handout_10_21

handout_10_21 - -5) 3 e x 2 +4 6. y = log( e 2 x ) 7. y = r...

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Math 1a: Derivatives of logarithmic functions October 21, 2009 1. Consider the function f ( x ) = x - e log x for x > 0. a. Find f ( x ) b. On which intervals is f ( x ) increasing/decreasing? c. For which x is f ( x ) minimum? d. Which one is bigger: e π or π e ? 2. Find the derivative of each of the following functions. Use logarithmic di±erentiation if you think it would help. 1. f ( t ) = log( t 2 + 1) 2. f ( x ) = log log( x ) 3. y = (sin x ) 6 x 1

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4. g ( u ) = log(1 - cos u ) 5. y = x 7 ( x
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Unformatted text preview: -5) 3 e x 2 +4 6. y = log( e 2 x ) 7. y = r (5 x +3)(9 x − 1) x 2 +1 3. In this problem, we show that lim x → (1 + x ) 1 /x = e . Consider the function f ( x ) = log(1 + x ). 1. Write a limit that represents f ′ (0). 2. Find f ′ (0). 3. Conclude that log p lim x → (1 + x ) 1 /x P = 1 , and hence lim x → (1 + x ) 1 /x = e. 2...
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.

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handout_10_21 - -5) 3 e x 2 +4 6. y = log( e 2 x ) 7. y = r...

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