I have instructed calculus I an II, each once, at the college level and would like to emphasize that math is not just about memorizing formulas and concepts for a test and that applied math is not a bunch of contrived word problems. I would like to encourage my students to do a a "final project" the next time I teach the course (or at least the next time I have a bit more power to set the curriculum). For those of you who have done this before, what types of questions have you asked? and what were the students solutions like? Did you find your projects were manageable enough with just the calculus they learned and some basic research about whatever topic they chose? The next time you teach calculus will you do it again? I'd like to have at least one or two theory and application choices, plus an option for them to choose their own topic (requiring my approval for this option).
2 Answers
I have done something like this with ordinary degree students. I will just show you what I have done it may give you some ideas.
I think the better students enjoyed the projects but some of them might have been too long.
The project titles were:
- Nowhere-Continuous Functions
- Intermediate Value Theorem
- Fixing Nasty Functions and making them Nice
- Continuity
- Leaving Cert Questions
- L’Hopital’s Rule
- Linear Algebra
- Differentiation
- Extrema of Functions of Several Variables with an Application to Statistics
- Riemann Sums
- Further Techniques of Trigonometric Integration
- The Natural Exponential Function as a Power Series and The Most Beautiful Formula in Mathematics
- The Area & Volume of familiar Shapes & Solids
- Euler’s Method of Numerical Solution of Differential Equations
- The Length of the Monza Circuit
- Dynamical Systems
You can find projects 1-9 here, projects 10-15 here and project 16 is question 3 in here.
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1$\begingroup$ This is great a good mix of pure and applied questions. $\endgroup$ Mar 14, 2014 at 18:11
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Here are some I have used that do not require precise measurements
Cooling of a hot cup of water. Take measurement every 5 minutes and deduce that the change is proportional to the average over 5 minutes. This fits in neatly with exponential functions, (don't tell them that it is a differential equation!)
Use numerical integration (Riemann sums) to find $$\int_0^1 \frac{1}{1+x^2} dx$$ Though I used this as part of a programming class. But they should be able to manually calculate this with say 10 divisions. Great way to emphasize Riemann sums.
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$\begingroup$ I like the idea of physical experiments, and I could see morphing something like (1) into a great final project. thanks! Maybe we should phrase (2) something like "Use the following integral to calculate the value of pi numerically. Provide a bound on the error for your estimate." That is a bit more open ended, and perhaps a bit more interesting way to phrase it, otherwise I imagine the students just sum up the rectangles and don't even think about the answer. Maybe I'm being a bit cynical here. $\endgroup$ Jan 9, 2014 at 4:06
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$\begingroup$ you may not want to let the cat out of the bag and mention $\pi$. It comes as a surprise to many of them (talk of Easter eggs). May be you can wink and nod and ask....do you recognize it? Some (not all) may get very excited and want to try 20 segments or more $\endgroup$ Jan 9, 2014 at 4:25
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$\begingroup$ Did a quick check...with 10 segments, the bounds are 3.04 and 3.24 and average is 3.14. That may be the time to bring in $\arctan$ into the picture. $\endgroup$ Jan 9, 2014 at 4:30