Increments

Concept	Quick Reference
Increment of $x$	$\mathrm{\Delta }x=x_{\text{new}}-x_{\text{old}}$
Increment of $f$	$\mathrm{\Delta }f(x)=f(x+\mathrm{\Delta }x)-f(x)$
$\mathrm{\Delta }y$ (when $y=f(x)$)	$\mathrm{\Delta }y=f(x+\mathrm{\Delta }x)-f(x)$
Sign of increment	Can be positive, negative, or zero

Increments

As in calculus we talk about changes of functions all the time, it would be convenient to introduce a concise way to express changes in variables and functions.

An increment of a variable is the difference between two values of the variable, and may be thought of as a change in value of the variable. We denote an increment of a variable by placing $\mathrm{\Delta }$ — the Greek letter for $D$ — before the variable. Therefore, $\mathrm{\Delta }x$, $\mathrm{\Delta }t$, $\mathrm{\Delta }y$, etc. denote increments of $x$, $t$, $y$, etc. They are read delta x, delta t, delta y, etc.

• The increment is always the second value of the quantity considered minus the first value.
• Notice that $\mathrm{\Delta }$ is NOT a multiplier of the variable. In fact, $\mathrm{\Delta }x$, $\mathrm{\Delta }t$, $\mathrm{\Delta }y$, etc. are single entities. We can think of $\mathrm{\Delta }$ as a difference operator that acts on what follows it and gives the new value minus the old one.

For example, if $t$ changes from $2$ to $2.3$, its increment is

$$\mathrm{\Delta }t=2.3-2=0.3,$$

and if $x$ changes from $2$ to $1.9$, its increment is

$$\mathrm{\Delta }x=1.9-2=-0.1.$$

Corresponding to an increment of a variable, there exists an increment of a function of that variable. If the independent variable $x$ is given an increment $\mathrm{\Delta }x$, then the value of $f$ receives a corresponding increment $\mathrm{\Delta }f(x)$, which is obtained by subtracting the value of $f$ at the old value of $x$ from the value of $f$ at the new value of $x$. That is,

$$\mathrm{\Delta }f(x)=f(x+\mathrm{\Delta }x)-f(x).$$

This increment $\mathrm{\Delta }f(x)$ will, in general, depend on both $x$ and $\mathrm{\Delta }x$.

If $y=f(x)$, then instead of $\mathrm{\Delta }f(x)$, we can equally write $\mathrm{\Delta }y$. That is,

$$\mathrm{\Delta }y=f(x+\mathrm{\Delta }x)-f(x).$$

Geometric Interpretation

A visual representation of related increments is helpful. In the following figure, the curve represents a function $y=f(x)$. Let $x_0$ be a given initial value of $x$, and let $P(x_0,y_0)$ be the corresponding point on the curve. If we consider a nearby point $Q$ on the curve, then in going from $P$ to $Q$, the increments of $x$ and $y$ are:

$$\mathrm{\Delta }x=MN=PR\text{and}\mathrm{\Delta }y=\mathrm{\Delta }f(x)=RQ;$$

the coordinates of $Q$ are $(x_0+\mathrm{\Delta }x,y_0+\mathrm{\Delta }y)$.

The figure above illustrates a case in which both increments are positive; however, $\mathrm{\Delta }x$ may be either positive or negative, and $\mathrm{\Delta }y$ may be positive, negative, or zero. In applications, $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ are usually numerically very small.

Comparing Increments

Having determined increments of $x$ and of $y$, the natural next step is to compare them by dividing the increment of $y$ by the increment of $x$. This is exactly what we do when finding average velocity: we divide an increment of distance by an increment of time. In finding average acceleration, we divide an increment of velocity by an increment of time. This ratio

$$\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}=\frac{f(x+\mathrm{\Delta }x)-f(x)}{\mathrm{\Delta }x}$$

is called the difference quotient and is the starting point for the definition of the derivative.

Frequently Asked Questions

Can $\mathrm{\Delta }x$ be negative?

Yes. The increment $\mathrm{\Delta }x$ is simply the new value of $x$ minus the old value. If $x$ decreases, then $\mathrm{\Delta }x$ is negative. There is no requirement that increments be positive.

Is $\mathrm{\Delta }x$ the same as $dx$?

No, not exactly. $\mathrm{\Delta }x$ denotes a finite (possibly large) change in $x$. The symbol $dx$, which appears in Leibniz notation for derivatives, represents an infinitesimally small change. In practice, $dx$ is the limiting idea that $\mathrm{\Delta }x$ approaches as it gets arbitrarily small.

Why write $\mathrm{\Delta }y$ instead of just "the change in $y$"?

The notation $\mathrm{\Delta }y$ is compact and unambiguous. It makes formulas like $\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}$ easy to read and manipulate, and it mirrors the Leibniz derivative notation $\frac{dy}{dx}$ that arises naturally when we take the limit as $\mathrm{\Delta }x\to 0$.