# Writing assignment about Calculus 1

A writing assignment report is not to be simply a list of responses to the questions outlined or asked in the body of the writing assignment. It should be an essay, in paragraph form, that discusses what the group did and found in the course of answering the questions in the writing assignment. Complete, coherent sentences are a must! The report must be typed on a google doc, double-spaced, and should include relevant graphs and equations. While it is preferred that the equations and supporting mathematical work are created electronically, they may be neatly hand-written and integrated in the report as pictures or snapshots of pdfs. Graphs must be created electronically.

*A good writing assignment report contains the following:*

- an introduction explaining to the reader what the writing assignment was about.
- a body that discusses the concepts involved, provides a description of the procedures your group used, and insights the group gained during the course of the work.

*a conclusion that gives an overview of the results and summarizes what you learned.

Calculus 1 Max/Min Writing Project (Landscape Design)

Reports must be neat and well organized.

They should be written in complete sentences and typed. The project will be graded both for mathematical quality and for expository quality. Look at the six examples in section 4.7 of our text for good ideas about expository style. Notice how the author mixes the necessary mathematical equations with sentences of logic and explanation.

Notice that the exposition is almost always in present tense. And, the author frequently

uses we referring to the reader and author working together on the steps of solution. These are all standard conventions in mathematical writing.

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*The Problem. *

You are a landscape designer. A client has asked you to design a plan to enclose a rectangular garden having 10,000 m^2 of area. The north and south sides are to be bounded by wooden fencing, which costs $20/m; the east and west sides are to be bounded by rhododendrons, which cost $50/m. Find the dimensions of the garden that minimize the total cost of the fencing and shrubbery. The client is getting estimates from several designers, so you need to prove that your plan is guaranteed to result in the lowest cost (given those per-meter costs for installing the fencing and the plants). Accordingly, when you write up your plan include a careful description of how you find these dimensions and how you verify that these dimensions actually minimize the cost.

Having finished your plan for the garden to be enclosed by rhododendrons and wooden fencing, you realize that rectangular gardens with different ”boundary material” on the north and south than on the east and west have become popular. To simplify the design process, you want to create a template for these problems. The general situation is as follows. A 10,000 m^2 gardenis to be enclosed on the north and south by material A, which costs a dollars per meter, and on the east and west by material B, which costs b dollars per meter. Find the dimensions that minimize the total cost of enclosing the garden. (The dimensions will be in terms of the constants a and b.) As before, in order to prove to your clients that your plan is the best possible, include a careful description of how you find these dimensions and how you verify that these dimensions actually minimize the cost.

What does your template tell you about the dimensions of the garden when a = b? When a = 4b?When a = (1/4)b? Does it agree with the dimensions you found for the rhododendron/wooden fence-enclosed garden? What happens if the area of the garden is to be something other than10,000 m^2?

The audience for your paper is the client, but assume this client is someone who is familiar with calculus. Begin with a brief introduction as a reminder of the problem. Incorporate into your report a computer-drawn graph of the cost function used in the first part of the analysis (with costs $20/m and $50/m). You will need to experiment with the window dimensions in order to include the important details of the function. The graph should exhibit the domain of the function and the minimum point.

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