Mathematics for Business and Economics - Fall 2015

Transcription

1 NAME: Mathematics for Business and Economics - Fall 2015 Final Exam, December 14, 2015 In all non-multiple choice problems you are required to show all your work and provide the necessary explanations everywhere to get full credit. In all multiple choice problems you don t have to show your work. 1

2 1. The manager of a bicycle shop has found that, at a x price (in dollars) of p(x) = 150 x 4 per bicycle, x bicycles will be sold. (a) Find an expression for the total revenue from the sale of x bicycles. (b) Find the number of bicycle sales that leads to maximum revenue. (c) Find the maximum revenue. 2

3 2. The graph on the following page from the U.S. Office of Management and Budget shows the federal debt from the year 2007 to the year 2012 (in billions of dollars), with x = 7 corresponding to the year Find the rule of a linear function g that passes through the two points corresponding to 2007 and A small business borrows $50,000 for expansion at 9% compounded monthly. The loan is due in 4 years. How much interest will the business pay? A $21, B $21, C $21, D $21, E None of the above 3

4 4. Shipping charges at an online bookstore are $4 for one book, $6 for two books, and $7 for three to five books. Last week, there were 6400 orders of five or fewer books, and total shipping charges for these orders were $33,600. The number of shipments with $7 charges was 1000 less than the number with $6 charges. How many shipments were made in each category (one book, two books, three-to-five books)? 4

5 5. Sarah Hendrickson needs to rent a van to pick up a new couch she has purchased. The cost of the van is $19.99 for the first 75 minutes and then an additional $5 for each block of 15 minutes beyond 75. Find the cost to rent a van for 2 hours, 1.5 hours, 3.5 hours, 4 hours. Then graph all ordered pairs, (hours, cost), for the function. 5

6 6. Suppose the supply function of a certain item is given by S(q) = 7 5 q and the demand function is given by D(q) = 3 5 q +10 (a) Find the consumers surplus. (b) Find the producers surplus. 7. Find the average rate of change of the function y = e 1 x over the interval [0,2]. 6

7 8. Use algebra and the properties of limits as needed to find the given limits. (a) lim x 1 x 2 +x 2 x 1 x 2 (b) lim x 2 x 2 (c) lim x 1 3x x (d) lim x 2 + x 2 4 A B 0 C D does not exist and neither or E None of the above 7

8 9. An insurance firm pays $4000 for a new printer for its computer. It amortizes the loan for the printer in 4 annual payments at 8% compounded annually. Prepare an amortization schedule showing the first four payments for the loan. Payment Amount of Interest for Portion to Principal at Number Payment Period Principal End of Period 0 $

9 10. A speculator agrees to pay $15,000 for a parcel of land; this amount, with interest, will be paid over 4 years with semiannual payments at an interest rate of 10% compounded semiannually. Find the amount of each payment. A $ B $ C $ D $ E None of the above 11. Let 1 if x 1 2 if 1 < x 2 f(x) = 3 x 2 if 2 < x 4 log 4 x if x > 4 At which point(s) f is discontinuous? Why? 9

10 12. Find the derivatives of the following functions (a) f(x) = 6x3 +x 1 x (b) f(x) = x 1 x (c) f(x) = ln(1+2 x ) 13. Let f(x) = lnx. Find the instantaneous rate of change of f when x = 2. 10

11 14. Solve the exponential equation 5(e x +1) = The profit (in millions of dollars) from the sale of x million units of Blue Glue is given by P(x) =.7x The cost is given by C(x) =.9x (a) Find the revenue equation. A 1.6x+50.1 B 1.6x C 1.6x 50.1 D 0.2x+50.1 E None of the above (b) What is the break-even point? A 28.3 million units B 36.4 million units C 32.5 million units D 255 million units E None of the above 11

12 16. Let f(x) = 1 x x2. (a) Find the intervals of increase and decrease of f. (b) Find local maximum and minimum values of f. (c) Find the intervals of concavity of f. (d) Find the inflection point of f. 12

13 17. Saltwater taffy can be sold wholesale for $45 per thousand individual candies. The cost of producing x thousand candies is C(x) =.001x x x. (a) What is the revenue function in this situation? (b) What is the profit function in this situation? (c) What number of candies will produce the largest possible profit? 13

14 18. Let f(x) = 3x Use the definition of the derivative to find f (x). Then find an equation of the tangent line at x = 2. 14

15 19. A hospital dietitian has two meal choices: one for patients on solid food that costs $2.75 and one for patients on liquids that costs $3.75. There is a maximum of 600 patients in the hospital. Thehospital always hasat least 100 patients on solid foods and at least 100 on liquids. What number of each type of patient would minimize food costs? 15

16 20. Determine the domain of the function f(x) = x+2 2x 5. A (, 5/2) B (5/2, ) C (, 2) D (2, ) E None of the above 21. Use the graph of the function to determine (if possible) the following: (a) lim x 0 f(x) (b) lim x 0 +f(x) (c) lim f(x) x 0 (d) f(0) (e) lim x 2 f(x) (f) lim x 2 +f(x) (g) lim f(x) x 2 (h) f(2) (i) lim x 4 f(x) (j) lim x 4 +f(x) (k) lim f(x) x 4 (l) f(4) 16

17 22. Sketch the graph of each of these functions: (a) y = x+1 2 x 2 (b) y = x if x 1 if x > 1 (c) y = x 17

18 (d) y = log 2 x (e) y = x 1 18

19 (f) y = x (g) y = 2 x 19

20 23. Find the vertical and horizontal asymptotes of f(x) = 1+ 1 x. 24. Jerry Ryan borrowed $8000 for nine months at an interest rate of 7%. The bank also charges a $100 processing fee. What is the actual interest rate for this loan? A 8.77% B 8.87% C 8.67% D 8.57% E None of the above 20

21 25. Solve the logarithmic equation ln(x+7) lnx = A grandmother opens an investment account for her only granddaughter on the day she was born, investing $500. Each year on her birthday, she deposits another $500, making the last deposit on her 25th birthday. If the account paid a return rate of 6.2% compounded annually, how much is in the account at the end of the day on the granddaughter s 25th birthday? A $30, B $30, C $30, D $30, E None of the above 21

22 27. Find the following integrals (a) x 2 e 1+x3 dx 2 (b) (x+1) xdx 1 22

23 28. Liz is working to raise money for breast cancer research by sending informational letters to local neighborhood organizations and church groups. She discovered that each church group requires 2 hours of letter writing and 1 hour of follow-up, while each neighborhood group needs 2 hours of letter writing and 3 hours of follow-up. Liz can raise $1000 from each church group and $2000 from each neighborhood organization, and she has a maximum of 16 hours of letterwriting time and a maximum of 12 hours of follow-up time available per month. Determine the most profitable mixture of groups she should contact and the most money she can raise in a month. 23