The difference quotient

Differentiation: The derivative

The difference quotient

Change plays a major role in the study of mathematics and in particular the study of functions. In the following example, we look at the average change at a chosen interval.

We calculate the average change on the interval $[2,4]$ for the function $f(x)=2x-2$ .

The horizontal change $\blue{\Delta x}$ is: $\blue{\Delta x} = 4 - 2 = \blue{2}$ The vertical change $\green{\Delta y}$ is: $\green{\Delta y} = f(4)-f(2)=6 - 2 = \green{4}$ So the average change is:

$\frac{\green{\Delta y}}{\blue{\Delta x}} = \frac{\green{4}}{\blue{2}}=2$

Note that the notation $[a,b]$ for an interval can also be used for coordinates.

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Physical application

This works in the same way when we want to calculate the average speed of an airplane. Imagine an airplane which covers $4800$ meters in $20$ seconds. The average speed of the plane would be $\frac{4800}{20}=240$ meters per second.

Linear functions

For linear functions, as in the example, the average change on each interval is the same. Suppose we have the function $f(x)=px+q$ . The average change on each random interval $[a,b]$ is given by
$\frac{f(b)-f(a)}{b-a}=\frac{p\cdot b + q - (p\cdot a +q)}{b-a}=\frac{p\cdot b + q - p\cdot a -q}{b-a}=\frac{p\cdot(a-b)}{a-b}=p$

We also call the average change on an interval the difference quotient.

Difference quotient

The difference quotient of a function $f$ on an interval $[a,b]$ is given by:

$\dfrac{\green{\Delta y}}{\blue{\Delta x}}=\dfrac{\green{f(b)-f(a)}}{\blue{b-a}}$

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Etymology

Literally quotient means 'the result of division'. The words 'difference' and 'quotient' put together convey the right essence: dividing one difference by another.

Consider the function $f(x)=2 x^3 + 2 x + 2$ . What is the difference quotient on the interval $[4,8]$ ? Simplify your answer as much as possible.

$\frac{\Delta y}{\Delta x}=$ $226$

$\begin{array}{rcl}\dfrac{\Delta y}{\Delta x}&=&\dfrac{f(8)-f(4)}{8-4}\\&&\phantom{xxx}\blue{a=4 \text{ and }b= 8}\\ &=&\dfrac{(2\cdot 8^3+2\cdot 8 + 2)-(2\cdot4^3+2\cdot4 + 2)}{8-4}\\&&\phantom{xxx}\blue{x=4 \text{ and } x=8 \text{ substituted in } f}\\ &=& \dfrac{904}{4}\\&&\phantom{xxx}\blue{\text{added}}\\ &=&226\\&&\phantom{xxx}\blue{\text{divided}}\end{array}$

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