Project 2: Derivatives

Instructions

Use Winplot and Google Docs to create a document with four problems on it. The problems should be similar to those in the book, but don't work out the indicated problems.

1. A problem similar to problem 2.2.3 with four graphs and four derivatives. Notes: This would be a good time to use a Winplot image size of 8.1 cm, then you can fit two graphs side by side. Make sure all graphs are centered at the origin so that the viewing window doesn't give away the answer. Hide the scale labels on the axes.
2. A problem similar to problem 2.5.55 that includes some questions that require the use of the addition/subtraction rule, product rule, quotient rule, and chain rule. Include 8 parts. You will probably also need to include more than 3 values of x, and you are certainly not limited to the values 1, 2, and 3 like problem 2.5.55. You can use any integer values for x, they do not need to be consecutive.
3. A problem similar to problems 2.6.19-24 and 2.6.29-32. Type the equation of the relation and include a graph in the question itself (instead of making one of the parts be "Graph the curve with the equation …" you could make the instructions say "Consider the curve with the equation …" and then include the graph). In the graph you make for the solution, include the tangent lines at the points you picked. Problems 19-24 and 29-32 include a lot of different things; what I'm mainly after are for you to (a) ask about the number of horizontal tangents and the coordinates where they occur and (b) pick a few points on the curve and find equations of tangent lines at those points. No more than one of them should be a horizontal tangent.
4. Answer this question: A thief steals a ladies purse and takes off running north at a rate of 7 feet per second. A former high school football cornerback is standing 200 feet due east of the lady, sees the robbery, and immediately takes off running towards the thief at 12 feet per second. The purse rescuer continually adjusts his direction so that he is running directly towards the attacker. For a visualization, visit the Pursuit Curve at MathWorld.

1. Create a Google Spreadsheet and share it with your group (and anyone with the link). Start with time t=0 seconds and use an increment of dt=0.5 seconds to create a spreadsheet that calculates the coordinates of the attacker and the rescuer at time t. Hint: you are essentially stringing local linear approximations together to recreate a curve. Insert a scatterplot to show the path followed by the rescuer. The rest of the questions should be answered in your main document.
2. How quickly is the distance between the thief and the hero changing at t=10 seconds.
3. How many seconds does the chase last?
4. How far does the rescuer travel while chasing the criminal?
5. Compare your answers to those given in the paper Pursuit Curves by Michael Lloyd, Ph.D.. Read the first paragraph in the section on the special case of pursuing a straight line target to get the definitions. In particular, k is the ratio of the fox's speed to the rabbit's speed. Then skip to the solution is given at the top of page 6 and the paragraph that follows (the paragraph incorrectly states k>1 means the rabbit is faster than the fox  -- it should be the fox is faster than the rabbit).

Additional Notes

• Add your names at the top of the document under the title.
• Include instructions with each of the problems; don’t just create a graph or describe the graph.
• Include an answer key at the end of the document for questions 1-3. Question 4 will be answered in the main part of the document.

To submit the assignment, click the Share button in Google Docs and change it so that anyone with the link can view the results. Then copy the share link and email it to the instructor at james@richland.edu with an appropriate subject and the names of the people in the group. This project will have two links to share: one for the document and one for the spreadsheet. Only one person per group needs to submit the project.

Winplot Tips

File Menu

• When appropriate, use Image size to create a square document. The measurements are in centimeters. While 16.5 cm will make it fill the entire page horizontally, it will also take a lot of space vertically. You may want something smaller like 11.7 or 8.1, which will allow two side-by-side images.
• Use Bitmap to clipboard to capture the image so you can paste it into Google Docs. The regular copy to clipboard doesn’t work.
• Be sure to Save your graphic so that you can re-edit later if you need to do so.

Equa Menu

• Use implicit to graph an implicit equation. Sometimes, implicit graphs have a problem displaying if you change the zoom or arrow around to change the viewing. A workaround is to edit the graph in the inventory and then change the color.
• The inventory has a derive button that will graph the derivative of the selected function. The graph button will show or hide a graph, so a good way to make matching graphs is to sketch the function and derivative to make sure both are going to fit reasonably well on in the viewing window and then show one while hiding the other when you copy and paste the graph.

View Menu

• Use Grid to control the visibility of the grid, axes, tick marks, and labels. I recommend using a rectangular grid with dotted lines in all four quadrants. Check the scale box to show the numeric labels on the axes.
• Use View to control the domain and range of the image.

Btns Menu

• Use Text to add labels to the graph. Use the right mouse button to add or edit text and the left mouse button to drag it and change its position.

Google Docs Tips

Google Docs

• Increase the font size on your document. The equations are the size of your normal font, but that may be too small to see the mathematical expressions while editing them. You can change the font size and then update the Normal text style to match. Alternatively, you can highlight the entire document and change the font size, but this may not apply to text added later.
• Click the Share button at the top right and change who has access so that anyone with the link can view the document. Then you can just submit the link.

Equations

• Use Insert > Equation to insert an equation
• Create the structure before you use it. For example, click on the fraction template before typing the numerator.
• There are some keyboard shortcuts, most notably the ^ and _ give you superscripts and subscripts.
• Matrices are not supported. That might not sound like a big deal, but the trick to displaying piecewise functions is to create a matrix to use as placeholders. In fact, multi-line equations are not supported either. Basically, you don’t want to use Google Docs for serious mathematical content, but it will allow for collaboration.