WHAT IS RATE OF CHANGE

WHAT IS RATE OF CHANGE
(Limit of function part 2)

Rates of Change

The next problem that we need to look at is the rate of change problem. As mentioned earlier, this will turn out to be one of the most important concepts that we will look at throughout this course.

Here we are going to consider a function, $f\left(x\right)$, that represents some quantity that varies as $x$varies. For instance, maybe $f\left(x\right)$ represents the amount of water in a holding tank after $x$minutes. Or maybe $f\left(x\right)$ is the distance traveled by a car after $x$ hours. In both of these example we used $x$ to represent time. Of course $x$ doesn’t have to represent time, but it makes for examples that are easy to visualize.

What we want to do here is determine just how fast $f\left(x\right)$ is changing at some point, say $x=a$. This is called the instantaneous rate of changeor sometimes just rate of change of $f\left(x\right)$ at $x=a$.

As with the tangent line problem all that we’re going to be able to do at this point is to estimate the rate of change. So, let’s continue with the examples above and think of $f\left(x\right)$ as something that is changing in time and $x$ being the time measurement. Again, $x$ doesn’t have to represent time but it will make the explanation a little easier. While we can’t compute the instantaneous rate of change at this point we can find the average rate of change.

To compute the average rate of change of $f\left(x\right)$at $x=a$ all we need to do is to choose another point, say $x$, and then the average rate of change will be,

$$\begin{array}{cc}A.R.C.& =\frac{\text{change in}f\left(x\right)}{\text{change in}x}\\ & =\frac{f\left(x\right)-f\left(a\right)}{x-a}\end{array}$$

Then to estimate the instantaneous rate of change at $x=a$ all we need to do is to choose values of $x$ getting closer and closer to $x=a$(don’t forget to choose them on both sides of $x=a$) and compute values of $A.R.C.$ We can then estimate the instantaneous rate of change from that.

Let’s take a look at an example.

Example 2 Suppose that the amount of air in a balloon after $t$ hours is given by
$$V\left(t\right)=t^3-6t^2+35$$Estimate the instantaneous rate of change of the volume after 5 hours.

 Solution 

Okay. The first thing that we need to do is get a formula for the average rate of change of the volume. In this case this is,
$$A.R.C.=\frac{V\left(t\right)-V\left(5\right)}{t-5}=\frac{t^3-6t^2+35-10}{t-5}=\frac{t^3-6t^2+25}{t-5}$$To estimate the instantaneous rate of change of the volume at $t=5$ we just need to pick values of $t$ that are getting closer and closer to $t=5$. Here is a table of values of $t$ and the average rate of change for those values.

$t$	$A.R.C.$	$t$	$A.R.C.$
6	25.0	4	7.0
5.5	19.75	4.5	10.75
5.1	15.91	4.9	14.11
5.01	15.0901	4.99	14.9101
5.001	15.009001	4.999	14.991001
5.0001	15.00090001	4.9999	14.99910001

So, from this table it looks like the average rate of change is approaching 15 and so we can estimate that the instantaneous rate of change is 15 at this point.