# Additional material

## Videos from Math 4D

Morten Nome, the lecturer for Math 4D (almost the same as Math 4N), made some interesting videos.

## Fourier series approximation

Fourier series approximation of the sawtooth wave Fourier approximation

## Wave Equation

Here are different animations of vibrating strings.

#### Normal modes

Fundamental mode Fundamental mode Second mode Second mode Third mode Third mode Fourth mode Fourth mode

#### Initial conditions (no initial velocity)

Quadratic function Quadratic function Triangular wave Triangular wave

#### Other videos

## Newton's method

You can create animations for Newton's method easily on Maple.

> with(Student[Calculus1]): > NewtonsMethod(f, x_0, view = [a .. b, DEFAULT], iterations = n, output = animation);

We encourage you to play with different functions and initial values x_0 to build your intuition.

Here is one example:

> with(Student[Calculus1]): > NewtonsMethod(x^3-5x^2+x+2, -2.5, view = [-3 .. 5, DEFAULT], iterations = 5, output = animation);

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## Gauss integration nodes and coefficients

Since we have not seen how to obtain the nodes (=zeros of the Legendre polynomial) and the coefficients in Gauss integration formulas, here is a Matlab code that computes them.