The significance of initial rates

Before we start investigating, how the order of a reaction affects the rate, we need to define, which rate we are looking at. Why  

In the last few chapters we always looked at the disappearance of reactants (or the generation of a produet) as a function of time. For example we looked at a first order reaction 

This equation allowed us to caleulate the coneentration of C at any partieular point of time, as long as we knew the initial eoneentration of C and the rate constant k. We confirmed that this indeed was a first order reaction - we did this by plotting In (C) versus time, and got a straight line. 

Looking at the data of the time course of the consumption of reactant C  

Our starting point for the development of the equation for such a first order reaction was the general rate 

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