MathsCasts

Introduces the idea of half-range Fourier series and addresses the question of why we use them.

Demonstrates that symmetry of functions helps us to shorten the calculation of the integrals needed for the Fourier coefficients.

Deals with the details of calculating Fourier series coefficients for a function.

We solve simultaneously the equations of a plane and a cone and show that the intersections are circles, parabolas, ellipses, hyperbolas, straight lines or just the origin.

Sketch of y = 2(x+1)^2-3.

Brief introduction to why we use Laplace transforms, with mention of tables.

Brief and simple introduction to the first shift theorem for Laplace transforms.

Detailed discussion of using Laplace transform to solve a 1st order ODE.

We solve a non-homogeneous 2nd order constant coefficient ODE with boundary conditions using the Laplace transform method. A check is applied to the solution. The process of solving is then repeated with a deliberate error and it is demonstrated that not all checks can identify the error. We identify the feature that boundary conditions can be more effective than the original ODE in identifying errors.

Sketch of y = 2(x+2)(x-2)(x+3).

Demonstration of using Laplace transform to solve a 2nd order, linear, non-homogeneous, constant coefficient DE with boundary conditions and with right hand side appearing in the CF.

Standard pair of linear, coupled 1st order DE's solved by Laplace transform method.

In this recording we look at an example of differentiation using the inverse function rule.

Sketch of y =(x+1)^3-2.

The quotient rule for differentiation is proved from first principles.

Demonstrates that symmetry of functions helps us to shorten the calculation of the integrals needed for the Fourier coefficients.

Simple demonstration of how to obtain a series to evaluate approximations to pi, using a specific and quite simple Fourier series.

The integral of sin(ax) sin(bx) is performed using integration by parts.

The product rule for differentiation is proved from first principles.

The chain rule for differentiation is proved from first principles. Use is made of the small difference formula, which is discussed first.

Integration by parts is used to integrate the product of two polynomials.

It is demonstrated how to perform differentiation using the chain rule, side-stepping the use of new function names such as u and v.

We calculate the Laplace transform of a rectified sine wave shape.

We find the Laplace transform of a triangle wave as an example of a periodic function.

The integral of exp(-x^2) from 0 to infinity is evaluated using a double integral technique.

We discuss the details involved in differentiating an integral, examine the notation used and outline the proof that the result is just the integrand.

Laplace transform is used to solve a second example of a 2nd order, constant coefficient ODE with boundary conditions.

We calculate explicitly the Laplace transform of e^(-t^2) and demonstrate the need for the error function erf(s/2).

We exploit the analogy between complex numbers and 2 by 2 matrices to perform complex number reciprocals and division.

We show how to antidifferentiate sec(x) and hence gain insight also how to antidifferentiate cosec(x).

We revise the method of logarithmic diferentiation applied to the function y = x^x then view the result in an unconventional way.

In this recording, we look at the process of integrating rational functions of the form a divided by (bx+c)^m.

In this recording we look at integrating proper rational functions.

In this recording, we look at integrating rational functions where both the numerator and the denominator are linear functions.

Demonstrates integrating an improper rational function, where first long division must be used to rewrite it as a polynomial plus a proper rational function. The proper rational function that results is then re-written as partial fractions, and it is then possible to find the integral.

For a specific example, this recording demonstrates the process involved in multiplying two binary numbers using long mutliplication.

In this recording we calculate j to the power of j by first converting j into exponential polar form. We then also see that there are an infinite number of values for j to the power of j.

This recording demonstrates how to integrate rational functions where the denominator is a quadratic function and the numerator is a linear function which is a constant multiple of the derivative of the denominator. Integration by substitution is used.

We use the Cayley-Hamilton method to find all four square roots of a given 2 by 2 matrix.

We show how the Cayley-Hamilton theorem can be used to give an expression for any power of a 2 by 2 matrix. The method is used to find the 4th power of a given matrix.

We investigate linear independence for a set of three vectors in four dimensions and demonstrate how to decide on their (in)dependence using Gaussian elimination.

This recording demonstrates the general method for solving an equation of form sqrt(ax^2+bx+c) = cx+d and then illustrates this with an example, including checking solutions that were generated by squaring both sides to see which ones satisfy the original equation.