Simple Harmonic Motion is a specific type of periodic motion where is any motion whose displacement-time equation, apart from the constants, is a single sine and/or cosine function. Of particular importance is a certain kind of rectilinear motion known as Simple Harmonic Motion. A particle is said to be undergoing in simple harmonic motion with […]

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# How to Find the Number of Roots by Stationary Points and Deduce for an Inverse Function Explained Quickly

The stationary points can be found by solving the derivatives to zero. Some cubic equations have one root if the stationary points are in the same sign, such as two stationary points are either above the x-axis or below the x-axis. In these cases, the cubic graph cuts the x-axis only once, so the cubic […]

# The Trajectory of Projectile Motion on an Inclined Plane with Maximum Range Explained

Projectile motion on an inclined plane is one variation of projectile motion and the trajectory can be used to determine the position of the landing object on the included plane. We will discuss; path of the trajectory of the projectile range of the projectile maximum range initial direction is perpendicular to the direction at which […]

# AREA of the Shapes Formula Explained with Clear Examples and 9 Comprehensive Video Lessons

What is the Area of the Shapes? A shape is defined as the figure surrounded by the closed boundary in geometry. The closed boundary consists of lines, curves and points. The amount of space inside the boundary that expresses the extent of a two-dimensional shape is defined as an area. The area is measured in […]

# Read This Simple Work for Definite Integrals by U-SUBSTITUTION Method Explained in 3 Distinct Examples and Video Lessons

U-substitution in definite integrals is very similar to the method in indefinite integrals, while the initial bounds (usually x-values) need to be changed to the corresponding u-values for upper and lower limits. You need to account for the limits of the integration. Alternatively, you can integrate the integral expressions using u-substitutions, then change u-expressions back […]

# Probability with Venn Diagrams – Ultimate Guide

It is quite often to visualise multiple events using Venn diagrams to find probabilities. Representing Data in Venn Diagrams There are 9 types of Venn diagrams for holding two events when they overlap each other. Let’s assume there are two elective subjects, Arts and Biology, in a certain school. \( P(A) \) The probability that […]

# Proof of Sum of Geometric Series by Mathematical Induction

Considerations of Sum of Geometric Series The sum of geometric series is defined using r, the common ratio and n, the number of terms. The common could be any real numbers with some exceptions; the common ratio is 1 and 0. If the common ratio is 1, the series becomes the sum of constant numbers, […]

# Easy Method of Integration by Substitution: U-Substitution

The substitution method of integration is useful when an integral contains some function and its derivative. Set a replacement letter, say u mostly (sometimes w) and in this case, and replace all letters x by u for getting the integration done easier. Once integrate the u-integral in terms of u, replace x-written expressions to get […]

# Proving Sum of Consecutive Cubes Formula

The sum of the first n consecutive cubes is equal to the square of the sum of the first n numbers. This post explains how to analyse the pattern of the sum of consecutive cubes and the square of the sum of the first n numbers, derive the sum formula and prove the formula using […]

# Divisibility Proof by Mathematical Induction: Sum of Three Consecutive Cubes

There are many ways to prove “The sum of three consecutive cubes is always divisible by nine”. This post explains how to determine the pattern of the sum of three consecutive cubes by listing the first few number series and to see how they relate to each other. Having some examples showing the sum of […]