Introduction to Matrices - What is a Matrix?

In this video, we delve into the wonderful world of mathematical matrices.

Polar Conics Example 2 - Sketch r = 12/(6+2sinθ)

In this video, we'll identify the features that will allow us to sketch the equation r = 12/(2+6sinθ).

Polar Conics Example 1 - Sketch r = 3/(2+2cosθ)

In this video, we'll identify the features that will allow us to sketch the equation r = 3/(2+2cosθ).

Solving Linear Inequalities - Example -3(x - 2) ≥ 18

In this video, we solve the linear inequality -3(x - 2) ≥ 18 and plot the solution on a number line.

Multiplication of Complex Numbers

Generally, multiplication of complex numbers follows the exact same principles as multiplication in algebra. In this video I demonstrate 3 common examples of multiplication of complex numbers.

Solving Linear Inequalities - Example 2x < x - 1 ≤ 3x + 5

In this video, I show you how to solve for a combined inequality 2x < x - 1 ≤ 3x +5. The first step is to separate the inequality into 2 parts. So we have...

Polar Form of Conic Sections - Part 2

In this video, we discuss the variations of the polar form of conic sections, which we derived in the previous video as...

How to Solve Simultaneous Equations Graphically

There are several ways to solve for a linear system of simultaneous equations and the ones that we've explored already are the method of substitution and the method of elimination.

Another method, and it's my favourite method, is to solve simultaneous equations graphically.

Integral 3x^2+8/x^3+4x by Partial Fraction Decomposition

In this video, I demonstrate how to integrate the quotient (3x² + 8) / [x(x² + 4)].

Now, because we have a quotient of 2 polynomials, that we will have to perform partial fraction decomposition first to turn this integral into a form that we can integrate.

Proof that ∑2^(n-1) = 2^n - 1 with Mathematical Induction

In this video I demonstrate that the equation 1 + 2 + 2² + 2³ + ... + 2^n-1 = 2ⁿ - 1 for all positive integers using mathematical induction.

Polar Form of Conic Sections - Part 1

In part 1, we derive the equation for the polar form of conic sections. This is easily done with the geometric definition of conic sections where the locus of a point P moves in a plane, such that the ratio of the distance from it to a fixed point F (the focus), to the perpendicular distance from it to a straight line called the directrix is a constant called the eccentricity.

Derivatives of Polynomials by First Principles - Example 1

In this video, I demonstrate how to find the derivative of f(x) = x² - 8x + 2 using by First Principles.

The derivative is defined as:

The Multiplicative Inverse of a Complex Number

The multiplicative inverse of a complex number exists such that z∙z^-1 = 1. To find the inverse, let z = a+ib and z^-1 = c+id.

Now, when we multiply z and z^-1, we get:

The integral of (4x+3)/(x^2+1) - Version 2

In this video, I demonstrate how to integrate (or find the antiderivative of) the expression (4x+3)/(x²+1) by using a trigonometric substitution.

Integral of 1/(a^2+x^2)

The term 1/(a² + x²) is purely algebraic. However, its integral is trigonometric. When you look this up in a table of integrals, you'll find that:

Sketching Polynomials - Part 2 of 3

In this video, I demonstrate how to find the information we need to be able to sketch or roughly graph the function:

f(x) = 3x^4 + 4x^3 - 12x^2

The first step is to find the stationary or critical points.

How to Sketch a Parabola - Example 2 (y = x^2 - 4x - 12)

In this video, we graph the trinomial / quadratic function y = x² - 4x - 12 by finding its concavity, y-intercept, using the turning point formulas and the quadratic formula for the x-intercepts. Remember, there are only 4 pieces of information you need to accurately sketch a parabola:

Equation for Hyperbolas Translated from Origin

The Cartesian form of the hyperbola not centred about the origin, but rather centred at the point (p, q) is similar to the standard form, however we must incorporate the coordinates of the centre point into the equation.

Cutting Tape - Application of Geometric Series

In this video I demonstrate how to use formulas for the summation of a geometric series to calculate the length of tape is required if I was to cut it first by 10cm, then 96% of that, then 96% of the previous cut and so on.

I first calculate how much tape is required for 10 cuts, then I calculate how much is required for any number of cuts.

So to work out how much tape is required, we can add up the first few cuts to see if there is a pattern or progression.

Complex Number Plane Geometry Problem - Example 1

Given that a complex number A = 1 + i, we need to find the complex number B that lies in the 2nd quadrant, such that on the Argand Diagram, the points O, B and A form an equilateral triangle (where O is the origin).

To make sense of what this is saying, we need to first draw a diagram - an Argand Plane. We first note that A has the coordinates (1,1); O has the coordinates (0,0) and let's give B the coordinates (x,y), which we have to solve.

We then use the polar form of complex number multiplication to find a point B(x,y) that forms an equilateral triangle with the point A(1,1) and the origin O(0,0).