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M147A4 - MATH 147 Assignment#4 Due Friday November 20 1 a...

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MATH 147 Assignment #4 Due: Friday, November 20 1) a) Find the derivative of each of the following functions. i) y = cos(3 x 2 ) + arcsin ( x ) ii) f ( x ) = tan( x ) e x iii) f ( x ) = x sin( x ) 2) a) Show that f ( x ) = x 2 sin( 1 x 2 ) if x 6 = 0 0 if x = 0 is differentiable every- where but its derivative is not continuous at x = 0. b) Let g ( x ) be such that | g ( x ) |≤ M for all x [ - 1 , 1] . Let h ( x ) = x 2 g ( x ) if x 6 = 0 0 if x = 0 . Show that h ( x ) is differentiable at x = 0 and find h 0 (0) . 3) Assume that f ( x ) is continuous on [3 , 5] , f (3) = 2 and that f 0 ( x ) = 1 1+ x 3 on (3 , 5) . Show that 127 63 f (5) 29 14 . 4) Linear Approximation Assume that f ( x ) is differentiable at x = a . The linear approximation to f ( x ) centered at x = a is the function. L a ( x ) = f ( a ) + f 0 ( a )( x - a ) a) Assume that f ( x ) is such that such that f 00 ( x ) exists and is con- tinuous on an inteval I containing x = a. Apply the Mean Value Theorem twice to show that if | f 00 ( x ) |≤ M for every x I , then | f ( x ) - L a ( x ) |≤ M ( x - a ) 2 for each x I. b) Let f ( x ) = xe - ( x 4 4 ) .
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