PART I: The Problem Statement The MoreBook company has decided to try publishing books as a way to make money for the company. A famous writer Daniel written a new book and she plans to sell an average of 100,000 books each month in the next year. You have been given the job of scheduling print runs to meet the anticipated demand while also minimizing total cost to the company.

A) Setup Costs: Each print run costs $5,000 B) Production Costs: Each book costs $1 to produce C) Storage Costs: Monthly storage costs for books awaiting shipment average 1 per book.

If you decide to print all 1,200,000 books (the total for the year, 100,000 books per month for 12 months) in a single run at the start of the year and sales run as predicted, then the number of books in stock would begin at 1,200,00 and decrease to zero by the end of the year, as shown in the graph below.

On average the company will be storing 600,00 books for 12 months at 1 per book giving a total storage cost of,

Storage Costs = 600,000 X 12 X 0.01 = $72,000

Recalling that the setup cost for the single print run is $5,000, the cost of producing 1,200,000 books at $1 each is $1,200,00, the total cost for the year would be given by,

C = (Setup Costs) + (Production Costs) + (Storage Costs) = $5000 + $1,200,000 + $72,000 = $1,277,000

1. To cut down on storage costs you will print the book in two print runs of 600,000 each, as shown in the figure below.

A) What are the setup costs for two print runs?

B) What are the production costs?

C) What are the storage costs?

D) What is the total cost for the year?

Remark You should have gotten a total cost for the year of $1,246,000 for the year, resulting in a savings of $31,000 compared to the first scenario. Check your work above and fix any errors before you go on with this exam.

2. It appears to you that increasing the number of print runs reduces total costs for the year. Motivated by this idea, you decide to divide the 1,200,000 books into a twelve monthly production runs of 100,000 books each.

A) Create a graph similar to the one in Problem #1 of this exam. Use graph paper and include the graph for your work in this part (You can glue the graph to your exam paper if necessary). Label both axes as in the graph above (you will have to modify the y-axis values for this part but should use the same x-scale).

B) What are the setup costs for twelve (monthly) print runs?

C) What are the production costs?

D) What are the storage costs?

E) Fill in the amounts from your work above and compute the total costs for the year:

C = (Setup Costs) + (Production Costs) + (Storage Costs)

Remark You should have gotten a total yearly cost of more than $1,260,000 above, which is more than the total yearly costs for two print runs. So increasing the number of print runs to twelve has resulted in more total cost than for two print runs. So the question is, do minimum costs result with two print runs or can we do better?

Lets use the methods of calculus to compute the number of print runs that will minimize our total costs for the year (and prove that we are correct in the process). Let,

x = Number of print runs (our variable)

3. Recreate the table below in your write-up. Fill in the missing entries in the table below. One or more of the missing entries will be expressions involving the variable, x.

Cost Category Information Total for in dollars Setup Costs x print runs @ $5,000 per run Production Costs 1,200,000 books @ $1 per book Average Number of 1,200,000/x/2 = books in Stock per month 1,200,000/(2x) Average number of Storage Costs books in stock per month @ 1 per month for 12 months

Now add up the totals in the right-most column and thus create the total yearly cost function, C(x):

C(x) = _____________________________________(Fill in)

4. Check your cost function. Fill in the missing entries in the table below, including your work in Problem #2 above. Recreate the table below in your exam write-up. If your cost function is correct, you should get the values in the table for x = 1, 2, and 12 (result from Problem #2).

x C(x) 1 $ 1,277,000 2 $ 1,246,000

6 ?

12 ?

5. Compute the derivative of your cost function.

6. Use the methods of calculus to find the (absolute) minimum cost over the interval (0, 8) [We cant let x = 0 in our cost function; do you see why?]. Include a sign analysis for the derivative function. Show ALL work in computing the critical points. Be sure that you explain why any relative minimum you find is also an absolute minimum (since this does not automatically follow). Round final values to 2-decimal places. Recreate the summary table below (with entries filled in) as part of your write-up for this part.

x-values of critical points x-value resulting in absolute minimum cost over the interval (0, 8) (2-decimal places) Absolute minimum cost (round to nearest penny)

7. Graph the cost function over the interval [0, 12] using www.desmos.com or other software. Include the graph for your work in the problem. Also, mark the absolute minimum point on the graph, rounding the x- and y-values to 2-decimal places.

8. We cant print a non-integer number of print runs in reality so in the real world x must be an integer. Evaluate your cost function and determine the integer number of print runs that minimize total yearly costs. Show what values you computed in determining your minimum yearly cost. How many (integer) print runs should we make per year? What will be the (minimum) total yearly costs for this number of runs? State final answer in sentence form. How much more do we have to pay if we use an integer number of print runs to minimize costs as compared to if we could do any decimal (fractional) number of print runs?

PART II: GENERALIZING THE PROBLEM

Our work in Part I is for one instance of the print problem. Lets see if we can generalize the problem for any number of print runs, where the associated costs could be any positive numbers.

9. Recreate the table below in your write-up. Fill in the missing entries in the table below. One or more of the missing entries will be expressions involving the variable, x. You may want to study the table in Problem #3 for help here.

Let,

N = Number of books produced for the year (N = 1,200,000 in Part I) P = Setup cost for a single print run (P = $5000 in Part I) b = Cost of producing a single book (b = $1 in Part I) c = average monthly cost of storing a book (in dollars) (c = 0.01 in Part I) x = Number of print runs (as before)

Fill in the missing entries in the table below using the letters x, N, P, b, and c.

Now add up the totals in the right-most column and thus create the total yearly cost function, C(x):

C(x) = _____________________________________(Fill in)

The missing expression will contain the letters x, N, P, b, and c. Only x is a variable; the other letters represent constants.

Remark Check your cost function is correct. From part I, C(2) = $1,246,000 if N=1,200,000; P = 5,000; b = 1; and c = 0.01 dollars/month.

10. Compute the derivative of your cost function in Problem 9. The derivative function will contain one or more of the letters x, N, P, b, c.

11. Determine any critical points of the cost function. SHOW WORK. Your answer should be of the form, x= expression, where expression is an expression involving one or more of the letters N, P, b, and c (not x). Remark Check that your critical point is correct by using the values N=1,200,000; P = 5,000; b = 1; and c = 0.01 and verifying you get the same value as the critical point you computed in Part I. If not check your work as you have an error.

12. You should have determined that the cost function has a single critical point in the interval (0, 8). We can deduce that this critical point is the location of an absolute minimum if we can establish it is a relative minimum. We might create a sign analysis to do this, but the problem is that the critical point is an expression involving letters and not a number. To get around this problem we use the Second Derivative Test. Compute the second derivative of your cost function and then evaluate the second derivative at the x-value of your critical point in Problem #11. Simplify the final expression algebraically. Given that the constants N, P, b, and c are all positive, what can you say about the second derivative evaluated at the x-value of your critical point (is it positive or negative and why)? What can you conclude from the Second Derivative Test? Recreate the table below in your exam write-up.

Description Value Conclusion

Critical point X = ? (Fill in) N/A

Second derivative function C(x)=? (fill in) N/A Value of second derivative evaluated at critical point (Fill in simplified expression) Positive? Negative?

Based on the sign of the second derivative in the table, what can you conclude from the Second Derivative Test?

13. Based on the success in publishing Professor Daniel's book, the company have received and order of 2,000,000 books to be printed for the year. The students have computed the production costs to be $1.25 per book, and will be charged 2 per month for storage, and the cost per print run will be $5,500. How many print runs should you run to minimize total yearly cost and what will be the minimum cost for the year? Put your results in the summary table below.

x-value to minimize cost (rounded to 2-places) Minimum total cost (to nearest cent)

14. As before, we can only print an integer number of print runs. What integer number of print runs will minimize total costs? What will be the minimum total cost for this number of print runs? Show work.