# Useful Formulae

48 CALCULUS differentiation and integration Calculus makes use of infinitesimalsand limitsto solve two problems, the instantaneous rate of change of a function and the exact area under a curve. The graph of the function yfx at position a,fa has gradient fa, where the function fx x is called the derivativeof f. For each x,fx is the rate of change of fat x. To directly calculate such a derivative at a,we consider the slopes of the lines through a,fa and a,fa for evertinier values of . If they tend towards a limit,then the rate of change of f at a can be defined to be this limit. If xt denotes the position of an object at timet,then its velocity vt at time t is xt t. Its acceleration at,the rate of change of its velocity at t, is then xt t t. Suppose we have a function opposite and seek the area beneath its graph between aand b. The interval between a and bis divided into an everlarger number of equal lengths.This produces an ever larger number of narrowing rectangles,the sum of whose areas can easily be found at each stage. The area under the curve is given by the limit of these sums, written fxdx. If Fx satisfies Fx fx,then remarkably, fxdx Fb Fa. Fx is called theindefinite integral or antiderivativeof f anddenoted f dx. As Fx c Fx,it follows that all of the antiderivatives given opposite involve an arbitrary constant. dx b a df dt dx dt dv dt2 d2x b a 49