Final Project

Objective

The goal of this project is to give you chance to explore a topic of your interest that has some connection to calculus, and to share what you learn with your peers. It can be something purely mathematical, or it can involve an application of calculus to an area of your choice.

Description

You’ll form groups of 3-5 and collectively choose a topic you want to work on. Your group and topic will need to be approved by me: one person in each proposed group should send me an email letting me know who you’ll be working with and what you’ll be working on, by 11:59pm on the third Tuesday. I’ll want each group to have a different topic, so if there’s one you’re very excited about, you should send me your proposal early.

For the last day of class, you’ll prepare a handout and a digital presentation about your topic.

• The handout should be one typewritten page (front and back). It should give an introduction to the topic, and incorporate a detailed and substantial proof and/or example. It need not be entirely original (ie, you don’t need to prove a brand new theorem about calculus), but it should be fully explained in your own words. If you can incorporate the results of some computer calculations, that would be awesome! The handout should also include a list of works that you referenced, so that someone who wants to learn more about your topic knows where to start.

  Bring some copies of your handout to the presentation (at least one for me, and then a few more in case any of your peers would like to take one).

• The presentation should convey the main ideas of your topic and some insightful examples. It should take about 10 minutes. We’ll use the big data visualization wall in the library, and you’re encouraged to be creative about the format (slides, digital poster, videos…). Pictures are always nice, if at all possible, but remember that, if you don’t make a picture yourself, you have to attribute it to whoever did. All of the nitty gritty mathematical details do not need to appear during the presentation (that’s what the handout is for), but make sure you do incorporate some mathematical content at least at a high level. You should be prepared to answer any questions me or your peers might have, after your presentation is over.

Keep in mind that the intended audience is one of your classmates: someone who knows something about calculus broadly speaking (specifically, the topics we’ve discussed in our class), but may not know anything specific about your topic.

Grading

Your project score will be based on the following criteria.

• Content (out of 4 points). There should be correct and substantial mathematical content involved (during the presentation, or on the handout).
• Clarity (out of 3 points). The topic should be clearly explained and logically organized, in a way that is appropriate for the intended audience.
• Participation (out of 3 points). You’ll get full credit on participation if it is clear that everyone in the group understands the topic at the same level (in particular, everyone should have a substantial role in the presentation).

Possible Topics

There are so many possibilities! I encourage you to be creative. I only ask that it not be something that we’ve spent a significant amount of time discussing as a class.

If you’re struggling to find things, here’s a short list of things you might look into. Some of these are discussed briefly in exercises in our textbook. Do remember, though, that solving a single exercise is not a project. The exercise is meant to serve as a springboard for you to do your own research and flesh out a full project. For example, you might decide to look into the derivation of an equation, or find some data that justifies a model, or… You can be creative about what direction you go!

If you like physics, thermodynamics, astronomy, chemistry…

• Newton’s Law of Cooling.
• The Kirchoff, Fresnel, and Fraunhofer diffraction equations.
• The relationship between Einstein’s special relativity and Newtonian mechanics.
• The Clausius-Clapeyron equation.
• The Stefan-Boltzmann equation.
• Snell’s law of refraction.
• The Hagen-Poiseuille equation in fluid dynamics.
• The Beer-Lambert law in spectroscopy.

If you like biology, biochemistry…

• The Michaelis-Menten equation.
• The von Bertalanffy growth equation.
• The Hagen-Poiseuille equation in fluid dynamics.
• The Gompertz function and its applications to the growth of tumors.

If you like psychology…

• Stevens’ Power Law.

If you like computers…

• Sigmoidal correction in image processing.
• Euler’s method for solving differential equations.
• Algorithms for numerical integration.

If you like economics…

• Elasticity of demand.
• Annuities.
• Monopoly pricing.

If you like sports…

• The Pythagorean theorem of baseball (also called Pythagorean expectation).

If you really like math…

• The Ham Sandwich Theorem.
• Formal definition of a limit.