In this work averaging methods are used to analyse the behaviour of an adaptive controller with slowly varying parameters. The incorporation of time varying parameters into a stability analysis of an adaptive controller leads to certain perturbation terms in the error system. The type of averaging results so far applied to adaptive control cannot deal with these perturbations. Thus, some new averaging theorems are needed. These results are known in the averaging literature. However, here a very simple proof technique has been found which gives finite and infinite interval averaging results under weaker conditions. The averaging analysis shows that, with regard to parameter tracking, the adaptive algorithm behaves like a first-order filter. It is also clear that successful tracking then requires knowledge of the true speed of parameter change.