Mathematics Projects
First Semester Calculus

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This is always under construction.

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Projects available:

· An Introduction to Derivatives TeX version (You will also need the figures for page 1 and page 2 if you use the TeX file.) OR, Postscript version.

Project contributed by John Quintanilla.
This project is an introduction to derivatives. Its purpose is to give students an opportunity to play with the idea of limit of secant lines in order to determine instantaneous speed. It is designed to be given to students at the very beginning of the semester, before they have even seen limits or derivatives.
Project teaches and provides practice in

• Introduction to limits
• Introduction to derivatives
• Computation of instantaneous speed

· Limits of Polynomials. TeX version or Postscript version.

Project contributed by Neal Brand.
This project requires students to use the epsilon-delta definition of limit to prove that polynomial functions are continuous. Before attempting this project, students should have a good idea of what the definition of limit says. In particular, they should be able to prove that limits of specific quadratic polynomials are what they think they are.
Project teaches and provides practice in

• Epsilon--Delta definition of limit
• Proofs using several steps

· Numerical Integration. TeX version or Postscript version.

Project contributed by Neal Brand.
This project requires students to compute an integral numerically within a certain maximum error. They are also required to derive (with many hints) an error estimate for Riemann sums, the trapezoid rule, and Simpson's rule.
Project teaches and provides practice in

• Numerical integration using a computer or calculator
• Using an error estimate to establish the number of subintervals required
• Approximating functions with polynomials
• Deriving error estimates
  
• Derivatives without Limits. TeX version, Postscript version, PDF version.
  Project contributed by Neal Brand.
  This project requires students to compute the formula for the derivative of polynomials without using the limit concept.  Instead the derivative is defined in terms of double roots.
  Project teaches and provides practice in
• Factoring
• Connection between roots and factors of polynomials
• Geometry of derivatives
• Derivative as a linear approximation to the function
• Proof of a substantial theorem by first looking at special cases and using what is learned to do the general case
  
• Summation Formulas. PDF version.
  Project contributed by Neal Brand.
  This project requires students to derive the standard formulas for the sum of the r^th powers of the first n integers.  They also use this to derive the integral of xⁿ from 0 to 1.Project teaches and provides practice in
• Algebra
• Proof by induction
• Riemann sums
  
• A Chubby Chef's Polynomial.  PDF version.
  Project contribute by Neal Brand
  This project requires students to do careful reasoning about a class of polynomials in order to approximate a polynomial of degree n with one of degree n-1.  Project teaches and provides practice in
• Trigonometric identities
• Proof by induction
• Proof using Intermediate Value Theorem
• Basic properties about roots of polynomials
• Graphing with Mathematica (or other grapher)
  
• Integrating Polynomials.   PDF version
  This project requires students to integrate x^m for positive integers m by computing a limit of Riemann Sums with partitions not equally spaced.  Project teaches and offers practice in:
• Proving the sum for a finite geometric series
• Using nonequally spaced partitions to compute an integral
• How to compute a limit of Riemann sums when the norm of the partition is approaching 0 as opposed to n approaching infinity
  
  
• Tippe Tops. PDF version
  This project requires student to determine how deep to drill into a sphere in order to minimize the vertical location of the center of mass. This project teaches and provides practice in:
• Setting up integrals for functions defined piecewise.
• Computing the center of mass.
• Minimizing a function.
  
  
• Support Spindles. PDF version
  This project requires students to design a support spindle. Spindles were used to spin yarn before the invention of the spinning wheel and many people still use spindles to spin their own yarn. The project teaches and provides practice in:
• Deriving an integral formula based on limits of Riemann sums.
• Computing the moment of inertial for various standard solids.
• Design a spindle that is both attractive and has a moment of inertial that makes spinning efficient.

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