09-05-2012, 01:19 PM
Differential Calculus
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Introduction
In day to day life we are often interested in the extent to which a change in one quantity
affects a change in another related quantity. This is called a rate of change. For example,
if you own a motor car you might be interested in how much a change in the amount of
fuel used affects how far you have travelled. This rate of change is called fuel consumption.
If your car has high fuel consumption then a large change in the amount of fuel in your
tank is accompanied by a small change in the distance you have travelled. Sprinters are
interested in how a change in time is related to a change in their position. This rate
of change is called velocity. Other rates of change may not have special names like fuel
consumption or velocity, but are nonetheless important. For example, an agronomist
might be interested in the extent to which a change in the amount of fertiliser used on a
particular crop affects the yield of the crop. Economists want to know how a change in
the price of a product affects the demand for that product.
Constant velocity
the graph of part of a motorist’s journey along a straight road. The
vertical axis represents the distance of the motorist from some fixed reference point on
the road, which could for example be the motorist’s home. Time is represented along the
horizontal axis and is measured from some convenient instant (for example the instant an
observer starts a stopwatch).
Other rates of change
The situation above described a car moving in one direction along a straight road away
from a fixed point. Here, the word velocity describes how the distance changes with time.
Velocity is a rate of change. For these type of problems, the velocity corresponds to the
rate of change of distance with respect to time. Motion in general may not always be in
one direction or in a straight line. In this case we need to use more complex techniques.
Velocity is by no means the only rate of change that we might be interested in. Figure 4
shows a graph representing the yield a farmer gets from a crop depending on the amount
of fertiliser that the farmer uses.
What is the derivative?
If you are not completely comfortable with the concept of a function and its graph then
you need to familiarise yourself with it before continuing. The booklet Functions published
by the Mathematics Learning Centre may help you.
In Section 1 we learnt that differential calculus is about finding the rates of change of
related quantities. We also found that a rate of change can be thought of as the slope of
a tangent to a graph of a function. Therefore we can also say that: