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: