If we haveas a function ofit is quite easy to findWe often need to find whenis a function ofor there are several occurrence of bothandIn these cases we need to differentiate implicitly. We shall start with a simple case. Differentiate We can differentiate both sides with respect toobtaining 1 on the left hand side but […]

## The Product of Two Binomial Expansions

The General Binomial Expansion takes the form Any expression of the formcan be evaluated using this expression whatever the values of A and B – negative, decimal factional or even complex numbers. We can expand the product of two binomial expansions by finding each individual binomial expansion. Example: Expandup to and including the term in […]

## Integrating Quotients of Algebraic Expressions

Integrating any quotient of the formcan be done by making the substitutionFor example to findsubandso the integral becomes(1) There is however an alternative method which is also useful for integrating quotients of higher order polynomials. Carry out long division first on the quotient and integrate the result. For the example above,and the integral becomes This […]

## Long Division of Polynomials

Long division of one number by another, if the divisor is not a factor, results in a decimal number, or a quotient plus a remainder. For exampleremainder 1 or3 is the quotient and 1 the remainder. It is not just pure numbers that can undergo long division. So can polynomials. In the example below it […]

## Partial Fractions Examples

Example: Expressas partial fractions. Clear all the fractions by multiplying by Example: Express as partial fractions. Multiply byto clear all the fractions.

## Finding the Point of Intersection of Two Lines in Three Dimensions

If two lines intersect, they are both in the same place at the same time, so to speak. We don’t know what the point is, but because they both meet at the same point, we can put the equations of the lines equal to each other. This will result in simultaneous equations for the parameters. […]

## Finding The Equation of a Plane Passing Through a Given Point Perpendicular to a Given Vector

If a vectoris perpendicular to a planethen it is perpendicular to every vectorand line drawn in the plane, henceWe can find an expression for u by subtracting one point in the plane from another. This is shown on the diagram above aswhereis an arbitrary point andis given, so by findingor we find the equation of […]

## Finding the Equation of a Plane From Three Points

A line, which is a two dimensional object, is fixed by two points on it – two dimensions, two points. The equation of a line can be written given by– this is the cartesian form of the line. The cartesian form of a plane iswhereandare constants To find the equation of a plane we need […]

## Angles Between Vectors and Lines

The anglebetween two vectors u and v is given by the equation Example: Find the angle between the vectorsand We can use the same formula to find the angle between vectors in two, three or more dimensions: Example: Find the angle between the vectorsand To find the angle between two lines we use the same […]

## Parametric Coordinates – Equations of Tangents and Normals

To fins the equation of a tangent to a curve at a point we have to find the gradientat that point. Ifandare functions ofwe findand substitute a value forinto this expression. If we have to find the equation of the normal to a curve at a pointwe findIf we have the coordinatesor only one of […]