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 […]

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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 […]

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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 […]

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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 […]

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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. […]

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