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Being a grad student I'm going to teach a whole class for the first time the coming summer and I'm looking for some motivational problems which I could use to introduce different topics. In other words, I would like to start a topic by giving an interesting problem and then motivating the study of that topic as something that will allow us to answer the problem.

The class is Calculus III and the syllabus for the class is roughly the following:

Linear algebra: Matrices, rank, determinant, eigenvalues, diagonalization.

Higher-order ODEs: Homogeneous/nonhomogeneous with constant coefficients, undetermined coefficients.

Systems of linear ODEs: Essentially constant coefficient stuff.

Series solutions of ODEs: Again, the most basic material.

Vector Calculus: up to divergence and Stokes thm.

A more exact syllabus can be found here:

http://hans.math.upenn.edu/ugrad/calc/m240/09aMath240Syllabus.pdf

For linear algebra the obvious choice is Google's PageRank of which there's quite a nice exposition given here:

http://www.rose-hulman.edu/~bryan/googleFinalVersionFixed.pdf

This problem is about as good as a motivating problem can possible be, since:

  1. It touches the students' daily lives.
  2. It requires precisely all the material covered in the linear algebra section of this class.
  3. It's a problem that on the surface might look like having nothing to do with the topic.

I'm curious if people on this board would know some other interesting problems that would motivate the other topics and make them seem relevant to students? Being an algebraist, I'm not too familiar with any interesting applications of ODE material.

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Have the kids derive the equation for the catenary. –  J. M. Apr 26 '12 at 16:26
    
@J.M. I think all of my friends from St. Louis have heard me lecture on that at least twice! +1 –  The Chaz 2.0 Apr 26 '12 at 16:40
    
In my opinion, linear algebra is "geometric" in nature. I like the question: - If you are submitting your picture to a online dating site, what kind of transformation would you apply to it? A friend said that in his case the transformation would be highly non-linear... I think the question is a good joke. Anyway, applying a linear transformation to a picture is a good way to get the felling on what a linear transformation is. –  André Caldas Apr 26 '12 at 17:16

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