Second Semester Calculus

#### A Probability Computation. TeX version or Postscript version.

Project contributed by Neal Brand.

This project requires students to compute a certain probability. No previous knowledge of probability or counting techniques is assumed.

Project teaches and provides practice in- Basic counting
- Limits
- Riemann sums
- Improper integrals

#### Fibonacci Numbers. TeX version or Postscript version.

Project contributed by Neal Brand.

In this projects, students will derive the formula for the nth Fibonacci number.

Project teaches and provides practice in- Partial fraction decomposition
- Algebra of power series
- Deriving a power series
- Radius of convergence of a power series

#### Trigonometry without Geometry or Noncircular Trigonometry. TeX version or Postscript version.

Project contributed by Neal Brand.

This project gives an alternate definition to the trigonometric functions which does not involve geometry.

Project teaches and provides practice in- Defining functions in terms of integrals
- Inverse functions
- Fundemental theorem of calculus
- Deriving trigonometric identities based on alternate
definitions

#### The Sound of a Guitar. Postscript format or PDF format.

Project contributed by Neal Brand.

This project is an introduction to Fourier series. Students should be able to compute integrals and have at least a basic ideas of what a series is.

Project teaches and provides practice in- Computation of integrals
- Introduction to Fourier series
- Trigonometric identities
- Use of Mathematica or Maple

**Trigonometry using Differential Equations.**PDF format.Project contributed by Neal Brand.

This project gives an alternative definitions of the sine and cosine functions involving a differential equation. Students are not expected to have any background knowledge of differential equations.

Project teaches and provides practice in- Defining functions using differential equations
- Chain rule
- Geometry of second derivative
- Periodic functions
- Mean value theorem implications

**The Exponential Function (Using Differential Equations)**PDF format.

Project contributed by Neal Brand.

This project is an introduction to differential equations. Students can generally do this project as soon as they know the basic computational techniques for taking derivatives. The purpose is to give students a rigorous way to define the exponential function without first defining the logarithmic function.

Project teaches and provides practice in- Using uniqueness and existence theorems
- Induction
- Rules of differentiation

**The Exponetial Function (Power Series).**PDF format

Project contribute by Neal Brand

This project is an introduction to differential equations. Students can generally do this project as soon as they know the basic computational techniques for taking derivatives. The purpose is to give students a rigorous way to define the exponential function without first defining the logarithmic function.

Project teaches and provides practice in- Computing interval of confergence
- Induction
- Taylor expansions
- Computations involving power series