By Swinburne University of Technology
MathsCasts
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.
Deals with the details of calculating Fourier series coefficients for a function.
Demonstrates that symmetry of functions helps us to shorten the calculation of the integrals needed for the Fourier coefficients.
Introduces the idea of half-range Fourier series and addresses the question of why we use them.
Sketch of y =(x+1)^3-2.
In this recording we look at an example of differentiation using the inverse function rule.
Standard pair of linear, coupled 1st order DE's solved by Laplace transform method.
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.
Sketch of y = 2(x+2)(x-2)(x+3).
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...
Detailed discussion of using Laplace transform to solve a 1st order ODE.
Brief and simple introduction to the first shift theorem for Laplace transforms.
Brief introduction to why we use Laplace transforms, with mention of tables.
Sketch of y = 2(x+1)^2-3.
The quotient rule for differentiation is proved from first principles.
Integration by parts is used to integrate the product of two polynomials.
The chain rule for differentiation is proved from first principles. Use is made of the small difference formula, which is discussed first.
The product rule for differentiation is proved from first principles.
The integral of sin(ax) sin(bx) is performed using integration by parts.
Simple demonstration of how to obtain a series to evaluate approximations to pi, using a specific and quite simple Fourier series.
Laplace transform is used to solve a second example of a 2nd order, constant coefficient ODE with boundary conditions.
We discuss the details involved in differentiating an integral, examine the notation used and outline the proof that the result is just the integrand.
The integral of exp(-x^2) from 0 to infinity is evaluated using a double integral technique.
We find the Laplace transform of a triangle wave as an example of a periodic function.
We calculate the Laplace transform of a rectified sine wave shape.
It is demonstrated how to perform differentiation using the chain rule, side-stepping the use of new function names such as u and v.
