07finsol - MAT1300D Solution to Final Examination Fall 2007...

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MAT1300D Solution to Final Examination Fall 2007 1 Solution to Final Examination MAT1300D, Fall 2007 Part I. Multiple-Choice Questions (30 marks) 1. Find the equation of the tangent line to the graph of y = 3 x - 1 when x = 4. (A) y = 1 1 2 x + ; (B) y = 1 3 2 x + ; (C) y = 3 7 4 x + ; (D) y = 3 2 4 x + ; (E) y = 3 1 4 x - . Solution . y' = 3 2 x . When x = 4, y = 5, and y' = 3 4 . Hence, the equation of the tangent line is y = 3 4 ( x - 4) + 5, or y = 3 4 x + 2. 2. Calculate 2 3 12 lim 3 x x x x →- - - + . (A) 1 5 (B) 3 (C) 7 4 (D) 4 9 (E) - 7. Solution . x 2 - x - 12 = ( x + 3)( x - 4). 2 3 3 3 12 ( 3)( 4) lim lim lim( 4) 7. 3 3 x x x x x x x x x x →- →- →- - - + - = = - = - + + 3. On what interval is the function g ( x ) = - 2 x 3 + 12 x 2 - 36 x + 3 concave down? (A) (2, ) (B) (2, 3) (C) ( - 1, ) (D) ( -∞ , - 1) (E) ( - 2, 4). Solution . g' = - 6 x 2 + 24 x - 36, g" = - 12 x + 24. g'' = 0 implies x = 2. When x < 2, g" < 0, the graph of g ( x ) is concave down. 4. Which of the following statements is true for the function g ( x ) = 2 x 3 + 3 x 2 - 36 x + 2? (A) x = - 1 is a local minimum. (B) x = - 3 is a local maximum. (C) x = - 3 is a local minimum. (D) x = 1 is a local minimum. (E) x = 1 is a local maximum. D 1. E 2. A 3. B 4. C 5. A 6. C 7. E 8. D 9. C 10.

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MAT1300D Solution to Final Examination Fall 2007 2 Solution . g' = 6 x 2 + 6 x - 36. Critical points are the roots of x 2 + x - 6 = 0, x = - 3, 2. When -∞ < x < - 3, g' > 0; when - 3 < x < 2, g' < 0; when 2 < x < , g' > 0. g attains a local maximum at x = - 3. 5. Calculate 4 0 (3 1) x dx + (A) 10 (B) 15 (C) 20 (D) 25 (E) 30 Solution . 4 4 4 1/ 2 3/ 2 0 0 0 2 (3 1) 3 3 4 4 16 4 20 3 x dx x dx dx + = + = + = + = . 6a. Suppose that for a certain product, the demand function is given by D ( x ) = 11 - x 2 and the supply function is given by S ( x ) = 2 x + 3. Calculate the producer's surplus. (A) 1; (B) 4; (C) 9; (D) 13; (E) 22. Solution . The equilibrium point is 11 - x 2 = 2 x + 3, x = 2, p = 7. The producer's surplus is 2 2 2 2 0 0 0 (7 2 3) (4 2 ) 4 4 x x dx x dx x x = - - = - = - = . 6c. Suppose that for a certain product the demand function is given by p = 180 x - , 0 x 180. For which values of x the demand is elastic? (A)
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07finsol - MAT1300D Solution to Final Examination Fall 2007...

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