MTH 151 Sections L, N

Writing Assignment 1

MTH 151 Sections L, N Due Date: April 23, 2018

In this writing assignment, you are part of a team (your group members) who work in the business analytics department of Big Ed’s Gas Farm, a moderately sized gas station. Your boss, Ed, is tasking your team to investigate a demand function that was modeled based on data gathered by a different department. Your boss has taken calculus before and recalls the notion of a limit and a derivative, but is hazy on some of its applications. He knows that your team has a good grasp on demand functions and elasticity and wants a report from your team that explores the properties of two demand functions and determines which demand function would be reasonable as a model for their gas station.

The following are the demand functions:
D1(p) = 2000e−p/2

D2(p) = 2000p

where D1(p) and D2(p) are the total gallons of gas that can be sold in a day at a price of p dollars per gallon.

Ed has broken up his requests into three sections, with specific questions that he wants answered about both demand functions. In Section 1 he wants to you consider some of the basic properties ofD1(p) and D2(p). Answer the following questions about both models, where the questions are phrased for a model D(p):

1. (i)   According to this model, what are the possibles prices for a gallon of gas? (In other words, keeping in mind that D(p) is a demand function, what is the domain of D(p)?). Does this range of possible prices seem appropriate? Explain.

2. (ii)   What happens when prices approach 0? Find limp→0+ D(p). Does your result make sense in the context of a demand function for a gas station? What is the general trend as prices get arbitrarily large? Find limp→∞ D(p). Explain what your result means in this context.

3. (iii)   Demand functions must always decrease as price increases (this is a law of economics). Is this true for D(p)? Explain by finding D′(p).

For Section 2, your boss wants your team to investigate elasticity for both demand functions. Answer the following questions for both models:

1. (i)   Find the elasticity function E(p). Simplify E(p) as much as possible. The price of a gallon of gas at Big Ed’s Gas Farm gas stations at the moment is $2.50 per gallon. Find E(2.50). Interpret this value. In terms of revenue, what would be beneficial, raising or lowering the price? Or would changing price even affect revenue? Explain.

2. (ii)   Find the price intervals where demand is inelastic and elastic. (If the demand function is never inelastic or elastic, show why). Find the prices where demand is unit elastic. For prices that are unit elastic, determine if they minimize or maximize revenue using the intervals of elasticity found above. Explain your results in the context of a gas station. Do your results make sense in this context? What price for a gallon of gas does this model recommend? Does this seem reasonable?

Finally, in Section 3, your boss wants your team to summarize the results of Section 1 and 2, and make a decision based on your findings. Which model does your team recommend as a demand function? Explain your team’s reasoning.

Note that your boss wants a report, so your team should structure your response more as a paper and not as answers to a math assignment. See the Mathematical Writing checklist on the back of this page.

Writing Assignment Instructions

Please work in groups of 2 or 3 with people from your section and hand in one report per group with all names clearly printed. Please type your report and hand in a stapled paper copy on the due date (Note that Word has a robust equation editor that can be accessed using the following keyboard shortcut: ’Alt’ and ’+’ together.) If you want to work in a group of 4, see the the bottom of the page for the extra work that needs to be done. Every member of the group should be equally involved. You may not discuss details of the writing assignment with students from other groups or sections. Doing so counts as academic dishonesty. This paper will be graded on both mathematical content and writing. Focus on writing your mathematics neatly in an easy to follow manner. The following is a checklist to follow when writing the project paper.

Mathematical Writing Checklist

1. 1   The paper uses correct grammar, spelling and punctuation. There are no sentence fragments or run-on sentences.

2. 2   All mathematical steps that play a role in the paper are shown and are correct.

3. 3   The paper is readable. The level of mathematics and explanation is suitable for an average Calculus I student who has studied as much as we have at this time in the course.

4. 4   Every decision made by the writer (e.g., to create an equation, to make a substitution, to differentiate an expression) is explained in sentences, unless it is the result of a straightforward algebra or calculus operation applied to a previous step. Mathematics and explanations are interspersed, so that the reasoning and purpose in each step is clear. See Section 4.7 in our textbook for examples of good mathematical writing.

5. 5   The paper makes full use of appropriate mathematical expressions and diagrams. All variables are defined, and all diagrams are explained in the exposition. Diagrams are as clear and uncluttered as possible, and they are labeled.

6. 6   There are appropriate references to any resource material (web sites, books, people, etc.) that assisted in the preparation of this work, together with a description of the help that was received.

7. 7   Notes are included in the text to acknowledge passages that have been taken or paraphrased from the work of other people, including textbook authors. However, the great majority of the thinking in this paper is your own.

   Extra work for Groups of 4

   For groups of 4, your boss wants your team to explore an extra demand function in addition to the

models on the first page. Consider the following demand function:

D3(p) = 300e2/p

Answer the same questions in Section 1 and Section 2 for this new demand function. Based on your results decide if D3(p) makes sense as a demand function for Big Ed’s Gas Farm. Explain your reasoning.