Calc III modeling projects - part 2

The blog Continuous everywhere and differentiable nowhere, (CEDN) by a high school math teacher in Brooklyn, New York, has a list of Calc III projects.  Here are the ones I felt were modeling projects.  See the blog site for more details, including the overall assignment sheet.

  • Volume of an n-dimensional sphere:  Define a sphere in 2D, 3D, 4D, etc.  Find the formula for the volume of an n-dimensional sphere.  Strangely, as the dimension increases, the volume of given radii decreases.  This is related to the problem with data modeling in multidimensions when the measure is a distance.  The higher the dimension, a larger percentage of points are more than a fixed distance from a given point.  Study this weirdness.  Resource: Rogawski, Section 15.3.
  • Fluid dynamics and hurricane modeling:  This blogger recommends studying material from Anton, pp. 1883-7, on fluid dynamics and how that study can be used to model hurricanes.
  • Lissajous Curves:  The prospect is to study Lissajous curves in 2D, and then consider them in 3D.  Maybe this would be a good 3D printing project.
  • River navigation problem:  The blogger describes a navigation problem where the boat has limited maneuvering capabilities and there are safe and unsafe places to land on the shore of a river.  It sounds like an interesting problem that can be modified to consider more variables, such as the difference in current along the shore versus the center of a river.
  • Maxwell's Equations:  Research Maxwell’s Equations and explain them. Resource: Daniel Fleisch’s A Student’s Guide to Maxwell’s Equations.
  • Ideal Gas Law:  Investigate the Ideal Gas Law (PV = nRT) focusing on what the 3D plotting of P vs. V vs. T looks like, given constants n and R.  The CEDN blog has a spreadsheet that can help, more details for the project, and some online resources. Murphy's note:  Students often have difficulty interpreting plots such as found on the CEDN website, and so I am particularly interested in this project.
  • Mathematica Demonstrations:   If you know a thing or two about programming or are willing to learn some basic Mathematica, explore the Wolfram Demonstration Projects for Multivariable Calculus here. Find a topic in Multivariable Calculus that hasn’t been illustrated yet, or one that might be poorly illustrated, and create your own Demonstration. 
  • Five Intersecting Tetrahedra:  This is Activity 11 of Thomas Hull's "Project Origami."  There is a problem in the activity that asks the reader to solve a multivariable optimization problem.

The CEDN blogger mentions harmonographs, which are covered in a previous post in this blog.

Again, these ideas came from the blog Continuous everywhere and differentiable nowhere, and more details, links, and resources can be found there.  

Comments

Post a Comment

Popular posts from this blog

3D printing for multivariable calculus

I keep running across references to 3D printing as projects in multivariable calculus.  On the one hand, this is a modeling project in that the code has to be written in order to produce a physical printout.  OTOH, it may be that it has all be done before. Regardless, here are the references I have found in my travels.j John Zweck , UT Dallas, has code and images of 3D printouts of sample curves. The Harvey Mudd multivariable calculus project page has references to programs for 3D printing. Triply Periodic Minimal Surfaces has a seemingly unending list of surfaces to be studied and printed.
Read more

Calc II - Integral Calculus - Projects - Part 1

After having taught Calc I and Calc III recently, now I find myself teaching Calc II, integral calculus, in the fall.  So again I am looking for projects. These ideas are on Dave Martin's Real teaching means real learning blog. A project that combines creating images on GeoGebra with functions and then calculating the areas created.  I think students would like the GeoGebra part, but there is little motivation for the need of calculating the areas.  Another page talking about the same project with some nice pics. Can Kobe Bryant jump over a moving car ?  A video shows him doing it.  Can we use calculus, including integration, to investigate this? Explain mathematically explaining everything you know about an irregularly shaped symmetrical object such as a bottle or a vase. Include detailed drawings and calculations. Create a video of an object in free-fall .  You must throw your object, and calculate the velocity of the object when it left your hand.  Appropriate measure
Read more