In order to evaluate the rate of the enlarging circular
area, the concept of derivative has been employed. Derivative is widely
used in science and engineering research. For example, how to calculate
the linear density of a non-homogeneous rod. The mass from the left end
to the point x is
mass = f(x)
The mass between x_{2} and x_{1} is
Δmass = f(x_{2})
- f(x_{1})
So the average density of the rod between x_{2} and x_{1} is
average density = Δmass/Δx
= ( f(x_{2}) - f(x_{1}))/(x_{2}-x_{1})
Letting x_{2} approaches x_{1}, the linear density ρ of
the rod at point x_{1} is the limit of the average density when Δx
approaches 0.
In other words, the linear density of the rod is the derivative of the
mass with respect to length.
Since the concept of derivative can apply to many problems in science
and engineering, it is important for students to understand its physical
meaning. |