Finding the Equation of the Line of Symmetry of a Graph

If a graph of a function has a line of symmetry, the function be even about line. This means that if the function is transformed so that the line becomes the y axis, we will be able to writewherefor some number Supposeshown below. The graph has a vertical line of symmetry with equation Suppose we […]

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Functions Which Cannot be Integrated

The usual methods of integrating – rearrangement, substitution, use of an identity of some form, or integration by parts – do not always work. If they do not work, it is often necessary to estimate the value of an integral. An estimation can usually be made using techniques of numerical integration. These techniques are usually […]

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Proof of the Compound Angle Formula for sin(A+B)

The compound angle formulacan be proved using simple trigonometry. Consider the triangles below. From this diagram we can deduce the three extra angles in the triangle below. Then the sides and finally Using the whole triangle gives Multiplying both sides bygives

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Proof of the Compound Angle Formula for cos(A-B)

The compound angle formulacan be proved by considering the sector of the unit circle between angles from the x – axis betweenand The coordinates ofareand the coordinates of L areThe distance,betweenandsatisfies From the Cosine Rule Since the circle is a unit circle,andso Equating these two expressions gives which simplifies to

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Proof of the Product Rule

The Product Rule is used to differentiate functions consisting of a product of simpler functions. It states: Ifthen Proof from first principles using the difference quotient is quite simple. The Product Rule can be used repeatedly to differentitate a product of any number of functions.

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Proof of the Chain Rule

The Chain Rule is used to differentiate a function of a function. It states, If a functionis the composite of two simpler functionsandsothen We can prove the Chain Rule from first principles using the difference quotient. The Chain Rule is probably the hardest of the three differentiation rules to understand. It is often hard to […]

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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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Transformations of Graphs

There are many ways that graphs can be transformed but many of them can be expressed as– transformations,– transformations, an– transformation followed by a– transformation or vice versa. The transformations together with their effects are summarised below. is transformed to or– transformation or both Effect Scaling byin the– direction Scaling by 3 in the– direction […]

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