Outline

# GeoGebra Apps for A-Level Further Pure Mathematics

## Table of Contents

### Matrices and transformations:

- Matrix representation for a rotation θ degrees anticlockwise about (0, 0)
- The matrix representation for a reflection in the line y = mx.
- The matrix representation of a shear.
- Matrix representation of a reflection in 3D.
- Finding invariant lines under a transformation given by a matrix.
- A line of invariant points
- Eigenvalues and eigenvectors
- Eigenvalues and eigenvectors
- Eigenvalues and eigenvectors in 3D
- Repeated Transformations of a matrix

### Introduction to complex numbers:

### Roots of polynomials:

### Complex numbers and geometry:

- Writing a complex number in modulus-argument form.
- Multiplication and division of complex numbers in modulus-argument form.
- Loci in the Argand diagram introduction.
- Loci of the form |z - a| =r.
- Loci of the form |z - a| < r.
- Loci of the form |z - a| < r.
- Drawing loci of the form arg(z - a) = θ.
- Inequalities on an Argand diagram of form β ≤ arg(z - a) ≤ α.
- Loci of Iz-z_1I=Iz-z_2I on an argand diagram.
- Loci of the form |z - z_1| > |z - z_2|.
- Combination of loci on an Argand diagram.
- Complex roots of a quadratic

### Matrices and their inverses:

### Vectors and 3D space:

- The scalar product proof
- Scalar product 3d
- How many points are required to define a plane?
- The vector equation of the plane
- The normal vector equation of the plane
- Converting the vector to the Cartesian plane equation
- Finding the Cartesian equation of a plane through 3 points
- A plane containing a point and a line
- Determining whether four points lie in a plane
- The angle between two planes.
- The intersection of two planes
- The intersection of three planes.
- The intersection of 3 planes at a point
- The intersection of three planes.
- Intersection of 3 planes using the inverse of a matrix
- The intersection of three planes using the inverse matrix
- Intersection of three planes by row elimination

### Vectors 1:

- The cross product or vector product
- The vector equation of the line in 2-dimensions
- The vector equation of a line in 3D.
- The point of intersection and angle between two lines in 3D
- Parallel, perpendicular and skew lines in 3D
- The distance between parallel lines in space
- The shortest distance between skew lines
- The intersection of a line and a plane
- The intersection of two planes
- The distance between two parallel planes
- The shortest distance between a line and a parallel plane.
- Intersection of 3 planes by row elimination including non-unique solutions

### Further calculus:

### Sketching polar curves:

- An introduction to polar coordinates
- Copy of Sketching curves with polar equations introduction
- Sketching polar curves: Circle centre (0, 1), radius 1
- Sketching polar curves: Cardioid
- Sketching polar curves: A four-leaved rose
- Sketching polar curves: Parabola
- Sketching polar curves: Limaçon
- Sketching polar curves: An eight-leaved rose
- Sketching polar curves: Lemniscate
- Sketching polar curves: Advanced
- Sketching polar curves: Conchoid
- Sketching polar curves: Spiral
- Finding the area enclosed by a polar curve

### Maclaurin series:

### Hyperbolic functions:

### Applications of integration:

- Volume of revolution introduction.
- A volume of revolution around the x-axis
- Volume of revolution of the area bounded by two functions
- A volume of revolution around the y-axis
- Area and volume using parametric curves.
- Gabriel's Horn
- Gabriel's Horn 01
- The arc length of a curve.
- The area of a surface of revolution.

### Complex numbers - De Moivre's theorem:

### Extra pure Mathematics: Multivariable calculus:

- The domain of the graph for a surface.
- Sketching contour curves of a function of two variables 01.
- Sketching contour curves of a function of two variables 02.
- The geometrical meaning of partial derivatives.
- Finding stationary points of surfaces 01
- Finding stationary points of surfaces 02.
- Tangent planes and normals to a point introduction.
- The tangent plane and normal line to a surface 01.
- The tangent plane and normal line to a surface 02.
- Cylindrical coordinates.
- Spherical coordinates
- Triple integral to determine volume
- Using a triple integral to determine volume

### 1st order differential equations

### Euler's method

- Euler's method for f'(t, x) = x-2t, x(0) = 1
- Euler's method for f'(t, x)= xt, x(0) = 2
- Euler's method for f'(t, x)=x/(2sqrt(t + x)), x(0.5) = 1
- Euler's method for f'(t, x)=(4 - t)/(t + x)
- Euler's method for f'(t, x)=1/xt, x(1)=1
- Euler's method for f'(t, x)=xt/(x^2+2), x(1)=2
- Euler's method for f'(t, x)=1/lnx, x(1)=1.2
- Numerical solution of a third-order differential equation 01
- Numerical solution of a third-order differential equation 02
- Numerical solution of a second-order differential equation

### 2nd order differential equations

- Second order differential equations damped oscillations
- Non-homogeneous 2nd order differential equations
- Modelling a bungee jump with a 2nd order differential equation.
- The steady state response of a damped second-order system
- Solving a 2nd-order differential equation using Laplace transformations
- Graphical solution of a 2nd-order differential equation
- A system of first-order differential equations

### Fourier series

- Basic sine wave
- Integrals of periodic functions
- Square wave using Fourier terms
- Sawtooth wave using Fourier terms
- Triangular wave using Fourier terms
- Fourier series for a periodic function 01
- Fourier series for a periodic function 02
- Fourier series for a periodic function 03
- Fourier series for a periodic function 04
- Fourier series problem
- Fourier series for a piecewise periodic function 01
- Fourier series for a piecewise periodic function 02
- Fourier series for a piecewise periodic function 03
- Fourier series for a piecewise periodic function 04
- Fourier series for a piecewise periodic function 05
- Fourier series for a piecewise function 01
- Fourier series for a piecewise function 02
- Fourier series for a piecewise function 03
- Fourier series for a piecewise function 04
- Fourier series for a piecewise function 05
- Fourier series for a piecewise function 06
- An even function
- An odd function
- Fourier half-range series
- Significance of the constant term in Fourier series
- How to change the period of a sine graph
- How to change the period of a cosine graph
- Fourier series with an arbitrary period of 2
- Fourier series with an arbitrary period of 4
- Fourier half-range cosine or sine series