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praexam2sp10141-v2

praexam2sp10141-v2 - = cos x x 2 − π 2 4 for-π 2< x...

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Math 141 (Section 04) Practice EXAM 2 (Version 2) 1. Let f ( x ) = e x + x for all x . a) Justify why f has an inverse and find the domain and the range of f 1 . b) Find ( f 1 ) (1) . 2. a) Simplify sec(tan 1 ( x )). b) Find f ( x ) if f ( x ) = ln(1 + 2 x ), for each x . 3. Evaluate the following integrals a) integraltext 1 0 e 1+ln x 1+ x 2 dx. b) integraltext 4 3 1 x 2 6 x +18 dx. 4. a) Find the following limits, justify your answers and clearly identify all indeterminate forms. (i) lim x →∞ x 1 / 2 sin 1 x . (ii) lim x 0 + ln x ln(sin
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Unformatted text preview: ) = cos x x 2 − π 2 / 4 for-π/ 2 < x ≤ 0. ±ind the value that should be assigned to f (-π/ 2) to make the function f continuous on [-π/ 2 , 0]. Justify your answer. 5. Consider the following linear Frst order di²erential equation dy dx + y cos x = cos x. a) ±ind the general solution of the equation. b) ±ind the particular solution for which y ( π/ 2) =-1. 1...
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