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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 89 x x x x o ± ²± (b) 2 3 56 7 lim 2 3 10 x xx of ² ± ± ± (c) 4 2 lim 4 x x x o ± ± 2. (7 points) Consider the function ³´ 2 36 i f 4 6 if 4 10 if 4 fx x ­ ² µ± ° ± ® ° ± ¯ . (a) Find 4 f ± . (b) Find 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 ´ 0 lim h f xh fx h o c . Use this definition to find x f c given ³ ´ 2 53 f x ±²± (no points will be given for using the rules of differentiation). 4. (4 points) Consider the function ³ ´ 32 397 4 f ²² ± . 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 ) 54 3 7 x ye x ¶± ( b ) 5 2 3 1 ln 1 x x ª º ± « » « » ² ¬ ¼ ( c ) ³ ´ ³ ´ 27 tan sin 2 cos 1 9 yx x x x ² ± ± 6. (6 points) Find dy dx given 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 ±² 4 7 2 27 x y x ³ ³ has a horizontal tangent. 9. (4 points ) A company’s profit function is ± ² 32 0.00002 0.009 6 1000 Px x x x ³ ³ ´ ´ 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 ´ ´ . ( a ) Find the derivative dy dx . (b) Find the equation of the tangent line to the graph of this relation at the point 0 , 1 . 11. (5 points) The demand equation for a particular commodity is given by 50 09 . 0 ´ ³ x p where p is the price per unit if x is the number of units manufactured and sold. ( a ) How m any un i t s w i l l b e so ld if the price per unit is \$32.00 ?
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math final 2007 fall - Dawson College Fall 2007 Mathematics...

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