Business Mathematics IISample Final ExaminationThis test is only presented as a sample of the types of questions which might appear on an examination. The instructor of each section will set his or her own examinations, which may differ from the sample in both length and content.Show all of the steps and all of the work that you use to solve each problem. A list of formulas that you may find useful is provided on the last page of this test.1. Different types of trendlines have been fitted to the data from six test markets.(i) Which trendline gives the best approximation for the actual demand function, D(q)?(ii) Use the trendline that you selected in part (i) to estimate the number, q, of units that would be sold ata price of $30.2. The fixed costs for a good are $12,000 and the marginal cost is $25 per unit. Give a formula for the total cost function, C(q).3. Graphs of the demand and profit functions for a good are shown below. (i) Use the graphs to estimate the number of units that should be produced and sold in order to maximizethe profit.Polynomial Trendline$0$20$40$600 100 200 300 400qD(q)Linear Trendline$0$20$40$600 100 200 300 400qD(q)Quadratic Trendline$0$20$40$600 100 200 300 400qD(q)Demand Function$0$10$20$30$40$500 100 200 300 400 500qD(q)Profit Function-$3,000-$2,000-$1,000$0$1,000$2,000$3,0000 100 200 300 400 500qP(q)© 2002 by The Arizona Board of Regentsfor The University of Arizona.All rights reserved.- Business Mathematics II, Sample Final Examination: page 2 -(ii) Use the graphs to estimate the price per unit for the number of units that you found in part (i).4. The graph of a demand function is shown at the right.(i) Shade in the area that corresponds to the ConsumerSurplus, when q = 200.(ii) Use the graph to estimate the revenue, R(200).(iii) Use the picture to estimate the Consumer Surplus,when q = 200.(iv) Give a formula, part of which will involve an integral, for the Consumer Surplus, when q = 200.5. The revenue function for a good is R(q) =  0.3q2 + 12q. Use a difference quotient, with an increment of h = 0.01 to approximate the marginal revenue at q = 10, R (10).6. This question refers to a good whose graphs of revenue, R(q) and cost, C(q), are shown atthe right. (i) Use the above graphs to estimate the value of q at which R (q) = C (q).(ii) Marginal function number ___ could be the derivative of R(q).D(q)0 100 200 300 400 500204060$$$Marginal Function 102468100 100 200 300 400 500q$/unitMarginal Function 2-0.20-0.15-0.10-0.050.000 100 200 300 400 500q$/unitMarginal Function 3-60-30030600 100 200 300 400 500q$/unitRevenue & Cost Function$0$1,000$2,000$3,000$4,0000 100 200 300 400 500qR(q) & C(q)RevenueCost- Business Mathematics II, Sample Final Examination: page 3 -(iii) Marginal function number ___ could be the derivative of C(q).7. Fill in the boxes of the screen capture in such a way that Solver would find a value in the Cell A2 which gives Cell E2 the value of $5,000, subject to the constraint that Cell B2 is less than or equal to $27.Questions 8 and 9 refer to the following data on test markets and costs for a good. All monetary amounts are in dollars and all quantities are single units.- Business Mathematics II, Sample Final Examination: page 4 -8. From the data in Test Market 2, compute the number of units that you would expect to sell in the entire national market, at a price of $84.95.9. Sketch a graph of the marginal cost function, MC(q) for 0  q  3,000.10. Let f(x) = x3. You are to approximate thearea under the graph of f, above the x-axis, and overthe interval from 1 to 5.(i) Find points x0, x1, and x2 that subdivide [1,5] into two subintervals of equal length.x0 = ____ . x1 = ____ . x2 = ____ .0 500 1000 1500 2000 2500 3000123456MC(q)qMarket Number Market Size PriceProjected Yearly Sales1 200 $89.95 302 300 $84.95 543 150 $79.95 324 350 $74.95 845 200 $69.95 546 400 $64.95 120Cost per unit$5$2$4QuantityFirst 1,000 units:Next 1,000 units:Further:Potential national market: 5,000Test MarketsVariable CostsCost DataFixed Cost: $1,5000501001502002500 1 2 3 4 5 6xf (x )- Business Mathematics II, Sample Final Examination: page 5 -(ii) Find the midpoints m1, and m2 of the subintervals.m1 = ____ . m2 = ____ .(iii) Compute the midpoint sum S2(f, [1, 5]).Questions 11 - 13 refer to a continuous random variable X, whose p.d.f., fX(x), and c.d.f., FX(x), are plotted below.11. (i) Show how you can use the graph of fX(x) to estimate P(X  0.6).(ii) Show how you can use the graph of FX(x) to estimate P(X  0.6).12. (i) Which of the following is most likely to be the mean of X, X? Circle your choice.a. X = 0.75 b. X = 0.65 c. X = 0.5(ii) Which of the following is most likely to be the standard deviation of X, X? Circle your choice.a. X = 1.0 b. X = 0.2 c. X = 0.0513. (i) Is X a normal random variable? Why or why not?(ii) Let m be the random variable that is the sample mean for a random sample of size 100 from X. Is approximately normal?14. X is a finite random variable whose p.m.f. is given below.x 1 2 3fX(x) 0.4 0.2 0.4(i) Compute the mean, X, of X.0 0.2 0.4 0.6 0.8 10.511.522.5fX( )xx0 0.2 0.4 0.6 0.8 10.20.40.60.81FX( )xx10XXmS- Business Mathematics II, Sample Final Examination: page 6 -(ii) Compute the variance, V(X), of X.(iii) Compute the standard deviation, X. of X.15. The following sample of size n = 4 was taken for the values of a random variable, X.7, 4, 4, 9(i) Compute the sample mean, (ii) Compute the sample standard deviation, s. 16. Let T be the random variable that gives the time, in hours, for a rush order of parts to arrive at your business. The parameters for T are T = 12 and T = 3. Let be the random variable that is the mean of a random sample of n = 100 delivery times.(i) (ii) (iii) Give a formula for the random variable, S, that is the standardization of .(iv) What is the approximate distribution of S?17. Graphs of the probability density functions for random variables X and Y, with the same means, are shown below.(i) Which is likely to give you a better estimate of the mean, a sample of size 4 from X, or a sample of size 4 from Y? Explain your answer.(ii) Could X be a normal random variable?(iii) Could X be a standard normal random variable?18. Let X be a normal random variable with X = 100 and X = 8. Fill in the information that would be .x?t?tttt0.10.20.30.4FX(