Name: CU ID#: Section #: Instructor s Name: Score:

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1 Name: CU ID#: Section #: Instructor s Name: Score: General Directions: Show work where possible. Answers without supporting work (where work is appropriate) may receive little credit. Do not round intermediate calculations. Answers in context ALWAYS require units. Give your answers to 3 decimal places UNLESS the answer needs to be rounded differently to make sense in the context of the problem OR the directions specify another type rounding OR the complete answer has less than 3 decimal places. When you are asked to write a model, include all components of a model: an equation, a description of the input, a description of the output, and the interval when known. When asked to write a sentence of practical interpretation, answer the questions: when?, what?, and how much? using ordinary, conversational language. Do not use math words, terms, or unnecessary phrases. Statement of Academic Integrity: I have not and will not give or receive improper aid on this test. Signature: Printed Signature: Future Value: T R(t)e r(t t ) dt Present Value: R(t)e rt dt T Consumer s Expenditure = p q Consumer s Surplus = D(p)dp Producer s Revenue = p q Producer s Surplus = S(p)dp Consumer s Willingness and Ability to Spend = Consumer s Revenue + Consumer s Surplus Producer s Willingness and Ability to Receive = Producer s Revenue Producer s Surplus p p 1 P p Total Social Gain = p S(p)dp + D(p)dp p 1 P p Average Value of f(x) = b a f (x) b a

2 (1) Between 199 and 1999, the rate of change of the closing price of a share of Texas Instruments stock was modeled by T (x) =.5715x x dollars per share per year x years after 199 Between 199 and 1999, the rate of change of the closing price of a share of Hewlett Packard stock was modeled by H (x) = x x x dollars per share per year x years after 199 Use T (2)=.4159 and H (2)= to check your calculator entry. (a) (3 pts) Find the input value at which T (x) and H (x) intersected between 199 and x= (b) (5 pts) Shade, on the graph below, the change in the closing price of TI stock between 199 and Write the mathematical notation for this change and evaluate the expression. Dollars per share per year T (x) H (x) Years since 199 (c) (7 pts) Shade, on the graph below, the region between the graphs of the two functions between 199 and the point of intersection. Write the mathematical notation for this area and evaluate the expression. Dollars per share per year T (x) H (x) Years since 199 (d) (2 pts) Interpret the answer to part (c) in context of the problem. (e) (2 pts) During the eight year period, when was the closing price of HP stock highest? Explain.

3 (2) The figure to the right shows graphs of the rate of change of revenue R (x), measured in thousands of dollars per year, and the rate of change of manufacturing costs C (x), measured in thousands of dollars per year, for a major corporation. The input x for each of these two functions is the number of years after The area of the shaded region is 2. (a) (2 pts) 5 [C '(x ) R '(x )]dx = (b) (3 pts) Give a sentence of practical interpretation (in context of the problem) for part (a) (3) The managers of a large corporation plan to start a scholarship program for children of employees. To set up this scholarship fund, the managers plan to invest $4, initially and to increase their investment by $15 each year. They plan to continue the investment for ten years. The interest rate will be 12% compounded continuously. (a) (3 pts) Write a function R(t) for the annual rate at which money flows into the investment account. R(t): (b) (4 pts) How much will the endowment be worth after 1 years? Show the mathematical expression you are using and evaluate that expression. Integral: Value after 1 years: (c) (3 pts) What is the 1-year present value of this endowment? Show the mathematical expression you would use. You DO NOT have to evaluate this expression. Integral:

4 (4) Suppose that the demand for mailboxes in New York can be modeled as D(p) = 25.3(.72 p ) thousand mailboxes where p is the price (in dollars) of the mailbox. Use D(5) = to check your calculator entry. (a) (3 pts) Is there a price at which New York consumers will no longer purchase mailboxes? If so, what is it? If not, explain why not. (b) (2 pts) Find the quantity of mailboxes consumers will purchase if the market price is $8. (c) (3 pts) On the axes below, sketch the demand function. Then shade the region whose area is the consumers surplus when the market price of each mailbox is $8. Thousand mailboxes Dollars per mailbox (d) (3 pts) Find the area of the shaded region in part (c). This may be done numerically or algebraically in either case show your work. (e) (3 pts) Calculate the consumers willingness and ability to spend when the market price of each mailbox is $8. Show your work.

5 (5) The demand for a 3-ring binder can be modeled by D(p) = 2-4p binders when the price is p dollars per 3-ring binder. The supply of such 3-ring binders can be modeled by binders if p <.5 S(p) = 466.7p 2 46p binders if p.5 where p is the price (in dollars) for one 3-ring binder. Use D(3) = 8 and S(2) = to check your calculator entry. 3-ring binders 3-ring binders Dollars per binder Dollars per binder (a)(3 pts) What is the price above which consumers will no longer buy these 3-ring binders? (b) (3 pts) At what price will the producer supply 16 3-ring binders? (c) (5 pts) On the appropriate graph above, shade the producer s revenue when 16 binders are supplied. Calculate the producer s revenue when 16 binders are supplied? (d) (4 pts) Find the point of market equilibrium (Be sure to include units with your answer.) (e) (2 pts) Interpret part (d) in a sentence that explains the idea of market equilibrium for this specific context.

6 (f) (2 pts) On the graph below, shade the total social gain at the equilibrium price. S(p) 3-ring binders D(p) Dollars per binder (g) (7 pts) Determine the total social gain when binders are sold at the equilibrium price. Show the integrals you use to find this amount. (6) In 1999, AT&T and British Telecommunications (BT) wanted to purchase Japan Telecom, the third-largest land-line and wireless carrier in Japan. AT&T and BT made their offer based on what they thought Japan Telecom s value was (using the 1998 profit numbers) over the next 5 years. Japan Telecom had 1998 profit of about $1.7 billion. Japan Telecom felt that their profit would increase by 4.5% per year, over the next 5 years, and could be reinvested in a continuous stream at 8.2% compounded continuously. (2 pts) AT&T and BT offered to pay $6 billion for Japan Telecom. Would Japan Telecom have considered this offer to be a fair price? yes/no (circle one) (4 pts) EXPLAIN your answer. Show the mathematical expression you would use to support your choice and evaluate that expression.

7 (7) During a summer thunderstorm, the temperature drops and then rises again. The rate of change of the temperature during the hour and a half after the storm began is given by t(x) = 9.48x x x 9.87 F per hour where x is the number of hours since the storm began. Use t(.75)= to check your calculator entry. (a) (5 pts) If the temperature was 85 F at the time the storm began, find the specific model for T(x), the temperature during the hour and a half after the storm began. T(x)= + F where x is the number of hours since the storm began. (b) (3 pts) Find the average temperature during the hour and a half after the storm began. (c) (4 pts) Using t(x), find the average rate of change of the temperature during the hour and a half following the beginning of the storm. Give the proper notation and evaluate it. (d) (3 pts) Using T(x), give the mathematical notation that would find the average rate of change of the temperature during the hour and a half following the beginning of the storm. You DO NOT have to evaluate this expression. (e) (3 pts) 1.5 t (x )dx practical interpretation for this integral. = Evaluate the integral and write a sentence of (f) (3 pts) T(1.5)-T() = Write a sentence of practical interpretation for this expression.