math final 2007 fall

math final 2007 fall - Dawson College Fall 2007 Mathematics...

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Dawson College - Fall 2007 Mathematics Department Final Examination Calculus 1 ( 201-103-DW ) 1. (9 points) Find each of the following limits. (a) 2 2 1 1 lim 8 9 x x x x o ± ² ± (b) 2 3 5 6 7 lim 2 3 10 x x x x x of ² ± ± ± (c) 4 2 lim 4 x x x o ± ± 2. (7 points) Consider the function ³ ´ 2 3 6 if 4 6 if 4 10 if 4 x x f x x x x ­ ² µ ± ° ± ® ° ± ! ± ¯ . (a) Find ³ ´ 4 f ± . (b) Find ³ ´ 4 lim x f x . (c) Use the definition of continuity to determine whether f is continuous at 4 x ± . 3. (5 points) The limit definition of the derivative is ³ ´ ³ ´ ³ ´ 0 lim h f x h f x f x h o ² ± c . Use this definition to find ³ ´ x f c given ³ ´ 2 5 3 f x x x ± ² ± (no points will be given for using the rules of differentiation). 4. (4 points) Consider the function ³ ´ 3 2 3 9 7 4 f x x x x ² ² ± . Find the equation of the tangent line to the graph of this function at 2 x ± . 5. (12 points) For each function find the derivative. Do not simplify . (a) 5 4 3 7 x y e x ± (b) ³ ´ ³ ´ 5 2 3 1 ln 1 x f x x ª º ± « » « » ² ¬ ¼ (c) ³ ´ ³ ´ ³ ´ 2 7 tan sin 2 cos 1 9 y x x x x ² ± ± 6. (6 points) Find dy dx given ³ ´ x x y 2 1 4 ± . 7. (6 points) Consider the function ³ ´ x e x x f 2 . Find the value of the second derivative of this function when 0 x . In other words, evaluate ³ ´ 0 f cc .
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8. (7 points) Find the x -values at which the function ³ ´ ³ ´ 4 7 2 2 7 x y x ± ± has a horizontal tangent. 9. (4 points ) A company’s profit function is ³ ´ 3 2 0.00002 0.009 6 1000 P x x x x ± ± ² ² where P is in dollars and x is the number of televisions manufactured and sold. At what rate is the profit changing if 100 televisions are manufactured and sold? 10. (6 points) Consider the implicitly defined relation 2 3 5 4 5 2 x y y x ² ² . (a) Find the derivative dy dx . (b) Find the equation of the tangent line to the graph of this relation at the point ³ ´ 0 , 1 . 11. (5 points) The demand equation for a particular commodity is given by 50 09 . 0 ² ± x p where p is the price per unit if x is the number of units manufactured and sold.
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