Functions and Rates of Change

Homework Set 1

Functions and Rates of Change

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.

Review from Forgotten Algebra

These are just minimal suggestions…problems that you have difficulty with should be followed up by extra practice. These are the skills that you are assumed to have before starting this course. I suggest looking at the problems before reading the unit, as you may already know how to do some of them. For those you do not know how to do, read as much of the unit as you need to in order to attempt the problem(s). Answers are provided in the back of the text.

Unit #

Problem #’s

Problems from Forgotten Calculus

These are just minimal suggestions…problems that you have difficulty with should be followed up by extra practice. Answers are provided in the back of the text.

Unit #

Problem #’s

Suggested Supplemental Individual Problems:

1)Let the following function be the cost function for the quantity, q, of products sold:

a.Use Graphmatica to graph this function in a suitable window. Remember that Graphmatica only recognizes the variables x and y.

b.How much will it cost to produce 60 units? (What is C(60)?) Show sufficient algebraic or computational work.

c.How many units can be produced for $1000? (Solve C(q)=1000) Show sufficient algebraic or computational work.

2)Let the following function be given:

a.Use Graphmatica to graph this function in a suitable window.

b.What is the value of G(4.4)? You may need to use a calculator for this.

c.When is G(t) = 10? Show sufficient algebraic or computational work. (Hint: You need to undo the power of 2/3 and make the exponent equal to 1. You can do this by raising each side to an appropriate power.)

3)Consider the following two functions:

a.Graph the two functions with Graphmatica being sure to label all points of intersection.

b.When is f(x) = g(x)? You should get a value of x as your answer. Show sufficient algebraic or computational work. You will probably need the quadratic formula to do this problem. When you are done, check your results with those of Graphmatica. You can check your answer by using the Find Intersection… command in the Point menu.

4)Suppose you have a Profit function, where x is the quantity, in tens of thousands, of units sold. For example, x=3.2 would represent 32,000 units sold. P(x) is millions of dollars.

a.Graph P(x) with Graphmatica. Use the View Grid Range… command to set your window to

Left: 5, Right: 110, Bottom: 10, Top: 35. Note that this gives a decent snapshot of the graph, eliminating most of the unnecessary parts of the graph. It also shows the general behavior of the Profit function…exponential.

b.How many items need to be sold in order to achieve a total profit of ten million dollars?

5)Consider the following system of equations:

a.Use Graphmatica to find the solution of this system. (The solution of a system is all points where the graphs of the equations in the system intersect.).

b.Use algebra to verify the graphical solution that you found in part a.

6)For a given item, the Supply Curve represents the quantity of that item that a producer is willing to provide at various prices. The usual assumption is that as price increases, the supplied quantity will increase. The Demand Curve represents the relationship between the quantity of the item that will be sold and the price at which it's offered. The usual assumption is that the quantity sold will increase as price decreases. Suppose that the Supply and Demand functions are given as follows:

In both equations, q represents units in hundreds and price is in dollars. Each function is valid in an appropriate domain. (q up to 500 units)

a.Graph the supply and demand function together in a good window. Label each graph clearly.

b.At a price of $55, how many units is the supplier willing to supply to the market and how many units will the public “demand” at that price?

c.The point where the supply and demand curves meet is often called the equilibrium point. What is the equilibrium point? (Remember, a point has two coordinates.) Print your graph and label the equilibrium point with the Annotate command.

d.Find the equilibrium point by hand, showing all algebraic work.

7)J. Pertalia looked at the relationship between investment as a percent of gross national product (x) and productivity growth (y), for several countries:

US, UK, Belgium(B), Italy (I), Canada (C), France (F), Netherlands (N),

Germany (G), Japan (J).

a.What is the bestfitting linear function for this data?

b.What does this model predict will be the productivity growth when investment is 30% of GNP? Justify your answer and show or explain where you got your answer.

c.What does this model predict will be the investment as a percent of GNP if productivity growth is 7%?

8)J. Dean made a statistical estimation of the costoutput relationship for a shoe chain in 1937. The data is given as follows:

where x is the output in thousands of pairs of shoes and y is the cost in thousands of dollars.

a.Find the bestfitting quadratic equation. Use Graphmatica to produce an appropriate graph…copy and center below.

b.Using Graphmatica, what does the model predicts about costs when output is 30,000 pairs of shoes?

c.What does the model predict the output will be if the cost is $4,000? Use Graphmatica.

9)According to a study by B. Glassman, the below gives the number of people with cell phone service, in millions of people.

a.Using Excel, what is the bestfitting exponential function that models this data? Let x = 0 correspond the year 1984 when you enter data in Excel.

b.Using this model, what is a good estimate of when the number of people with cell phone service will reach 50 million?

c.Using this model, what is a good estimate of how many will have service in 2000? Is the number reasonable? Justify your answer clearly.

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:

Results

Problem 1:

Problem 2:

Problem 3:

Name of person who typed this problem up:

1.GROWTH OF THE UNITED STATES POPULATION

The following table gives the U.S. population in 10-year increments.

a.Give the best fitting exponential trendline for this data. (Set your x variable to be years since 1790!)

b.Use your equation to predict the 2000 US population and compare it to the actual value reported above. Are the numbers close? By what percent is the equations prediction off compared to the actual value reported above? Show relevant computations. Explain the difference in these numbers and offer some reason why they would be so different.

c.Use the data in the table above to find the AVERAGE rate of change between 1880 and 1890. Give your answer in complete sentence form, including appropriate units.

d.Use the data in the table above to find the AVERAGE rate of change between 1890 and 1900. Give your answer in complete sentence form, including appropriate units.

e. Use the average of the average rates of change for 1880 to 1890 and 1890 to 1900 to estimate the instantaneous rate of change in 1890. Show all relevant computations and use data your trendline equation, not the table above.

f.Use Graphmatica and your trendline equation to find the INSTANTANEOUS rate of change of the US population in 1890. Give your answer in complete sentence form, including appropriate units.

g.Compare your results from parts (e) and (f). How close are they? What conclusions can you make about the two methods? Briefly explain your conclusions, if necessary.

h.Another way to model the US population is with a “logistic” equation. The relevant equation in this case is:

where t is years since 1790 and P is population in millions. First enter this into Graphmatica along with the trendline equation you got in part (a). Choose an appropriate domain and range, label your axes, and show the graph.

i.Use your graphs to comment on which model you think is more realistic for planning purposes. Explain clearly why you choose one over the other.

j.What does the logistic equation predict for the 2000 US population? Are the numbers close? By what percent is the equations prediction off compared to the actual value reported above? Show relevant computations. Compare the results to the trendline results for the year 2000.

k.Suppose you were asked to predict when the population was going to reach 300 million people in the U.S. How would you go about doing it and how would you qualify your result, if at all. Please indicate or show how you get your results and explain your reasoning and any qualifications carefully.

Name of person who typed this problem up:

2.PROFIT, REVENUE, AND COSTS OF A COTTON GIN PLANT
Fuller and coworkers (1997) estimated the Costs of a medium-sized cotton gin plant as shown in the following table:

x represents the annual number of bales produced in thousands.

y represents the total cost in thousands of dollars.

a.Determine the best-fitting linear and quadratic (called an “Order 2 polynomial” in Excel) and the square of the correlation coefficient of each. Which is a better mathematical fit, the linear or quadratic model? (Use the “better” cost model for the rest of the parts of this problem.) Please answer this part of the question in complete sentences, explaining what you did and how you determined which one was “better?”

b.The study noted that revenue was $63.25 per bale. At what level will production break even? Please show all algebraic work for this problem and state your final result in a complete sentence with appropriate units.

c.Determine the production level (in units) that will maximize the profits for such a plant and the actual amount of maximum profits. Do this only using algebra. (Hint: The profit function is a quadratic that opens downward so its maximum is its vertex; see Forgotten Calculus, Unit 6.) Please show all algebraic work for this problem and state your final result in a complete sentence with appropriate units.

d.Use Graphmatica to plot the Revenue, Cost, and Profit functions on one graph. Make sure that all three graphs are annotated with their appropriate names, that an appropriate domain and range is chosen, that the point of maximum profit is shown, and that the point where all profits are “lost” is also shown.

e.What is the AVERAGE rate of change of total costs between 6000 and 8000 bales produced? Use the data from the table above. Please show all relevant computations.

f.What is the AVERAGE rate of change of total costs between 4000 and 6000 bales produced? Use the data from the table above. Please show all relevant computations.

g.Use your results from parts (f) and (e)to find an approximate to the instantaneous rate of change at 6000 bales? Use the “average of averages technique.”

h.What is the INSTANTANEOUS rate of change of total costs at 6000 bales? Use Graphmatica and your trendline to accurately compute this rate. Include units in your answer.

i.Compare your estimate from parts (g) and (h). How close are they? (Try to quantify the closeness with percents.) In this case, is the result “average the averages” adequate? Justify your response.

Name of person who typed this problem up:

3.INVESTMENT AND GROWTH OF A RETIREMENT FUND
TIAA CREF is the largest retirement program in the world. They cater specifically to educational and research institutions. Every year, they send out “Benefit Reports” which give participants an idea of what their future retirement will be if certain factors (inflation, contribution levels, rates of return) stay fixed. Suppose a participant receives a Benefit Statement which has the data given in the table. The Rate of Return is that earned on contributions paid by the participant and First Year Income is the amount of first-year income the participant would receive (in 30 years) on that rate of return, all other things being equal.

a.Use Excel and Trendlines to come up with an exponential equation that models this data. Graph it with Graphmatica, labeling axes and choosing an appropriate domain and range.

b.Using Graphmatica, what does this model predict for the First-Year Income with a rate of return of 10.5%?

c.10.5% is halfway between 9% and 12%. Is the predicted first year income for a 10.5% annual rate of return half way between 183,550 and 250,700? Why or why not? Explain clearly. (Think about your explanation…there is a simple mathematical reason for the result.)

d.Over the last five years, the CREF growth account has averaged an annual return of 5.83%. (The last two years of market madness have brought that down from over 25% as a five-year average!) What would the First-Year Income be if the account continues to grow at this rate?

e.The values in the First Year Income column are actual dollars paid out 30 years from now. Inflation will steadily erode the value of these benefits so that they are worth less in the future than they are today. Take the 12% benefit, $250,700: What will the benefit be worth thirty years from now in terms of today’s dollars? Assume an average annual inflation rate of 3.5% over the next thirty years. Please clearly explain how you got your result. (Please check your result to make sure it is reasonable before you submit your answer. The result you get should be lower than the stated benefit, not higher. You cannot simply discount the 12% rate be 3.5%…this does not work…think about why!)

f.What is the AVERAGE rate of change of yearly income between 3% and 6% annual returns? Use the data from the table above. Please show all relevant computations. Your final result should be reported as a complete sentence with appropriate units.

g.What is the AVERAGE rate of change of yearly income between 6% and 9% annual returns? Use the data from the table above. Please show all relevant computations. Your final result should be reported as a complete sentence with appropriate units.

h.Use your results from parts (f) and (g) to find an approximate to the instantaneous rate of change at 6%?

i.What is the INSTANTANEOUS rate of change of total annual income at 6% annual rate of return? Use Graphmatica to compute this rate. Your final result should be reported as a complete sentence with appropriate units. Include units in your answer.

j.Compare your estimate from parts (h) and (i). Record any observations or conclusions you make.

k.Suppose that the goal of the holder of this account is to have their first year income be $125,000 per year. What annual rate of return should he/she try to attain to meet this goal? Explain how you would compute this and any cautions you might give to the holder of the account. Your explanation should include details on how you got your result.

PART 3: Answers to Supplemental Individual Problems

(Please let me know if you think there might be an error…this is a relatively new format so typos and other strange errors may exist, although most have been caught by now.)