In general a trigonometric equation of the formfor example may have more than one solution, and may have in fact an infinite number of solutions. We often have to find the solutions within a certain range eg 0 – 360o ordepending on whether we are working in degrees or radians. All the trigonometric graphs – […]

## Solving Absolute Inequalities

It is very easy to solve linear inequalities of the formalmost as easy as solving linear equations:Solving an absolute inequality, where the equation includes modulus signs is a little bit trickier. When we remove the minus sign it can be hard to work out which way the inequality signs point. Given this, the safest way […]

## Rates of Change

When quantities depend on other quantities that are changing, for example the volume of a sphere depends on the radius which is increasing at 1 cm per second, we have to be very methodical in our approach if we want to find the rate of change of volume of the sphere. We use the chain […]

## Odd and Even Functions

Odd Functions Rotation symmetry order 2 about the origin even function*odd function=odd function even function*odd function=odd function Even Functions Reflection symmetry in the– axis even function*even function=even function odd function*odd function=odd function illustrated above is neither odd nor even. It has no symmetry.

## Definite Integrals

The definite integral ofbetween the valuesandcalculates the area under the graph between the valuesandThis is writtenwhere andare the limits of integration andis the integral ofThe larger value ofis the upper limitand the smaller value ofis the lower limit Example: Find the area under the curvebetweenand To evaluate this integral we use the identityrearranging it to […]

## Integration by Parts

Integration by parts is used to integrate a product. It is derived from the product rule for differentiating a product: We subtract a term from the right hand side to give and then integrate to give which is usually written as It is important to chooseandthe right way round. If there is anterm, thenis usually […]

## Proof of Simpson’s Rule

Simpson’s Rule is used to numerically estimate the value of integrals that either cannot be or are difficult to evaluate analytically. The rule approximates a function with a collection of arcs from quadratic functions and integrate across each of these. Proof: Let P be a partition of [a,b] into n subintervals of equal width, , […]

## Practical Exponential Questions

Exponential decay is a real phenomenon with many practical applications – radioactive decay, Newton’s Law of Cooling, measuring and controlling the thickness of metals – since the intensity of radiation decays exponentially with penetration into the metal.. Radioactivity is used in smoke alarms to – too low a level of radiation detected will set off […]

## The Mid Ordinate Rule

Also called the midpoint rule – the mid ordinate rule is another method to numerically estimate integrals. It states: If an area of integration is divided into n strips, the area of the strip betweenandis given byso that the width of the strip is multiplied by the– value at the midpoint. We do this for […]

## Inverting Functions

Given a functionwe can easily find a value ofgiven a value ofby substituting the value ofinto the functionIf however we want to find the value ofgiven a value ofthen at some stage we will have to either invert the function or solve an equation to findThe first method is usually preferable because it is general: […]