Choose appropriate assessment instruments and strategies to assess the literacy development of the student. Use the data to plan a comprehensive program of instruction. Write a final case study for your assessment and proposed instruction of this student.
The text for using the case study is Leslie, L. & Caldwell, J. (2010). Qualitative Reading Inventory
Let’s 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 writeup. 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 rightmost 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 writeup. 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 can’t 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 2decimal places. Recreate the summary table below (with entries filled in) as part of your writeup for this part.
xvalues of critical points
xvalue resulting in absolute minimum cost over the interval (0, 8) (2decimal 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 yvalues to 2decimal places.
8. We can’t print a noninteger 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. Let’s 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 writeup. 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 rightmost 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 xvalue 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 xvalue 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 writeup.
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 Erstwhile’s book, the Associated Students of Foothill College 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.
xvalue to minimize cost (rounded to 2places)
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.
Date: ___________ Course: ____________________
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