A-LEVEL MATHEMATICS Topic : Tangents and Normals

Tangents

Given a curvewe can find the equation of the tangent toat a point whereby

1. Finding
2. Differentiating f(x) and substituting x=p to find
3. Usingto find the equation of the tangent.

Example: Find the equation of the tangent to the curve

Multiply the brackets out to getand differentiate to get

and now find

Normals

Give a curvewe can find the equation of the normal toat a point whereby

4. Finding
5. Differentiating f(x) and substituting x=p to find
6. Usingto find the equation of the normal.

Example: Find the equation of the normal to the curve

Multiply the brackets out to getand differentiate to get

and now find

A-LEVEL MATHEMATICS Topic ; Rates of Change

Intuitively the rate of change of something is the rate of change with respect to time:
The acceleration is the rate of change of velocity with respect to time.
The velocity is the rate of change of distance with respect to time.
In general we have a quantity that is a function of time and we find the rate of change of that quantity by differentiating with respect to time. For example
A model helicopter takes off from a point O at timeand moves vertically so that its
height,cm, above O after timeseconds is given by

1) Find the rate at which the height is increasing.
2) Verify thathas a stationary value whenand determine whether this stationary
value is a maximum value or a minimum value.
3) Find the rate of change ofwith respect towhen
4) Determine whether the height of the helicopter aboveis increasing or decreasing at
the instant when
1)
2)When
is stationary therefore y is stationary.
3)
When
4)
If the height of the helicopter is increasing,
If the height of the helicopter is decreasing,
When
The height of the helicopter is increasing.

A-LEVEL MATHEMATICS Topic : Finding the Area Between Lines and Axes

The graph is of the line(1). We have to find the area of the region A, which is a triangle. Towards this end, we find the base and the height of the triangle.
To find the base, note that at the point where the line crosses the– axis,Substitute this value ofinto (1):

To find the height, note that at the point where the line crosses the– axis,Substitute this value ofinto (1):

The base is 8 and the height is 4 so the area is

To find the height of the triangle now we have to solve the simultaneous equations
(1)
(2)
(2)-(1) gives
Subinto (1) to give
To find the base we have to find where the lines cross the– axis. This will be where y=0.


The base of the triangle is 12—8=20 and the area is