m21am2sol - Problem 1 (10 points) . Compute the following...

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Unformatted text preview: Problem 1 (10 points) . Compute the following derivatives. d dr 4 3 r 3 = 4 r 2 d dx ( e x cos x ) = e x (cos x- sin x ) d dx x x 2 + 1 = x 2 + 1- 2 x 2 ( x 2 + 1) 2 = 1- x 2 ( x 2 + 1) 2 When a > , find d dx ( a x ) = a x ln a d dx (sin(tan x )) = cos(tan x ) sec 2 x d dx sin- 1 p 1- x 2 = 1 p 1- (1- x 2 )- 2 x 2 1- x 2 =- 1 1- x 2 ( x > 0) d dx ( tan- 1 e x ) = e x 1 + e 2 x d dx ( ln e 2 x ) = 2 d dx (ln | cos x | ) =- sin x cos x =- tan x d dx ( x + 1) 3 ( x 2 + 1) 5 ( x 2 + x + 1) 7 = ( x + 1) 3 ( x 2 + 1) 5 ( x 2 + x + 1) 7 3 1 x + 1 + 5 2 x x 2 + 1- 7 2 x + 1 x 2 + x + 1 1 2 Problem 2 (10 points) . Let us consider the function f ( x ) = x 2 sin ( 1 x 2 ) x 6 = 0 x = 0 . (1) Find the derivative f ( x ) for x 6 = 0. f ( x ) = 2 x sin 1 x 2 + x 2 cos 1 x 2- 2 1 x 3 = 2 x sin 1 x 2- 2 x cos 1 x 2 (2) Find the derivative f (0) using the definition of the derivative. f (0) = lim h f ( h )- f (0) h = lim h h sin 1 h 2 = 0 (3) Is f ( x ) a differentiable function for all values of...
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m21am2sol - Problem 1 (10 points) . Compute the following...

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