sheet5 - x = 0 of the function f ( x ) = tan(2 | x | + x )...

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CM221A ANALYSIS I Exercise Sheet 5 Calculate the derivatives of each of the following two functions. Find all of the local and global maximum and minimum values, if there are any, of the functions on the intervals given. Sketch the graphs of the functions in these regions and explain your results! 1. f ( x ) = 1 1+log( x ) 2 on (0 , e]; 2. f ( x ) = xe - x 2 / 2 on [0 , ); 3. Evaluate the following derivative (you must state the conditions under which the result holds) d d x f ( x ) 2 g ( x ) 3 ! . 4. Evaluate the derivatives of the functions f ( x ) = x x for all real numbers x > 0. You will need to use the standard rules for differentiating the exponential and log functions. 5. Calculate the derivative from the left and derivative from the right at
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Unformatted text preview: x = 0 of the function f ( x ) = tan(2 | x | + x ) . 6. The function f ( x ) is dened for all x R by f ( x ) = ( x 2 (sin(1 /x ) if x 6 = 0 otherwise. Calculate the derivative of f ( x ) at all non-zero x . Using only the , de-nition of dierentiation prove that f is dierentiable at x = 0 and nd its derivative at that point. Sketch the graph of the derivative. Is the derivative continuous at x = 0? 7. Let g ( x ) = e-1 /x 2 if x 6 = 0 and g (0) = 0. Calculate the rst two derivatives of g ( x ) at x = 0. Explain your answer! 1...
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This document was uploaded on 01/31/2011.

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