Chapter 2
Models of Growth: Rates of Change
2.1 Rates of Change
2.1.3 Mathematical Notation: The Difference Quotient
The phrases you see over and over — “rate of change,” for example — are clearly important. In fact, we are repeating them frequently to stress that importance. However, once we have made our point, it becomes equally important that we reveal the standard notation for the important concepts.
The next concept we shorten to a symbol is “change,” specifically, the change in a variable from one value to another.
| Definition If \(x_1\) and \(x_2\) are values of the variable \(x\), the change from \(x_1\) to \(x_2\) is the difference, \(x_2 - x_1\). We write
to abbreviate “The change in the variable \(x\) is the difference between the second value and the first.” |
In a similar manner we may abbreviate \(P_2 - P_1\) by \(\Delta P\), \(s_2 - s_1\) by \(\Delta s\), \(t_2 - t_1\) by \(\Delta t\) and so on. We combine these symbols to obtain a notation for the average rate of change of one variable with respect to another.
| Definition If \(s = f(t)\),
then the average rate of change of \(s\) as \(t\) changes from \(t_1\) to \(t_2\) is the ratio of the change in \(s\) to the change in \(t\), i.e.,
. This quotient of differences is called, naturally enough, a difference quotient.
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