In Calculus we'll be working with many different
types of functions. One of the major things we'll be interested
in is how these functions change. Fortunately this idea is very intuitive and
you may have started to do this already in previous math courses.
Let's take a look at a quick example to see how we do this.
If we read a function from left to right we can see that sometimes it increases, and other
times it decreases. I like to imagine the function as a roller
coaster so that I can easily see places where it is going up, versus where it is going down.
Knowing how a function increases and decreases certainly tells us more about the behaviors
of a function, but it's still not precise enough for what we will need later.
To see why, let's look at two more functions. Here are two function that are both increasing.
But one is increasing more rapidly. After all it looks steeper.
But how steep is it? Where does it increase the most?
From these questions you can see we need some way to describe how rapidly the function is
increasing or decreasing so we can better compare their behavior.
Let's take a step back. If these two function were lines, this would
be a much easier problem. When we are working with lines we can use
the notion of slope to accurately describe how quickly a line is increasing or decreasing.
This comes from measuring the rise of the function in the y direction, versus the run
of the function in the x direction. With slope we can say how much it is increasing
or decreasing by simply giving this ratio of numbers.
So we can say that the line on the right is increasing more, simply because it has a larger
value for its slope.
Now back to our other functions. These function are definitely not lines so
unfortunately we can't use the idea of slope to describe how they are changing... or can
we?
To build a line all we need are two points. If we choose two points on our function then
we can certainly draw a line right through them.
Note how the line works as a good approximation when describing how the function is changing.
If the line we draw is steep, then we can expect that our function is increasing fairly
quickly as well. But it is only an approximation right now.
If we change the two points we are using, we will get a different value for the slope
of the line. What this line really tells us then is the
average change of the function. How much the function has changed in the y-direction
versus the x-direction between the two given points.
Let's play around with this idea so that it makes a bit more sense.
Suppose we choose two points on a function. Drawing our line like earlier, we can find
it has a slope of 2/3. This tells us that as we are moving on the
function between the two points, we may go up, we may go down, but on average we'll increase
in the y-direction by 2 as we move in the x-direction by 3.
Or, maybe a better way to say this is that the average change of the function between
the two points is also 2/3.
If we choose two other points, our line is now horizontal.
This has a slope of zero. If we move along the function now, you'll
notice that despite the ups and downs, we will end up at the other point at the same
y-value. So on average, there is no change in the function.
So by describing the slope of a line between two points on a function we can get a lot
of information about the behavior, specifically it average rate of change.
Depending on what the function describes this could give us information like the average
speed of a car or the average flow rate from a tank of water.
The possibilities are endless.
But that's not all we can do. Next you'll see how we can use this average
rate of change to describe how a function is changing at a single point, rather than
between two points, and along the way we'll pick up our first major tool of calculus,
the limit.
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the average rate of change of a function, you can find those here.
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