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).

North-South Opening Hyperbolas

The standard equation of a hyperbola describes one that is centred about the origin, symmetrical about the y-axis and opens from the vertices in an east-west direction. So what about hyperbolas that open in the north-south direction?

Well, the equation of north-south hyperbolas is very similar to the standard - just the variables x and y are swapped around. Hence we have...

Problem Solving using Simultaneous Equations - Finding a Person's Age

In this video, I demonstrate how to use modelling and application of simultaneous equations to solve a simple, real world problem. The question is Cindy's mother is 8 times as old as Cindy is. In 3 years time, she will be 5 times Cindy's age. Use simultaneous equations to determine their present ages.

Problem solving is much easier when you have properly defined variables and well set up equations.

Hyperbolas - Example 1: Sketching x^2/16 - y^2/48 = 1

Previously, we've learned how to find all of the features of the standard hyperbola

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1

In this video, we use this knowledge to sketch x²/16 - y²/48 = 1.

Firstly, we convert the equation to standard form:

How to Integrate ∫sin^3(x)cos^4(x)dx

In this video, I demonstrate how to integrate sin³xcos⁴x by reserving a factor of sin(x) and converting the integrand into powers of cos(x). This then turns this integral into a simple power integral.

The first step is to write sin³(x) as sin(x)sin²(x).

Then by the Pythagorean identity, sin²(x) = 1 - cos²(x). So the integral becomes: