The key idea of Calculus at level 8 is that there is a link in both directions between the gradient of a function and the original function.
While this key idea is the same as the key idea of Calculus at level 7, it now covers a broader range of functions. In addition, students at this level will also explore some fundamental ideas of Calculus, namely discontinuities and limits; and differential equations. This latter topic leads to the modelling of some simple real world problems.
A large part of this thread is spent on techniques of differentiation and integration. These both reinforce the connection between differentiation and anti-differentiation and allow functions to be integrated that would be difficult or impossible using only this connection. Some numerical methods are also introduced. The importance of these is due to the fact that not all functions are able to be integrated.
This key idea develops from the key idea of Calculus at level 7 by adding a number of functions to the polynomials operated on. It also depends heavily on all of the algebra that is found both at earlier levels of the curriculum as well as at this level. Algebra is used continually in Calculus at Level 8 especially in the techniques of differentiation and integration.
From this level, Calculus continues and opens up at University and beyond into applications with functions of more than one variable. These applications cover many disciplines from Physics, Chemistry, and Biology through to Economics. A great number of these applications are founded on the differential equations that are introduced at this level. Much of these real world problems require numerical techniques to solve them.