Find d dx f 1 ( x ) at x = 4. 2. Find the global maximum of f ( x ) = x 1 /x in the domain x > 0. Hint: x 1 /x = e 1 x ln x . 3. Let f ( x ) = sin 1 (tan x ). (a) What is the natural domain for f ( x ) so that the inverse function f 1 ( x ) exists? (b) Calculate the inverse function f 1 ( x ) and specify its domain. 4. Suppose the rate at which a rumor spreads - that is, the number of people who have heard the rumor over a period of time - increases with the number of people who have heard it. If y is the number of people who have heard the rumor, then
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Unformatted text preview: dy dt = ky, where t is the time in days. (a) If y is 1 when t = 0 and y is 5 when t = 2, ±nd k . (b) Using the value of k from (a), ±nd y when t = 3. 5. True or false? Give a reason for each answer. a) e log 2 x = O ( x ) . b) tanh x = o ( √ x ). c) If f = o ( g ), g = o ( h ), then fg = o ( h ). 6. Evaluate the following. a) sin(tan − 1 x ) . b) sin(sec − 1 ( x/ 6). 1 7. Evaluate the following: [28] (a) Z x √ − 3 + 4 x 2 − x 4 dx (b) Z √ 2 dx x 4 + 3 x 2 + 2 (c) Z 1 + tan 2 x 1 − tan 2 x dx (d) Z dx 5 x p ln(3 x ) dx 2...
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