Increments

The increment of a variable in changing from one numerical value to another is the difference found by subtracting the first value from the second. An increment of $x$ is denoted by the symbol $\Delta x$, read ``delta $x$''.

The student is warned against reading this symbol ``delta times $x$'', it having no such meaning. Evidently this increment may be either positive or negative^4.2according as the variable in changing is increasing or decreasing in value. Similarly,

        $\Delta y$ denotes an increment of $y$,

         $\Delta \phi$ denotes an increment of $\phi$,

         $\Delta f(x)$ denotes an increment $f(x)$, etc.

If in $y = f(x)$ the independent variable $x$, takes on an increment $\Delta x$, then $\Delta y$ is always understood to denote the corresponding increment of the function $f(x)$ (or dependent variable $y$).

The increment $\Delta y$ is always assumed to be reckoned from a definite initial value of $y$ corresponding to the arbitrarily fixed initial value of $x$ from which the increment $\Delta x$ is reckoned.

Example 4.2.1   For instance, consider the function

$$y = x^2.$$

Assuming $x = 10$ for the initial value of $x$ fixes $y = 100$ as the initial value of $y$. Suppose $x$ increases to $x = 12$, that is, $\Delta x = 2$; then $y$ increases to $y = 144$, and $\Delta y = 44$. Suppose $x$ decreases to $x = 9$, that is, $\Delta x = - 1$; then $y$ increases to $y= 81$, and $\Delta y = - 19$.

It may happen that as $x$ increases, $y$ decreases, or the reverse; in either case $\Delta x$ and $\Delta y$ will have opposite signs.

It is also clear (as illustrated in the above example) that if $y = f(x)$ is a continuous function and $\Delta x$ is decreasing in numerical value, then $\Delta y$ also decreases in numerical value.