First Semester Calculus

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

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

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 x^{n}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.