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ws_4 - Mathematics for Engineers...

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Mathematics for Engineers I (Math103)... student-intranet/faculties/math/ 2009 - Fall/math103 Class Work Sheet 4 1 . 1. Use the definition of the derivative to find f 0 ( x ): i) f ( x ) = x 3 ii) f ( x ) = x + 1 iii) f ( x ) = 1 x iv) f ( x ) = 1 x v) f ( x ) = cos x vi) f ( x ) = sinh x . 2. (i) Show that the function f ( x ) = 3 x is continuous but not differen- tiable at x = 0. (ii) Show that f ( x ) = x 2 + 1 if x 1 2 x if x > 1 is continuous and differen- tiable at x = 1. Sketch the graph of f ( x ). 3. Find the equation of the tangent line for: (i) f ( x ) = x + 1 at x = 3. (ii) f ( x ) = sin x at x = π 4 . 4. If the tangent line to y = f ( x ) at (4 , 3) passes through the point (0 , 2), find f (4) and f 0 (4). 5. Find the average rate of change of f ( x ) = 1 x , on [2 , 3] and the instan- taneous rate of change at x = 2. 6. Find the derivative of the following: y = (2 x 3 + 3)( x 4 - 2 x ) y = x ( x - 1) y = x - 1 x +1 y = x + 1 x y = cos 3 x - 12 sin x 2 y = tan(8 x ) + 2 sec 3 (12 x ) y = x 3 tan 2 (6 x ) y = cot 3 (7 x ) + csc 4 (6 x ) 1 prepared by: Hany El-Sharkawy using “L A T E X 2 ε 1
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y = cos(2 x 2 ) - cot 3 (9 x ) y = ( x - 4) cos 2 (9 x ) y = 1 - tan x 1+tan x y = ( x + cot x ) 5 2 y = (tan x 2 ) 5 y = p sin( x ) y = cot 2 x + cot x 2 y = sin ( tan( sin x ) ) y = cos( 1 + x 2 ) y = cos - 1 ( sin - 1 x ) y = sin - 1 (3 x + 1) y = x sec - 1 (3 x 2 ) y = sin - 1 ( tan - 1 x ) y = cot - 1 ( 1+ x 1 - x ) y = x +sec - 1 x x 2 +tan - 1 x y = 1 x 2 + sin - 1 (2 x - 1) y = tan - 1 ( x 3 ) y = ( cot - 1 x ) 3 y = sec
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