We want to find the shortest distance from the planeabove to the pointActually we will be finding the distance between the pointsin the plane andThe vectorfrom towill be parallel tothe normal to the plane. The component ofin the direction ofisso the distance is The least distance from the planeto the pointis

## Using the Trapezium Rule Twice to Obtain an Improved Estimate for the Value of an Integral

The trapezium rule gives an estimate for the value of an integralaswith accuracy proportional toso that ifand the true value of the integral isthen We can obtain an improved estimate for the value of the integral using the trapezium rule twice with different values ofSuppose that we use the trapezium rule twice with different values […]

## Finding the Acute Angle Between a Vector and a Plane

By the ‘angle between a vectorand a plane’ we mean the angle between the vectorand a particular vector drawn in the plane. In the plane there are actually many vectors, and the particular vector we pick is the vector that makes the smallest angle with the vectorThis is the angle shown as in the diagram […]

## Finding the Point of Intersection of a Line with a Plane

The line P has equationand we want to find the intersection of this line with the planeWe substitute the values offrom the line in terms of into the equation of the plane. We solve this to findthen substitute this value ofback into the equation of the line to find the point of intersection. Example: Find […]

## Trigonometric Identities

The best way to learn how to prove trigonometric identities is to do lots of example, there being ever so many identities to prove. Example: Prove Now divide every term in the numerator and denominator byto give Example: Prove the identity Divide every term in the numerator and denominator byto give Now perform long division […]

## The Binomial Theorem and Estimation

The binomial expansion can in certain circumstances give highly accurate estimates of certain powers or roots. This can happen for example when we want to find not too high a power of a number that is close to an integer. The binomial expansion is given by Suppose we are asked to findto 5 decimal places. […]

## 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 […]

## 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 […]

## 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 […]

## Differentiating Exponentials When the Base is Not e

Ifthenis a very familiar result. If howeverorthen differentiating is not so easy. We have to change the base first toand then differentiate. We do this using the relationship The last expression is of the formwhich differentiates toApplying this example towe obtain Differentiatingonly introduces another factorto giveWe can then find tangents and normals in the usual […]