PART 1: Suggested Individual Work

Homework Set 2 Derivatives and Optimization

PART 1: Suggested Individual Work The Individual portion of homework is primarily for preparation for the Final Exam. These problems are not due and should not be submitted. Do them on your own and come to class with questions about particular problems. We will deal with those during our break out times.

Problems from Forgotten Calculus Here a list of practice problems from Forgotten Calculus that you can try if you feel you need more practice with basic computational type of problems.

For Unit 12, only read the section the Product Rule

Unit # Problem #’s 3 4 6 10 13 9 17 20 23 24 25 10 6 12 14 16 18 1 3 5 6 8 11 10 12 12 2 4 18 19 13 1 4 7 11 12 14 1 4 7 8 16 1 2 11 13 16 19 1 5 8 11 3 4 7 9 22 Use Graphmatica for Unit 22 Problems 7 11 13 14 23 Use Graphmatica for Unit 23 Problems

Suggested Supplemental Individual Problems: This is a list of problems that are more applied and can be done if you need more time and practice with this kind of problem.

Find the equation of the line that is tangent to the graph of y x5 4 x 3  2 x 2  3 when x = 2.

Joel Dean gave a statistical estimation of the average-cost-output relationship for a shoe chain for 1938. He discovered that if x is the output (thousands of pairs of shoes) and y is the average cost (dollars), y was approximated by the model: y  0.0602x 2  4.063x 112.76 a. Use Graphmatica to graph the model in an appropriate window. Copy and center it below. b. Using the tangent line tool , what is the slope of the tangent line at the point x = 12? Explain your results in complete sentences. c. What is the derivative dy/dx? Show all work and steps. d. What is the value of the derivative when x = 20? Interpret your result.

Gerard Pfann found that the total cost C(x) in thousands of guilders incurred by a company for hiring (or firing) x workers was approximated by the model: C(x)  .0071x 2 a. What is the rate of change of costs with respect to workers hired when 100 workers have been hired? Use the marginal cost function to find your answer and show all computations by hand. b. Using the marginal cost function, what is an estimate for the cost to hire the 50th employee? c. What is the actual cost to hire the 50th employee? Show all work.

If C(x) is the total cost of producing x units of some product, then the average cost per unit is given by C(x) A(x)  x>0. x The cost function, in dollars, is defined by C(x)  20x  15000 a. Draw a graph of A(x) in an appropriate window in Graphmatica. b. What is a general formula for A¢(x)? Show all algebraic work. c. What are the values of C(150) and A(150)? Interpret each of them in complete sentence form including proper units. Show all algebraic work. d. What are the values of C¢(150) and A¢(150)? Interpret each of them in complete sentence form including proper units. Show all algebraic work.

When an initial amount A is invested into an account that earns interest “continuously,” the total amount in the account after n years is given by the formula: T(n)  Aern where r is the annual interest rate earned by the account. Let A = $5000 and r = 8% for the following: a. Draw a graph of T(n) in an appropriate Graphmatica window. Copy and center it below: b. What is T ¢(n)? Show all algebraic work. c. What is the value of T (10)? What does your answer mean? Show all algebraic work. d. What is he value of T¢ (10)? What does your answer mean? Show all algebraic work.

The number of units, n(t), produced per employee per day over the last ten years in a certain plant is given by the function: 30t n(t)  t 1 where t is the number of years from 10 years ago. a. What is the rate of change of n with respect to t? Show all computations by hand. b. Graph n(t) and n’(t) on the same axis, choosing an appropriate domain. Copy and center it below: c. How can you use the graph of n’(t) to explain what is happening to production rates over time?

Krueger and Grossman examined the relationship in a variety of countries between per capita income and various environmental indicators. The goal was to see if environmental quality deteriorates steadily with growth. The table gives the data collected relating units y of coliform in waters to GDP per capita income x in thousands of dollars.

x 1 3 5 7 9 11 13 y 1.8 2.8 2.5 3.7 1.0 3.5 6.0 a. What is the best-fitting cubic (third-degree) polynomial for this data? Graph your model with Graphmatica in an appropriate domain. Copy and center it below. b. What is the derivative of your resulting model? Show all algebraic work. c. What is the value of the derivative when x = 10? What does this value mean?

Barton et al studied how the size of agricultural cooperatives was related to their variable cost ratio (variable cost over assets as a percent). In the table, x represents assets in millions of dollars and y represents the “margin ratio.” The data is given in the following table:

x 2.48 3.51 2.5 3.37 3.77 6.33 4.74 y 11.7 16.8 25.4 16.9 17.9 22.8 16.7 a. Find a quadratic to model the data. Graph in an appropriate domain. b. What is the derivative of your model? Show all algebraic work. c. What is the value of the derivative when x = 3? What does this value mean?

3 2 Assume that the total cost function of an item is given by C(q)  q  6q 12q , in thousands of dollars, where q is also measured in thousands of units. Let 0£ q £5. a. For what value of q is cost minimized, if at all? Show all algebraic work. Be sure you run the first derivative test on all critical points. b. For what value of q is the average cost minimized, if at all? Show all algebraic work along with and appropriate graph.

In one model for the annual expenditure on rental equipment one can use the equation P E(x)   Rx c x where P is the replacement cost, R is the average repair cost for the first year, and c is a positive constant reflecting increased repair costs as x years pass. Consider the case of a Laundromat. a. Given that a replacement machine costs about P = $500, and yearly repairs for the first year average $30, and c has been calculated to be 4/3, how many years should you keep the equipment to minimize E, the annual expenditures? You may want to use Graphmatica on this problem…if you do, include your graphs, clearly labeled. b. What is the value of E’(1)? Indicate how you got you’re your result and be sure to interpret it! c. Managers have decided to automatically reconsider replacing the machine when the rate of expenditure is increasing at the rate of $50 per year. When does this take place? Show or explain how you got your answer.

q  300  2 p Monthly demand for a small television set is where p is the dollar price per set. At present, the price of each television is $109. a. Is the demand inelastic, of unit elasticity, or elastic at this price? What does this mean about what happens to demand and revenue as price changes? Be specific. b. For what price is demand of unit elasticity? Interpret this result and use it to determine what should be done to the price to maximize revenue. Be specific in your answer. Show all algebra on this problem.

Suppose the data given in the table is known about the value of an antique t Value wardrobe. Let t be the number of years since the collection was 0 950 purchased, and suppose the prevailing interest rate is 8.1%. Use Excel to 1 1050 find a linear function that describes the value of the wardrobe as a 2 1150 function of time a. What is the Optimal Holding time for the wardrobe? Show all algebraic 3 1250 work. Recall that the Optimal Holding Time is that time t that solves the 4 1350 V '(t) 5 1440  r 6 1550 V (t) equation, , where r is the prevailing interest rate. 7 1630 b. Given a prevailing interest of 5.0%, the value of a collection of rare items 8 1725 has value of y100 0.2 x2  .04 x  3 dollars. Use Graphmatica and a 9 1825 printed graph to find the optimal holding time. 10 1925 In economics the point of diminishing returns is the point when increased expenditure of labor, money, etc. does not bring about a proportionate rise in output. This x P essentially happens when the rate of change of a function reaches its 2 227 maximum. Hence, it’s the derivative that reaches a maximum. This 5 330 necessitates the use of a second derivative (the derivative of a derivative) to 8 500 determine this point. In the following table, x represents the number of 11 600 thousands of dollars spent on advertising and P represents profit from sales 14 620 less advertising expenditures. a. Find a third degree polynomial that models this data. (You can paste your Excel graph and equation if that’s easiest.) b. What is the point of diminishing returns? Show all algebraic work. c. Interpret what your answer to part b means in real terms. d. Show a graph (on Graphmatica) that shows a reasonable domain and range for this model.

Joel Dean found that the cost function for direct materials in a furniture factory was approximated by the mathematical function: C(x)  0.667x  0.00467x 2  0.000185x3 where x is output in thousands. This model is good for x up to 50. What is the approximate value where marginal costs are at a minimum? Show all algebraic work.

Assume that a firm has data to support the fact that the percent, p(t), of a product’s market that can be attained by spending a certain amount on advertising each year for the next ten years is given by: t 4 125  P( t ) 0.14  t 135  where t is given in years. a. Graph this function in an appropriate domain. Be sure your graph shows the basic nature of the curve. b. Using Graphmatica, what is the function’s point of diminishing returns? Explain what your answer means in real terms.

Profits from a new product introduced into the market can grow quickly at first and then level off. This behavior is usually modeled with what is called a logistic curve. Suppose the annual profits P(t) in millions of dollars t years after the introduction of a product is modeled by: 10 P(t)  1 20e0.2t a. Graph P(t) in a suitable domain and range. b. What is the point of diminishing returns? Use Graphmatica but explain how you obtain your final answer. c. What is the “limiting value” of annual profits? (This is the value where profits are leveling off.) Explain why the answer from the graph can be seen in the equation rather easily. A small electronics chain has been selling 200 DVD players a week at $350 each. A market survey indicates that, on average, for each $10 in price reduction offered, the number of players sold should increase by 20 per week. a. What are the weekly demand and revenue functions? b. How much should they reduce or increase the price to maximize its revenue? Show all algebraic work. PART 2: Required Team Mini-Projects

The following problems are to be done and submitted in the teams you were assigned to for the course. All members of the group should attempt all parts of all problems and the team should meet in and/or out of class to compare solutions and then decide what the best write-up should be. For each problem, one person on the team should be designated as the person who prepares the solution in Microsoft Word. This responsibility should be rotated equally throughout the quarter. All problems are due in print form by the end of class on the due date specified.

Please start each new problem on a new page using page breaks where appropriate.

Please type up computations and answers after each part on the following problems. Include this cover page with your work.

Group Number:

Group Names:

Last Name First Name

Results

Problem 1: CR+ CR CR NC

Problem 2: CR+ CR CR NC

Problem 3: CR+ CR CR NC Name of person who typed this problem up:

1. MAXIMIZING PROFITS FOR A TOURIST COMPANY

A tourist company charged $60 per person for a sightseeing trip and got 40 people to sign up for the trip. Past experience and data tells them that for the same trip, each $2 increase in the price above $60 results in the loss of one customer. The company has fixed costs of $300 per trip and variable costs of $4 per person. a. Build a spreadsheet in Excel that gives the Revenue, Profit, and Cost totals for all trips that contain anywhere between 5 and 40 people. Arrange these numbers in columns, adding any other columns as necessary. The entries in each row should be computed using formulas rather than manually entered. Copy and paste the relevant cells below. Indicate for what price per person the company will realize maximum profits. Run a Trendline with Profit as the y-variable and number of people on the trip as the x-variable. Report your trendline and, table, and a graph of your trendline below.

Trendline =

Table of values:

Graph of trendline in Graphmatica…

b. Now take an algebraic approach. For this part of the problem, let x be the number of price increases. Find simplified expressions in terms of x for Revenue, Cost and Profit. (Do not change the meaning of x! It should be used consistently throughout your expressions.)

Show computations and algebraic simplifications here…

Final results:

R(x) 

C(x) 

P(x)  c. Use calculus by hand to find both the price they should charge to maximize profits for a trip and the maximum profits they can expect to realize in that case. Make sure your result is stated as a complete sentence with appropriate units. (In particular, the role of x should be clear.) Show all algebraic work.

Show computations here…

Final interpretation in sentence form…

e. Show the graphs of all three functions (R, C, & P) on one set of axes. Make sure you have an appropriate domain, range, axes labels and annotations.

Graph…

f. Now take one more algebraic approach. For this part of the problem, let x be the actual cost per person for the trip. Find simplified expressions in terms of x for Revenue, Cost and Profit. (Do not change the meaning of x! It should be used consistently throughout your expressions. To do this, you may want to look at the table you did in Excel to look for linear patterns between x and the other variables and then use those to find expressions the number of people, revenue, cost, etc. This is algebraically “delicate” so proceed cautiously.)

Show computations and algebraic simplifications here…

Final results:

R(x) 

C(x) 

P(x)  g. Use calculus by hand to find both the price they should charge to maximize profits for a trip and the maximum profits they can expect to realize in that case. Make sure your result is stated as a complete sentence with appropriate units. In particular, the new role of x should be specified. Show all algebraic work.

h. Show the new graphs of all three functions (R, C, & P) on one set of axes. Make sure you have an appropriate domain, range, axes labels and annotations.

Graph…

i. Compare your overall results, equations, maximum profits, etc from the two methods (Excel and algebra). What similarities, differences, or other observations can you highlight? Explain this part very carefully and clearly!

Your explanation…

j. Now compare your overall results, equations, maximum profits, etc from the three methods (Excel and algebra). What similarities, differences, consistencies, or other observations can you highlight? Explain this part very carefully and clearly!

Your explanation…

Name of person who typed this problem up:

INCOME DISTRIBUTION IN ECONOMIES

For a given population, let L(x) be the proportion of total income that is received by the lowest paid 100x% of income recipients. Thus, L(0.3) = 0.2 means that the lowest 30% of income recipients receive 20% of total income. The graph of the function L(x) is called the y Lorentz Curve. A sample curve is shown here. 1 % Inc. This curve is used to compute the Gini Index, which is a measure of income distribution in an economy. The U.S. Census Bureau closely tracks the Gini Index from year to year. In 0.5 1993, D. Chotikapanich suggested the Line of Equality following particular function as a Lorentz curve with k > 0: x kx 0 e -1 y= L( x ) = k 0 0.2 0.4 0.6 0.8 1 e -1 % of Households a. Let k = 2 in the equation given above and use Graphmatica to carefully display a graph.

Graph…

b. Find L(.20) for your equation with k = 2 and interpret it clearly.

c. Find L(0.8) for your equation with k = 2 and interpret it clearly.

Final interpretation in sentence form… d. Compare and contrast your results for parts (b) and (c)…what do they say about income distribution in this case? Note that L(0.8) gives you the amount of income earned by the upper 20% of the population, which can be compared with L(0.2) .

Your comparison/contrast and comments…

e. Use Graphmatica to find L'( x ) . Report the algebraic expression here after you simplify it appropriately.

Graph…

Algebraic expression (please simplify as much as possible)…

f. Use Graphmatica to find L'(.2) (the derivative of L( x ) at x = 0.2) and interpret it clearly, including units. Compare this to what it would be in a totally “fair” distribution.

Final result in sentence form…

g. The graph obviously shows that L( x ) does not have a relative maximum or minimum point. Use the expression from Graphmatica from L'( x ) to algebraically prove that this function cannot have a relative maximum or minimum point.

Your “proof”/explanation…

Here is an example of actual data and its use to construct a Lorentz Curve: In 1960, the small Norwegian city of Moss exhibited the following data. (Lee Soltow, Toward Income Equality in Norway, University of Wisconsin Press, 1965 p.10) Income # of Males % of Males Estimated % of Income (in Krona) (Non-cumulative) (cumulative) (cumulative) 0-5000 753 11 2 5000-10000 967 25.2 9.6 10000-15000 2347 59.7 40.4 15000-20000 1786 85.8 73.3 20000-25000 493 93.1 84.9 25000-30000 202 96 90.8 30000-35000 270 100 100 h. Compute and compare the quadratic, power and exponential Trendline equations for income distribution in Moss, Norway. Which one is mathematically the “best” one to use? We will call it M( x ) from now on. Graph M (x) below.

Your equations:

Quadratic =

Power =

Exponential =

Your choice and justification for M (x) …

Graph of M (x) …

i. What is the value of M (50) and what does it mean?

j. What is the value of M '(50) and what does it mean? (Don’t forget units!) Does the result mean they have “fair” distribution of income? Why or why not? Explain.

Final results and explanation in sentence form…

k. Use M (x) to determine what top percent of this town’s population makes the top 5% of income. Please explain how you get your answer, including any graphs that you may use to arrive at a conclusion.

Final results and explanation, graphs…

Name of person who typed this problem up:

ELASTICITY OF DEMAND Units Monthly demand for a small television set is studied by small chain of Price ($) Sold stores and the following data results. 55 14 60 13 a. Suppose that experience tells us that the Demand equation for this 65 13 kind of item in this market is typically a cubic polynomial. Use 70 13 Trendlines for this data to determine the Demand Function. (When 75 12 3 Excel gives a number like –2E-05x , this is its way of displaying 80 12 scientific notation. The E here is not the natural base e. Hence, this 85 11 -5 3 3 number is -2� 10x - 0.00002 x .) 90 11 95 10 Demand Function = 100 10 105 9 110 9 b. Find the Elasticity function, E( p ) , and simplify it completely. Do 115 8 not use Graphmatica for this part of the problem. Show all work. 120 8 125 7 Elasticity Function = 130 6 135 5 140 4 c. If the chain of stores is currently pricing the units at $80 each, is the 145 3 demand here elastic or inelastic? Give your numerical result, showing 150 0 computations on how you got it. Clearly interpret what the elasticity value means in terms of revenue and the specific numerical effect on demand caused by a price increase of 1%.

Final result and interpretation in sentence form…

d. Repeat part (c.) for a price of $120.

Final result in sentence form… e. Plot E( p ) in its own window and use the graph to determine what prices results in unit elasticity. Clearly indicate what this means in terms of total revenue and use your result to advise the chain on where it should set its price for this item and what the total maximum revenue they can expect to attain on this item.

Graph…

Your advice and results…

f. Use your equation for Demand from part (a.) to find a simplified expression for Total Revenue. (Use the fact that total revenue equals price per unit times number of units sold.) Plot the resulting function it its own Graphmatica window.

Revenue function =

Graph…

g. Use Calculus by hand to find the price at which maximum revenue is achieved in this case, assuming you are using your Revenue function from part (f.). Also include the total revenue that can be expected in this case. Compare your results to part (e.). If they don’t match, you’ve done something wrong…go back and find/fix it.

Final results and comparison in sentence form… h. If the company pays $92.50 per unit for this item, what will be their total percent profit margin on this item, assuming they take your advice to set the price as you specified in part (e.). (By percent profit margin here, I mean profits as a percent of revenue.)

Part 3: Answers to Supplemental Individual Problems

1) y=40 x - 75

2a.) y 120 Dollars

Pairs of Shoes in Thousands 0 x -10 0 10 20 30 40 50 60 70 -20 b. The slope of the tangent line when x = 12 is –2.62. For each additional unit over 12,000 the cost will decrease by $2.62 per unit. c. y'= 0.0602 x2 - 4.063 x + 112.76 =.0602(2x ) - 4.063(1) + 0 =0.1204x - 4.063 d. f '(20)= .1204(20) - 4.063 =2.408 - 4.063 = -1.655

At 20,000 pairs of shoes, cost is decreasing by 1.655$ per thousand pair.

3a.) C'(x)  0.0071(2x)  0.0142x C'(100)  0.0142(100) 1.42 The rate of change of costs with respect to workers hired when 100 workers have been hired is 1,420 guilders per additional worker. b. C '(49) 0.0142(49)  0.696 Guilders The marginal cost to hire the 50th employee is about 696 guilders. c. C(50)- (49) = 17.75 - 17.0471 = 0.7029

Thus, the actual cost to hire the 50th employee is the difference, or $702.90

4a.) 600 y

Dollars 500

0 Units x -50 0 50 100 150 200 250 300 350 400 450 500

20x + 15000 -15000 b. A( x )= = 20 + 15000 x-1 . So A'( x )= - 15000 x-2 = x x2 c. C(150)  (20)(150) 15000  3000 15000  18000 The total cost of producing 150 units is $18,000

18000 A(150)  150  120 The average cost per unit of producing 150 units is $120

d. C'(x)  20x 15000  20

C'(150)  20 At the 150th unit, costs per unit are increasing by $20 per unit.

15000 A'(x)  x 2

15000 A'(150)  22500  .667

st The average cost to produce the 151 unit will decrease by 0.667 dollars.

5a.) y 6x10^4

Dollars

4x10^4

2x10^4

Years 0 x -5 0 5 10 15 20 25 30 35

b. Using Graphmatica, we can get the derivative to be 400e.08n , after simplification. c. T(10)  5000e.8 11,127.70

When $5,000 is invested with continuous compounded interest, the total amount in the account after 10 years will be $11,127.70 d. T'(10)= 400 e.8 = 890.22

A $5,000 investment, which earns interest continuously, will grow by approximately $890.22 every year after the initial 10-year investment period

30 6a.) You need Graphmatica to do this. After simplifying, the derivative is 2 (t +1) b. y 30 Units Per Employee Per Day n(t)

5 n'(t) Years 0 x 0 2 4 6 8 10 12 14 16 18 20

c. n'( t ) gets smaller which means n( t ) must grow more slowly and eventually look like it’s going to level out. 7a.)

Dollars in Thousands

6 y  0.0174x 3  0.3238x 2 1.7184x  0.2738

Coliform in Water 00 2 4 6 8 10 12 14x b. f( y )= 0.0174 x3 - 0.3238 x 2 + 1.7184 x + 0.2738 f( y )= (0.0174)(3) x 2 - (0.3238)(2) x + 1.7184 + 0 =0.0522x2 - 0.6476 x + 1.7184

c. When the GDP per capita income is $10,000, the units of coliform is increasing by 0.4624 coliform per thousand dollars in GDP 8a.) y=0.939 x2 - 7.2374 x + 30.785 y 25

20 Mar gin Rati o

Do llar s in Mil lion s x 0 1 2 3 4 5 6 7 b. y(x)  0.939x 2  7.2374x  30.785 y'(x)  (2)(.939)(x)  7.2374 y'(x)  1.878x  7.2374 c. y'(3)= (1.878)(3) x - 7.2374 = -1.6034

At $4 million, of agricultural assets, the marginal ratio decreases by 1.6% per million dollars. 9a) C(q)  q 3  6q 2 12q C'(q)  3q 2 12q 12  3(q 2  4q  4)  3(q  2)(q  2) 0  3(q  2)(q  2) q*  2 C'(1)  312 12 112  3 C'(3)  3 32 12 3 12  3

The critical point occurs at q = 2. However, since C’(1) and C’(3) are both positive we know that q = 2 is a stationary inflection point, not a relative minimum or maximum. Therefore, given the domain 0  q  5 total cost will be minimized at q = 0 since C’(q)=0 at q = 0 and total cost increases production increases to the maximum value of 5000 units.

q3-6 q 2 + 12 q b. A( q )= = q2 - 6 q + 12 . So A'( q )= 2 q - 6 . Thus, the critical point is q = 3 . q A'(2)= - 2 and A'(4)= 2 which shows that q = 3 is a minimum.

Average Cost 10

00 1 2 3 4 5x Units x 1000

Average cost will be minimized when production is at 3000 units. At this point average cost will be equal to $3/unit. 10)

y 1400

x -2 0 y =2 Annual4 Expenditure6 8 ;10 x = years12 14 16 18 a. Annual cost will be minimized when marginal cost is equal to 0. ( 500 1 E'( x ) 0   40 x 3 ). This takes place at x = 2.95 years. x2 b. E’(1) = -460. Expenditures are decreasing by $460 per year.

c. Expenditures are increasing by $50 per year at about 5.1 years. Take the derivative in Graphmatica and do a Point Evaluate on the derivative when y = 50.

11) Using Graphmatica, we can find that E(109)= 1.329 . Since E(p)=1.329, which is greater than 1, demand is elastic when the price of each television is $109. If the price of the televisions is increased 1% to $110.09 demand would decrease 1.329%. Therefore, revenue is decreasing. b. Using Graphmatica, we can find that E( p )= 1, when p =100 . Demand will be of unit elasticity when the price of the televisions is $100. This means that the price of the televisions should be reduced to $100 to maximize revenue. 12) a. Excel gives the equation, V = 96.909t + 955.91

V ' 96.909 V '(t)  r V (t) 96.909  .081 96.909t  955.91 96.909  .081 (96.909t  955.91) 96.909  7.850t  77.429 7.850t  19.48 t  2.48

Optimal Holding Time for the antique wardrobe collection is approximately 2.48 years. b.

Gr Rate

Time 0 5 10 15 20 25 30 35 40 45 The optimal holding time is 39.52 years, per Graphmatica. 13a) y = -0.4537x3 + 9.0159x2 - 8.0595x + 208.87 b. y' 1.3611x 2 18.0318x  8.0595 y'' 2.7222x 18.0318 0  2.7222x 18.0318 2.7222x  18.0318 x  6.6240 The point of diminishing returns for dollars spent on advertising is approximately $6,624. c. This means that the company gets the biggest “bang for the buck” when spending $6,624 on advertising. Beyond this amount, the rate of profit begins slowing down.

C'(x)  .667  .00934x  .000555x 2 C''(x)  .00111x  .00934 .00111x  .00934 x  8.4144

Marginal costs are at a minimum at approximately 8,414 units produced. 15a) y 0.1

% o f P rodu ct's Mar ket Att aine d

Year s 0 2 4 6 8 10 b. The function’s point of diminishing returns is at 3.0 years. It is at this point that the dollars spent on advertising are no longer increasing the rate in which the product is capturing the market. 16a) y 10

Prof its in M ill ions of $ 8

Year s 00 10 20 30 40 x b. The point of diminishing returns occurs at approximately 14.98 years. Use Graphmatica to calculate the derivative (Marginal Profit) of the Profit equation and then find the critical points to determine the maximum rate of increasing profits. c. The limiting value of annual profits is $10,000,000. This can easily be seen in the equation because the denominator will approach the value of 1 as t gets larger.

17 a. Let x = # of times the store reduces the price of DVD players by $10. D( x )= 200 + 20 x R( x )= (200 + 20 x )(350 - 10 x )

R( x )= 7000 - 2000 x + 7000 x - 200 x2 R( x )= - 200 x2 + 5000 x + 7000 b. R'( x )= - 400 x + 5000 400x = 5000 x =12.5 To simply maximize revenue, the store should lower the price by $125.