Example of using the multivariate chain rule where in this case z=f(x,t) and x is a function of t. In this example the expression relating x and t is an implicit expression so implicit differentiation is required in finding dx/dt for that part of solving the problem.

Integration by parts using the formula is briefly revisited, then we look at how to code use of the formula into a visual process of adding terms that might be easier to remember.

Integration by parts using the formula is briefly revisited, then we look at how to code use of the formula into a visual process of adding terms that might be easier to remember.

We show an unusual approach to finding the exponential of a 2x2 matrix - using the Laplace transform. This method was suggested by a bright Swinburne student.

The 2nd shift theorem is proved using a change of integration variable then a simple example is presented.

Laplace transforms are used to solve a pair of linear, 1st order ODE's with boundary conditions. Cramer's rule is used along the way.

Laplace transforms are used to solve a pair of linear, 1st order ODE's with boundary conditions. Cramer's rule is used along the way.

The 2nd shift theorem is proved using a change of integration variable then a simple example is presented.

We show an unusual approach to finding the exponential of a 2x2 matrix - using the Laplace transform. This method was suggested by a bright Swinburne student.

We ask what happens in the 1st principles differentiation if the angle x is measured in degrees. The question is answered by referring to the limits calculated previously.