Thanks again for all of your fantastic work!

http://demonstrations.wolfram.com/TheTangentLineProblem/, Sam and Mike L.

http://demonstrations.wolfram.com/HyperbolicQuadrilateral/, Ron

http://demonstrations.wolfram.com/HyperbolicTriangle/, Ron and Mike R.

http://demonstrations.wolfram.com/GraphingSystemsOfInequalities/, Amber and Tori

http://demonstrations.wolfram.com/SummingAnArithmeticProgression/, Mike R.

http://demonstrations.wolfram.com/ConicSectionsEquationsAndGraphs/, Kathy, Kelly and Jenn

http://demonstrations.wolfram.com/BicycleGearRatiosAndMetersOfDevelopment/, Ivan and Tom

http://demonstrations.wolfram.com/SlopeAndLinearEquationsOfLinesThroughPoints/, Tom

http://demonstrations.wolfram.com/MultipleFerrisWheels/, Ben and Charles

http://demonstrations.wolfram.com/AreaOfATriangleInThePoincareDisk/, Steven

In any case, I just wanted to thank you for a great summer program and wish you an excellent school year.

1. A copy of your PowerPoint presentation (if you are using one). If you are using a presentation made in Mathematica, please export it to .pdf and submit that.
2. A .pdf export of your Mathematica Demonstration file.

To export a .pdf in Mathematica, please select File->Save As… in Mathematica with your file open and then change the format to “PDF Document (.pdf)”.

On Thursday, we examined the process of creating a Demonstration file which would later be uploaded to the Demonstration Project authoring page.

Next week, we will:

1. Finish our demonstrations.
2. Submit our demonstrations to Wolfram.
3. Finish our presentations for the Symposium on August 6.
4. Practice our presentations.
5. Complete our “resumés” and submit them.
6. Complete additional projects as necessary.
7. Debrief and pre-evaluate.
8. Have a BBQ on Friday afternoon.

Here are a couple maps showing first the route from ND to St. Pat’s Park (our proposed location) and second once you are in the park the way to the parking lot. With either map, you can click on the link immediately below the map which will take you to a full-on map that also has step by step directions.

1. How do you create a demonstration?

   I promise we will cover that tomorrow.  I have been meaning to discuss this for the past three days.

2. At the presentations on August 6, should we be doing Power Point presentations using screenshots taken from our demonstrations along with explanatory blurbs, or should we doing the actual demonstrations themselves?

   I expect that you will introduce your projects using a PowerPoint presentation, then switch to the demonstration to complete the presentation going back only if necessary.  That being said, it is also possible that you could develop the presentation in Mathematica (we saw videos about that on Day One).  Or maybe you will think of some other compelling way to present the material you want to present.  But remember, you only have 10-12 minutes (at least that is what I was told by Nevin Longnecker).

3. If we don’t do Power Point presentations on the 6th, what do the Powers That Be want to be turned in to be posted on the web – a Power Point version or the demonstrations themselves?

   I talked to Alex Hahn about this back in March and to Nevin Longnecker last Thursday.  I want you to submit your demonstration as your final project.  However, I also want you to submit a .pdf file of your demonstration along with it (so that it can be printed and included in printed material easily).  My understanding is that this will be accepted as constituting your project.

If you have any other procedural questions about the completion of the project, feel free to email me or post them as comments on this post.

An artist's rendering of a multiple ferris wheel

1. Description of Project
This project will model the rotation of a double Ferris wheel, i.e. Ferris wheels within a Ferris wheel, both as a Cartesian graph and as parametric graphs.
The Cartesian graph will have circles (representing the Ferris wheels), and points (representing the riders). The parametric graph will be a sinusoid representing the vertical position in relation to time. Through animation of the graphs, a “ride” can be tracked in two ways: as a path on the Cartesian graph and a height off the ground on the parametric graph.

2. Description of Concept
The position of a point on the ride is calculated using the sum of sinusoidal functions. The movement along the large wheel is modeled by the sinusoidal function with the larger period; the movement along any of the small wheels is modeled by the sinusoidal function the smaller period.

3. Description of Method
The demonstration project involves a graphic of multiple circles for the wheels and lines for the spokes. The main implementation consists of using a θ value to rotate the wheels/spokes; also, a bar of checkboxes can display the trail(s) of a rider on each wheel. In doing so, Mathematica constructs included Manipulate[], Show[], Graphics[], Circle[], Line[], ParametricPlot[], and Checkbox.

4. Description of Goal
First, students will be able to follow the graph representing the path of a ride. Second, students will be able to follow a graph of the height of the rider over time. Third, students will be able to relate the two graphs of a trail or path of a rider with the height-time graph. Finally, students will be able to relate an associated sum of sinusoids with a ride.

5. References
Dr. Steven Broad, University of Notre Dame
http://modstothemax.net/community/index.php?topic=16171.0

We also spent some time looking at models of hyperbolic space which were made in crochet.

Tomorrow, we will meet again in DeBartolo 331.  According to the schedule, we have this room every day for the rest of our program.  We will continue to work on proposals, refining them with a few last edits.  If you have changes you would like to make to what is posted on the website, please email me the changes.  It seems this would be simplest at the moment.  Please look to see if there are any comments on your groups’ proposal.

We will also review the Demonstration file.  Thus far, we have used Mathematica notebooks to produce “demonstrations.” The official demonstration must be submitted in a Mathematica Demonstration file, which is slightly different.

Approximating the average value of a function can be done graphically or with a table of values. The value approach illustrates the use of a limit, while the graphical approach suggests why a definite integral leads to the exact average. This Mathematica demonstration displays both methods of finding the average value of a function and illustrates the correlation between the number of partitions taken to the accuracy of the average.

In Mathematica, building a table of approximations is easy to obtain using a FOR loop. Once the function, the interval, and the number of partitions to be assessed are specified, approximations can be taken from the left end, the right end, or the midpoint of each partition. The location of the point in the interval can even be chosen at random. These approximations would then be used to calculate the average of the function. Mathematica can be used to find the exact average using a definite integral as well.

Both the table and graphical approaches are to be displayed on the same screen. Mathematica will graph the sine function from 0 to π with the PLOT command. (We chose a periodic function so that the interval will have meaningful endpoints, rather than one chosen arbitrarily.) The number of partitions n will be chosen by the user through the MANIPULATE command. A sample point in each interval is displayed using POINT, and a vertical line segment from that point to the sine function is drawn using LINE. At the point of intersection, the co-ordinates will be displayed using TEXT. A FOR loop will total the sample y values. That total will then be divided by the number of partitions, and the obtained quotient will be the approximate average for the number of partitions specified. Below the graph will be printed the number of partitions, the approximation, and the exact answer (calculated by Mathematica using an integral). As the user increases n, the generated results will form a table which will suggest that increasing the number of partitions increases that accuracy of the approximation. In an advanced version, the user could choose whether the sample point comes from the left end of the interval, the right end, the midpoint, or even a randomly selected point.

This exercise will introduce Calculus students to concepts such as successive approximations, the technique of partial sums, limits at infinity, the structure of a Riemann Sum, and a visual relation between area and average value. It is designed to be done as early as the second day of a Calculus course. Given a specific function and interval, students can be assigned to approximate the average value of the function by hand using several increasing values for the number of intervals.

References

Indiana State Standards

http://dc.doe.in.gov/Standards/AcademicStandards/StandardSearch.aspx

C.1.1 Understand the concept of limit and estimate limits from graphs and tables of values.

C.1.6 Find limits at infinity.

C.4.2 Calculate the values of Riemann Sums over equal subdivisions using left, right, and midpoint evaluation points.

C.4.3 Interpret a definite integral as a limit of Riemann Sums.

C.5.5 Use definite integrals to find the average value of a function over a closed interval.

Web Resources

http://www.ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=ma&chap_sec=09.4&page=theory

http://prepcalcabb0607.blogspot.com/2007/02/65-average-value-of-function.html

www.sheltonstate.edu/userfiles/File/faculty/…/ch8_2average_val.ppt

http://mathdl.maa.org/mathDL/47/?pa=content&sa=viewDocument&nodeId=2639

http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/averages.html

http://mathcentral.uregina.ca/QQ/database/QQ.09.00/esther2.html

http://archives.math.utk.edu/visual.calculus/5/average.1/index.html

www.math.ucla.edu/~tat/MicroTeach/average.ppt

http://www.lightandmatter.com/calc/calc.pdf

http://www.tutorvista.com/content/math/calculus/definite-integrals/definite-integral-limit-sum.php