ws13 - WORKSHEET 13 Fall 1995 1 Find the domain and range...

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WORKSHEET 13 - Fall 1995 1. Find the domain and range of the following functions. a) f ( x )=ln x 2 b) f ( x )=2ln x c) f ( x )=ln | sin x | d) f ( x ) = ln(sin x ) . 2. a) Use a calculator to estimate lim s 0 (1 + s ) 1 /s . This limit exists and its value is very important in mathematics and in almost any ±eld which uses mathematics. We call this value e . b) Show that lim h 0 (1 + h x ) x/h = e. (Hint: Here x is ±xed, so when h is small, we may think of h/x as some small number, say s .) 3. a) Below are several steps to rewrite the di²erence quotient for the function f ( x )=log b x .F o re a ch step, state exactly what was done and justify it. log b ( x + h ) log b x h = log b [( x + h ) /x ] h (1) =log b (1 + h x ) 1 /h (2) =log b [(1 + h x ) x/h ] 1 /x (3) = 1 x log b (1 + h x ) x/h . (4) b) Use the de±nition of the derivative to show that d dx log b x = log b e x. (
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