Cambridge Additional Mathematics

Introduction to differential calculus (Chapter 13) 337

Speedis a commonly used rate. It is the rate of change in distance per unit of time.

We are familiar with the formula:

average speed=

distance travelled

time taken

However, if a car has an average speed of 60 km h¡^1 for a journey,

it does not mean that the car travels at exactly 60 km h¡^1 for the

whole time.

In fact, the speed will probably vary continuously throughout the

journey.

So, how can we calculate the car’s speed at any particular time?

Suppose we are given a graph of the car’s distance travelled against

time taken. If this graph is a straight line, then we know the speed

is constant and is given by thegradientof the line.

If the graph is a curve, then the car’s instantaneous speed is given

by thegradient of the tangentto the curve at that time.

Historical note

The modern study of differential

calculusoriginated in the 17 th century

with the work of Sir Isaac Newton

andGottfried Wilhelm Leibniz. They

developed the necessary theory while

attempting to find algebraic methods

for solving problems dealing with the

gradients of tangents to curves, and

finding the rate of change in one

variable with respect to another.

Discovery 1 Instantaneous speed

A ball bearing is dropped from the top of a tall building. The

distance D it has fallen aftertseconds is recorded, and the

following graph of distance against time obtained.

We choose a fixed point F on the curve when t=2seconds.

We then choose another point M on the curve, and draw in the line

segment orchordFM between the two points. To start with, we

let M be the point when t=4seconds.

time

distance travelled

time

distance

time

distance travelled

1 h

60 km

Isaac Newton 1642 – 1727 Gottfried Leibniz 1646 – 1716

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_13\337CamAdd_13.cdr Tuesday, 7 January 2014 2:34:37 PM BRIAN