ReviewF17 - Calculus I Review Nov 28, 2006 Complex Numbers...

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Calculus I Review Nov 28, 2006 Complex Numbers Perform the operations and write the result in the form a + bi . (a) 2 ) 3 1 ( i + (b) ) 6 ( 3 1 i i i + + (c) 3 4 7 4 3 1 2 3 4 + + + + + i i i i Limits 1. Give the δ ε definition for the limit statement; c x lim L x f = ) ( . 2. Use the definition of limit to find a suitable that shows (a) 2 lim x 10 3 2 = x x (b) 1 lim x 1 4 3 2 = x x 3. Evaluate the limit (a) + + x x x 6 6 cos lim π (b) ) 4 sin( 1 lim x x x (c) x x x 3 sin 3 cos 1 lim 3 (d) + + 2 1 6 5 lim 2 2 x x x x (e) ) 3 ( 1 2 0 ) ( lim x x x x e + + (f) + x x x x x 2 tan 3 tan lim 0 (g) ( ) [ ] x e x x x + 7 2 ln lim (h) 1 ln lim 2 2 x x x x Differentiation 1. State the definition of the derivative of a function then use the definition to find the derivative of the functions. (a) x x x f = 2 4 ) ( (b) x x f 5 sin ) ( = (c) 3 7 1 ) ( + = x x f Page 1 of 5
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2. Find the indicated derivative and simplify. (a) , ) 5 4 ( 2 3 x x y = 2 2 dx y d (b) ( ) 3 2 3 2 3 1 x x y = , dx dy (c) + = 3 2 cos ) ( 2 3 x x f , (d) ) ( ' x f 1 2 ) 9 ( 2 5 3 / 1 + = x x x y , dx dy (e) , x e y x 3 sec 2 = 2 2 dx y d (f) y x x y arctan ln = , dx dy (g) ( h ) 2 sin x x y = = ) ( x F x tan 1 1 3 t dt , ) ( ' x F 3. Related rates (a) Assume x and y are differentiable functions of t. Find dt dx given ; ; 2 = x 1 = y 9 = dt dy , and 57 2 3 3 2 = + y x (b) A 5-meter-long ladder is leaning against the side of a house. The foot of the ladder is
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ReviewF17 - Calculus I Review Nov 28, 2006 Complex Numbers...

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