In each of these cases, we are interested in how one measurement varies as another one does. That is, there is a dependent variable that varies with respect to an independent variable. However, we’re not interested in the measurements themselves but in how they vary. We call this the rate of change.

Exploration

Let’s focus on the first scenario. We can’t directly measure speed, but we can measure distance and time. Notice that speed is the amount that distance changes per unit of time. That is, speed is the rate of change of distance with respect to time. So we can use our measurements of distance and time to figure out the speed.

Suppose that the distance ($x$x, in metres) is related to the time ($t$t, in seconds) by the relationship $x=3t$x=3t. Let’s plot this relationship first:

Notice that this is a linear function. Since speed is the rate of change of distance over time, we want to find out how the distance changes over any amount of time. Let’s start by picking two points on the line. We’ll use $\left(1,3\right)$(1,3) and $\left(3,9\right)$(3,9):

First we find the change in the independent and dependent variables:

Change in time	$=$=	$3-1$3−1 m
	$=$=	$2$2 m
Change in distance	$=$=	$9-3$9−3 s
	$=$=	$6$6 s

And then we divide the change in distance by the change of time to get the rate of change:

Rate of change	$=$=	$\frac{6}{2}$62 m/s
	$=$=	$3$3 m/s

So the speed is $3$3 m/s. We will get the same result no matter which two points we choose. Notice that this is the same as the gradient of a linear function. In fact, this is always the case when the function is linear.

Using this we can come up with a general formula for the rate of change in a linear function. If our two points are $\left(x_1,y_1\right)$(x1,y1) and $\left(x_2,y_2\right)$(x2,y2), then the rate of change is:

Rate of change	$=$=	$\frac{\text{change in dependent variable}}{\text{change in independent variable}}$change in dependent variablechange in independent variable
	$=$=	$\frac{x_2-x_1}{y_2-y_1}$`x`2−`x`1`y`2−`y`1
	$=$=	gradient of the linear function

Rates of change

The rate of change of a dependent variable with respect to an independent variable is how much the dependent variable changes as the independent variable changes.

In the case of a linear function, the rate of change is the gradient.

Note that the independent variable is most often time, but can be anything else.

Practice questions

Question 1

$Sarah$ is planning to donate some $scarves$ to the local charity for their winter drive next month. She already has $8$8 old $scarves$ that she is willing to give away and plans on adding to this amount by knitting $2$2 each week.

Complete the table.

Week Number of $scarves$

$0$0 $\editable{}$

$1$1 $\editable{}$

$2$2 $\editable{}$

$3$3 $\editable{}$

$4$4 $\editable{}$
What amount is added to the number of $scarves$ in each row?
What does this amount represent?
the number of $scarves$ that $Sarah$ will buy each week
A
the number of $scarves$ that $Sarah$ will knit each week
B
the difference between the number of $scarves$ that $Sarah$ will have knitted and the number of $scarves$ she will have bought
C
the number of $scarves$ that $Sarah$ will have knitted after $20$20 weeks
D

Week	Number of $scarves$
$0$0	$\editable{}$
$1$1	$\editable{}$
$2$2	$\editable{}$
$3$3	$\editable{}$
$4$4	$\editable{}$

Question 2

The graph shows the progress of two competitors in a $cycling$ race.

Loading Graph...

Who is travelling faster?
$Justin$
A
$Oliver$
B
How much faster is $Oliver$ travelling?

Outcomes

11.C.LD.1

Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit. Definition of derivative, relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.

Linear Rates of Change