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INTEGRATION

INTEGRATION. CHAPTER 15 FINAL. p. S ( q ). 600. D ( q ). q. 0. The demand function of a certain item is given by D ( q ) = 600 – 2 q – q 2 and the supply function is given by S ( q ) = q 2 + 8 q .

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INTEGRATION

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  1. INTEGRATION CHAPTER 15 FINAL

  2. p S (q) 600 D (q) q 0 The demand function of a certain item is given by D(q) = 600 – 2q– q2 and the supply function is given by S(q) = q2 + 8q. 1. Find the point at which supply and demand are in equilibrium.2. Find the consumer’s surplus.3. Find the producer’s surplus. Before solving the problems draw a sketch of the two equations: q stands for quantity and p stands for price. Since q and p can only be positive, sketch the graph in quadrant one only. Both of these graphs are parabola’s. If the squared term is negative, the graph curves downward. If the squared term is positive, the graph curves upward. Allowing q to be zero in both equations gives us a point of reference from which to draw each curve. Point of equilibrium

  3. Finding the point of equilibrium: this is where supply is equal to demand. S(q) = D(q) q2 + 8q= 600 – 2q– q2 2q2 + 10q – 600 = 0 Notice all of these numbers are divisible by 2. q2 + 5q – 300 = 0 Factor or use the quadratic formula. (q + 20)(q – 15) = 0 q + 20 = 0 q – 15 = 0 q = – 20 q = 15 not valid valid Find the price of the object by substituting the answer into either equation. S (15) = (15)2 + 8(15) = 345 The point of equilibrium is (15, 345).

  4. (15, 345) Consumers’ surplus. Read page 891 and 892 of your textbook. The red area in the figure below is the consumers’ surplus. We obtain this answer by using the following: CONSUMER’S SURPLUS IfD(q) is a demand function with equilibrium price p0 and equilibrium demand ofq0, then Note: Consumers make demands. Always relate demand to consumer in the problem. Answer: 2925

  5. Producers’ surplus: The green area in the figure below is the producers’ surplus. Producers supply. Always relate the supply function and its curve to the producers. PRODUCERS’ SURPLUS IfS(q) is a supply function with equilibrium price ofp0 and equilibrium supply ofq0, then Notice the use of the parentheses. Notice the sign change. Answer: 3150 See problem 33 page 895.

  6. A company is considering a new manufacturing process. It knows that the rate of savings from the process, S(t) = 400(t + 5), where t is the number of years the process has been in use. 4. Find the total savings during the first two years.5. Find the savings during the fourth year.6. If the new process costs $19200, when will it pay for itself? Answer: 4800 There is a difference between four years and fourth year. Four years is zero to four while the fourth year is from the end of year three until the end of year four. Answer: 3400

  7. ( x– 6 )( x + 16 ) = 0 x– 6 = 0 x + 16 = 0 x = 6 x = – 16 valid not valid The answer –16 is invalid is because time can not be negative. Answer: 6 years

  8. y C (x) 15 3 S (x) x 0 Suppose a company is introducing a process that will produce an annual savings (in thousands of dollars) given by S(x) = – 2x2 + 5x + 153 where x is the numbers of years of using the process, while producing a rate of annual costs (in dollars) ofC(x) = 7. For how many years will it pay to use the new process?8. What will be the total net savings during this period. 7. C(x) = S(x) Problem 7 could have been solved using the quadratic formula. Make certain if you use the formula, you know how to key it into the calculator properly. Also write the formula out in your solution but you do not have to show each calculation. 4x2 = 9(– 2x2 + 5x + 153) 4x2 = – 18x2 + 45x + 1377 22x2– 45x – 1377 = 0 ( x – 9 )(22x + 153 ) = 0 x – 9 = 0 22x + 153 = 0 x = 9 22x = – 153 valid x = – 153/22 not valid Answer: 9 years.

  9. 8. Net savings is savings minus cost. Answer: $98550 Make certain you understand why there appears to be two different answers. This is not always the case in every problem. That is why you must read.

  10. 9. The cost of maintaining a particular piece of equipment is given byM(x) = 34x + 57 (in dollars), where x is the age of the equipment in years. The company has allotted a maintenance budget of$645. Approximately how many years did the company budget? Give the answer to the nearest tenth of a year. Notice the upper interval is not x. We must use a variable different from the one in the equation. Unfortunately this is not factorable. So we will have to use the quadratic formula and because time can only be positive only the + sign will be used. a = 17 b = 57 c = – 645 Practice using your calculator, making certain to use the parentheses keys correctly. Answer: 4.7 years

  11. 10. Maintenance costs for an appliance generally increase as the appliance gets older. Past records indicate the rate of increase maintenance costs for a particular appliance is given approximately by M(x) = 6x2– 14x + 17. Find the break – even yearly fee a company should charge for a four year maintenance agreement. This however is not the answer to the original question. The original question asked for the yearly fee. Answer: The yearly fee should be $21 a year.

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