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.
Thanks for watching.
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If you want to see some example problems about
the average rate of change of a function,
you can find those here.
You can also skip ahead to the next lecture
where I cover approximating the instantaneous
rate of change of a function.
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