Supply and Demand and Optimal Selling

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Math 116 Review 1
1. Suppose that the total cost of manufacturing q units of a certain product is C  q  thousand dollars, where C  q   q3  30q2  500q  200

a) Find the total cost and the average cost of producing 10 units. b) Find the cost of producing the 10th unit.

2. Let f  x   4 x 2  3x  2 , evaluate and simplify the difference quotient

f  x  h  f  x , where h  0 . h

3. The average scores of incoming students at an eastern liberal arts college in the SAT mathematics examination have been declining at a constant rate in recent years. In 2000, the average SAT score was 605. The score is decreasing at a rate of 6 points per year. Form a linear function of time for the average SAT score. Let t=0 for 2000.

4. The owner of a toy store can obtain a popular board game at a cost of $15 per set. She estimates that if each set sells for p dollars, then sets will be sold each week. a) Express the owner’s weekly profit as a function of selling price p. b) Estimate the optimal selling price. c) How many sets will be sold each week at that optimal price?

5. A missile is projected vertically from an underground bunker in such a way that t seconds after lunch, it is s feet above the ground, where s  t   16t 2  800t  15 a) How deep is the bunker? b) Determine when the missile is at its highest point. c) What is the missile maximum height?


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6. George runs a copying service, and he charges 7 cents per copy. The cost of the copy machine is $8000, the cost of a life time maintenance service is $4000, and the cost of making a single copy is 3 cents. Find the cost function, the revenue function, the profit function, and the break-even point.

7. A manufacturer buys $28,000 worth of machinery that depreciates linearly so that its trade-in value after 12 years will be $1,000 a) Express the value of the machinery as a function of its age. b) Compute the value of the machinery after 7 years.

8. A company has been selling lamps at the price of $50 per lamp, and at this price consumers have been buying 3000 lamps a month. The company wishes to lower the price and estimate that for each dollar decrease in the price, 1000 more lamps will be sold each month. The company can produce the lamps at a cost of $29 per lamp. a) Find the optimal selling price per lamp that will maximize the profit. b) What is the maximum profit at that optimal selling price?

9. The supply function S  x  and the demand functions D  x  for a particular commodity are as follows: S  x   2 x  15 and D  x  

385 , where x is the level of production of that commodity. x 1

a) Find the equilibrium production level xe and the corresponding equilibrium price pe . b) Draw a rough sketch of the supply and demand curves on the same graph. c) For what interval of x is there a market shortage? A market surplus?

10. a) A closed box with a square base is to have a volume of 1,500 cubic inches. Express its surface area as a function of the length of its base. b) Find the interval for the possible lengths of the sides of the square base. c) What length of the side of the square base will minimize the surface area?


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11. An open box with a square base is to be built for $48. The sides of the box will cost $3 per square meter, and the base of the box will cost $4 per square meter. Express volume of the box as a function of the length of its base x.

12. A business manager determines the total cost of producing x units of a particular commodity may be modeled by the function C  x   7.5x  120,000 (dollars). a) Find the average cost A  x  . b) Find lim A  x  , and interpret your result. x 

13. Find the limit (if it exists).

14. Find the limit (if it exists).

15. Find the limit (if it exists). If the limiting value is infinite, indicate whether it is  or  . a) lim x 2

x2  x  6 x 2  3x  2




16. Find the...
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