## Overestimates and Underestimates of Integrals

We can estimate an integral using the trapezium or the mid – ordinate rule. Sometimes however it is desirable to find underestimates and overestimates for an integral. The closer these are, the more accurately can the value for the integral be found and we can also quantify the error for the estimate. The estimate below […]

## Radius of a Circle Inscribed in an Isosceles Right Angled Triangle

The diagram below shows a circle inscribed in an isosceles right angled triangle. The vertical distance from the base of the triangle at O to the point A where the circle is a tangent to the triangle in the diagram below is By symmetry the vertical distance from O to A is half the height […]

## Calculating Equal Monthly Payments on a Loan

Loans are usually arranged to be repaid at a constant fixed monthly rate. This means that to start with a large fraction of the repayment made each month is used to pay interest. As time progresses the amount used to pay interest decreases and with the last payment the loan is paid off completely. It […]

## Angles in the Quadrilateral Formed By Centres and Intersections of Two Circles

Two circles of raddi R and r respectively. Intersect in the point A and B as shown below. From the Cosine Rule, for the triangle CAB,and for the triangle OAB, Setting these equal gives Now use the identity cos 2x = 1-2 sin^2 x rearranged asto give Dividing by 4 and square rooting gives

## Savings Schemes

Many saving schemes require a regular deposit to be made in each set time period – typically week, month or year. A very good example is saving for a pension. Money is deducted from your wages and invested. It is hoped that over a working life enough will be saved to provide a good income […]

## Proof of the Remainder Theorem

The remainder theorem states that when a polynomialis divided by a linear expression the remainder is Example: Whenis devider bysubstituteto obtain the remainder We can prove it quite easily by performing long division ofbyto obtain the quotientand remainderWe can write or equivalently (1) If the degree ofisthen the degree ofis and the degree ofmust be […]

## Growth and Sustainability

‘Geometric’ is perhaps the wrong term for a sequences with terms defined by the ruleEach term is being multiplied by a constant factor to obtain the next term. A quantity is said to grow or decay exponentially if the quantity at the start of each time period is multiplied by a constant factor to obtain […]

## Proof of the Cosine Rule

With a triangle labelled as below the Cosine Rule states to prove the rule, drop a perpendicular fromto the side Pythagoras Theorem gives Equating these expressions gives hence From the above diagram,so

## The Factor Theorem

The Factor Theorem Ifis a factor ofthensois a root ofor equivalently, a solution of the equation Example: Show thatis a factor of henceis a factor of Example:andare both roots of the quadratic expressionFind Sinceandare both rootsandare both factors. A quadratic equation has at most two real factors/roots hence More complicated questions may involve simultaneous equations: […]

## Bearings

Bearings involve using trigonometry, generally the cosine or sine rules: Cosine Rule: Sine Rule: For the above diagram, find a)The distance BC b)The bearing of A from B and the bearing of A from C. a)Label the triangle as above, with sides labelled by little letters opposite angles labelled by big letters. so b) We […]