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FALL 2006 MATH 1050 FINAL EXAM Name____________________________ PART I: MULTIPLE CHOICE. Each problem has only one correct answer. Each problem is worth 7 points. PLEASE PLACE YOUR ANSWER IN THE SPACE PROVIDED.
____1. Simplify the expression, and the find the complex conjugate : i ( 6 i i^2 )
(a) 6 i (b) 6 i (c) 5 0 i (d) 6 i (e) 6 i
____2. Which interval represents the solution to the inequality: 0
x
x
(a) (^ ,^0 ] (b) (^ ^2 ,) (c) (^2 ,^0 ] (d) (^ ,^2 ) (e) (^ ,^2 )[^0 ,)
____3. How many real solutions does the equation have: (^) x^2 2 3 x 0 (a) 0 (b) 1 (c) 2 (d) 3 (e) 4 ____4. Which interval represents the solution to the inequality: |^ x ^6 |^1
(a) [5, 7] (b) ( - , - 5] (c) (-, 7] (d) [ - 7, 5] (e) The solution set is empty. ____5. Determine the domain of the function:
x
f x
(a) The set of real numbers x 40 (b) The set of real numbers x 40 (c) The set of real numbers x < 40 (d) The set of real numbers x > 40 (e) The set of real numbers x 0, 40 ____6. How many complex solutions does the equation have: 0 1 1 x^3 ^ x ? (a) 0 (b) 1 (c) 2 (d) 3 (e) 4
____7. Describe the sequence of transformations on the graph of f ( x ) x^3 that result in the graph of g ( x ) ( x 2 )^3 1.
(a) 100 (b) 6000 (c) 200 (d) 150 (e) 4000
____11. Given that x 2 is a factor of the polynomial function p ( x ) x^3 2 x^2 9 x 18 , find all zeros of p^ ( x ).
(a) x^ ^2 ,^ x ^3 i (b) x^ ^2 ,^ x ^3 i (c) x^ ^2 ,^ x ^3 i (d) x^ ^2 ,^ x ^3 i (e) x 2 , x 3
____12. Given that x 2 is a zero, with multiplicity of 2, of the polynomial function p ( x ) x^4 4 x^3 13 x^2 36 x 36 , determine
the completely factored form of the function.
(a) p ( x )( x 2 )^2 ( x 3 i )( x 3 i ) (b) p ( x )( x 2 )^2 ( x 3 i )( x 3 i ) (c)
p ( x )( x 2 )^2 ( x 3 )( x 3 )
(d) p ( x )( x 2 )^2 ( x 3 i )( x 3 i ) (e) p ( x )( x 2 )^2 ( x 3 )( x 3 )
____13. Which of the following statements is TRUE concerning the graph of the rational function: 200 50 ( ) 2 2 x x f x (a) The graph has a slant asymptote given by the line y^ ^50 x.
(b) The graph has a vertical asymptote given by the line x 50.
(c) The graph a horizontal asymptote given by the line y ^50. (d) The graph has a horizontal asymptote given by the line y ^0. (e) The graph has 2 vertical asymptotes. ____14. Which of the following identifies all vertical asymptotes of the function
x x
x
f x
(a) x^ ^0 ,^ x ^50 (b) x^ ^0 ,^ x ^50 (c) x 0 (d) x 50 (e) x 50
____15. Which of the following expressions is equivalent to: )?
ln( 2 ) ln(
x
x
(a) ln( x^ ) (b) 2 ln( x^2 ) (c) )
ln( 2
x
x (d) ln( x^3 ) (e) 3 ln( x )
____16. Solve the equation for x : 6 4 e^2 x 30.
(a)
ln 6
x (b) x ln 3 (c)
2 ln 4
ln 24
x (d)
ln 20
x (e) The equation has no solution
____20. Which of the following would be one of the terms in the partial fraction decomposition of? 1 2 x x x (a)
x^2 x (b)^
x
(c)
x
(d)
x
(e)
x
PART II: SHOW YOUR WORK in a clear and organized format if you expect to receive full or partial credit. IDENTIFY YOUR ANSWERS. Each problem is worth 12 points.
- Find ALL points of intersection for the graphs of the following equations: x y^2 12 2 x 4 y^2 0
- You have invested $20,000 in a stock portfolio at time t=0 years. The value, V, of the portfolio is assumed to be growing
according to an exponential growth model given by: V ( t ) 20 , 000 ebt. At time t=2 years later, the value of the portfolio is
(a) (6 points) Determine the appropriate value for the constant b in the model given above. (Round answer to 3 decimal
places.)
(b) (6 points) You have decided to sell all the stocks in your portfolio when the value of the portfolio reaches $30,000. According to this model, at what time (number of years) will the value of the stock portfolio be equal to $30,000? (Round answer to 2 decimal places.)
3. At time t=0 days, health officials in a small town have found 125 people infected with a virus that appears to be spreading throughout the community. The officials have determined that the number of people with the virus, N, t days after they first detected the virus is given by the function:
N ( t ) t^2 20 t 125 people; t 0
(a) (6 points) According to this model, what is the maximum number of people that will be infected by the virus?
(b) (4 points) If you are going to use X=150 minutes during a month, which option is preferred? Provide computations to support your answer. (c) (4 points) Complete the following statement: Option (ii) would be the cheaper option if the number of minutes you are going to use is less than __________. Provide computations to support your answer.
5. An investor wants to purchase “units” of stocks and bonds, where each unit is measured in thousands of dollars (that is, one unit of stocks is worth $1000). She wishes to purchase X units of stocks, and Y units of bonds according to the follow criteria: 1. To avoid excessive risk, she wants the number of units of bonds to be at least 3 times the number of units of stocks; 2. She wants to purchase at most 6 units of bonds; and 3. She wants the total number of units purchased to be no more than 8. (a) (6 points) Determine a system of inequalities appropriate for the restrictions on X and Y as given above.
(b) (4 points) Graph the inequalities in part (a) and shade the solution to the system. (c) (2 points) Is the decision to purchase 1 unit of stocks and 5 units of bonds acceptable according to the investor’s criteria given above? MATH 1050 FINAL EXAM KEY FALL 2006 PART I 1 b 2 c 3 a 4 a 5 d 6 a 7 b
