# Dr. Nixon

Nonlinear Waves, Optics, Asymptotic Analysis

# Mathematica

## Functions

Basic examples for creating functions, solving equations, and plotting curves in Mathematica.

## Limits

Automating the process of finding a tangent line and applying this to the 0th order Bessel function of the first kind. An interactive illustration of the definition of a limit.

## Derivatives

Basic code for taking the derivative. Interactive examples to illustrate the role of the first and second derivative on the shape of a curve. Generalizing the Product Rule.

## Approx.

Polynomial approximations. First found by matching the derivatives and then found again using an algebraic approach. An interactive illustration of the improvement of approximation as the number of terms is increased.

## Newton's Method

Newton's Method implemented in Mathematica with diagrams of its derivation from linear approximations. Error analysis for example problems.

## Mean Value Theorem

An interactive illustration of the Mean Value Theorem. Using the MVT to bound the error associated with a linear approximation.

## Numerical Integration

Basic code for calculating indefinite and definite integrals and basic code for calculating sums. Interactive examples for Riemann sums. Using Riemann sums to approximate areas.

## Arclength

Diagrams and code for using integration to find the length of a curve. The relationship between averages and integration.

## Differential Equations

Solving the differential equations governing a mass-spring system: simple harmonic motion, damped, and with a periodic forcing. Videos of the motion of the spring where the mass, spring constant, damping, or forcing can be adjusted.

## Sequences

Creating and plotting sequences in Mathematica by using Lists. Illustrations for the limit of infinitely long sequences.

## Series

Creating sequences of partial sums for a series and calculating infinite series. Examples include the Harmonic Series, the Basel Problem, and the Liebnitz Formula for Pi.

## Power Series

Interactive examples of the convergence of power series and the difference between functions with an infinite radius of convergence and those with a finite radius of convergence. Bonus: The series definition of the zeta function.