Відео

This video explains the concept of Least Squares regression. It provides a full proof of the Regression Line Formula. It derives the Total Error as the sum of the squares of the residuals. It shows the minimising of the Total Error with respect to a and b. It solves the simultaneous equations.

This video investigates the long-term probabilities created by Markov Chains - represented by Probability based Transition Matrices. It demonstrates that there are Equilibrium Probabilities towards which probabilities converge. These Equilibrium Probabilities represent the Invariant Points of the Matrix. Convergence is demonstrated using Invariant Line analysis. This video goes beyond the Furth...

This video introduces Markov Chains - Transition Matrices using Conditional Probabilities used to predict future events. The example used is whether tomorrow will be Fine or Wet using conditional probabilities based on whether today is Fine or Wet. The example shows the solution using a Probability Tree, and then converts the method to Matrices.

This video investigates long-term populations over time for different initial conditions. The example investigates the Adult and Juvenile Population over 1, 10 and 100 years. The initial population ratio is varied Adult:Juvenile, to show that the long-term ratio is independent of the initial ratio. The investigation shows that the long-term ratio follows the dominant Invariant Line for the Tran...

This video introduces Transitionary Matrices. The predict future populations from initial populations. This video predicts the population of Adult and Juvenile Meerkats 'year on year' from an initial population. This is a popular Further Maths topic.

This video shows how to find Invariant Lines using Mx = λx. The video finds the Invariant Lines for Stretches, Rotations and Reflections - and the special cases.

This video demonstrates how to find Invariant Point from 2x2 matrix equation: Mx = x. The video finds the invariants points for general stretches, rotations and reflections - along with the special cases.

This video deduces the general form of the reflection matrix. This is in terms of the angle the Lines of Reflection (mirror line) is from the horizontal axis.

This video proves the general solutions for combinations of: -Reflection then Rotation. -Rotation the Reflection. These both result in Reflections - but the reflection angles are different.

This video finds the general results to combining the same transformations: stretch stretch, rotation rotation, reflection reflection. Remember: transformation matrices operate right to left - as the vector will be on the right.

This video demonstrates using Euler's Method to create a numerical approximate solution for a 2nd Order Differential Equations. The method uses a Quadratic Fit to predict the next point - using the 2nd Differential from the Diff Eqn. The video demonstrates improving the accuracy of the results by reducing the size of the increment (increasing the number of iterations).

The video demonstrates how to use the Mid-Point method to find a numerical approximate solution to a 1st Order Differential Equation. It demonstrates that this method is more accurate than the Euler Method.

This video demonstrates Euler's method of finding a numerical approximation to a 1st Order Differential Equation. It demonstrates improvement by reducing the interval.

Video proving and using L'Hopital's Rule to find the limit of Division of Function f/g at x=a, where the values result in 0/0 or infinity/infinity - using the division of the Differentials. Two examples to demonstrate for sine/x and lnx/1/x. Extension shown where the gradients are also 0 such that 0/0 - then use the 2nd Differential. Note: method repeats for 0/0 to nth differentials.

LIMITS - Solving for 0/0 (using Taylor Series)

LIMITS - The Basics

LEIBNIZ THEOREM

DIFFERENTIAL EQUATIONS - APPROXIMATE SOLUTION WITH TAYLOR SERIES

POWER SERIES - Taylor Series

DIFFERENTIAL EQUATIONS - 2ND ORDER - REDUCE DIFFERENT DEGREES

DIFFERENTIAL EQNS - 2ND ORDER - REDUCE SAME DEGREE

DIFFERENTIAL EQNS 1ST ORDER - REDUCING BERNOULLI EQUATIONS

DIFFERENTIAL EQNS 1ST ORDER - REDUCING MIXED EQUATIONS

DIFFERENTIAL EQUATIONS 2ND ORDER - RESONANCE

DIFFERENTIAL EQUATIONS 2ND ORDER - DAMPING

NEGATIVE BINOMIAL DISTRIBUTION - PART 2

NEGATIVE BINOMIAL DISTRIBUTION - PART 1

BERNOULLI TRIALS

SYSTEMS OF DIFFERENTIAL EQUATIONS - REAL-LIFE APPLICATIONS