The general gradient of a graph is given by the gradient function To find the turning points of a graph we solve the equation To find the set of values for which the graph is going up or increasing we solveIn general we will get an interval or set of intervals. To find the set […]

## Differential Equations

We well know how to find the gradient function of a curve,We just differentiate the function we have forWe also need to be able to find y if we are givenTo do this we have to integrate. When we integrate we obtain a function ofbut the function we obtain is not fixed until we have […]

## Summation Notation

If we have to add up all the terms of a sequence, for example it is convenient to be able to express this in a compact form. We can write the sum of the above terms asIn English this means, “add up the k+1 powers of 2 starting with the zeroth power.” The summation has […]

## Finding the Area Between the Axes and a Curve That Crosses the Axes

The graph above shows the curveTo find the shaded area it is not enough to integrate betweenandThe area betweenandwhich is below the axis, will make the whole answer negative. Since the area is always positive our approach to the question will be this: Split the area into two areas.will be that part of the graph […]

## The Trapezium Rule

The trapezium rule is a numerical method for estimating integrals. It is most useful when there is no analytical answer to an integral, and only a number is needed. It works by approximating the area under the curve by a series of trapezia, then evaluating the areas and adding them up. The area under the […]

## Binomial Expansion

We can use the binomial theorem to expand and simplify expressions of the form where n is any positive whole number. The general expansions is Example: We evaluate the coefficient of each power of and simplify: Example: which simplifies to then Suppose we have an expansion. We know and but not n. If we have […]

## Quadratic Trigonometry

Trigonometry often reduces to quadratic equations using one of the formulae: For example, to solve , we look at the formulae: the second one substitutes for . We obtain . We take from each side to get . This factorises to give Either or We may have to use the quadratic formula instead of factorising. […]