The cost in dollars to produce x computers in a factory
is given by the function. Determine and interpret the
average rate of change from x equals zero,
to x equals two hundred,
and from x equals four hundred to x equals six hundred.
To determine the average rate of change,
we need to determine the slope of the secant line
passing through the function
at x equals zero, and x equals two hundred,
as well as at x equals four hundred, and x equals six hundred.
So we'll start by determining the slope of the secant line
from x equals zero to x equals two hundred.
But to do that we'll first have to determine the point on the function,
when x equals zero and x equals two hundred.
So we'll start by doing that.
So we know one x-coordinate is zero,
we know the other x-coordinate is two hundred.
And now we'll use the calculator to determine the y-coordinates, or the function values.
So the first thing we'll do is type the function into y equals.
I've already done that.
And now we'll make sure the table set is set on ask.
So we'll press second, window. We want to make sure that
the ask feature is highlighted here.
Once it is we can press second, graph, and then type in our x values.
So we have zero and we have two hundred.
So our y-coordinates are zero and one hundred thousand
Now here's the equation to determine the slope given two points,
so we'll call these the ones. X sub-one, y sub-one,
and we'll call these the twos.
So the slope of the line passing through these two points will be our average rate of change
on this interval.
So y-two minus y-one will be one hundred thousand minus zero,
that's one hundred thousand
Divided by x-two minus x-one,
that's two hundred.
One hundred thousand divided by two hundred is five hundred.
So this is the average rate of change
from x equals zero to x equals two hundred.
But to interpret this, it's helpful if we write this as a fraction,
and then take a look at what the change in y and the change of x
is actually measuring.
Well y is the cost in dollars, so this would be five hundred dollars.
And x is the number of computers, so this would be
five hundred dollars per one computer.
And this helps us interpret the meaning.
So the average rate of change is five hundred dollars per computer,
on the interval from zero to two hundred.
So the average rate of change in cost is five hundred dollars per computer
on the interval, x equals zero to x equals two hundred.
And that's important because if this interval changes we would
expect the average rate of change in cost to change as well.
So before we consider the interval from four hundred to six hundred,
let's take a look at this result graphically.
We just found the slope of the secant line,
which would be the line passing through these first two points on the left,
so it would look something like this.
And again the slope of this green line represents
the average rate of change
on the interval from zero to two hundred.
Now let's determine the average rate of change on the interval from
x equals four hundred to six hundred.
So again we'll start by determining the two points on the function
at these values of x.
So one x-coordinate is four hundred
and the other x-coordinate is six hundred.
And now we'll go back to the calculator.
And now we'll type in x equals four hundred,
and x equals six hundred.
So we have one hundred sixty thousand and one hundred eighty thousand.
And now we'll determine the slope of the line passing through these two points.
Again we'll call these the ones, and we'll call these the twos.
The slope is the average rate of change on this interval,
so we have y-two minus y-one,
divided by x-two minus x-one
So it looks like we have twenty thousand divided by two hundred,
which is equal to one hundred.
And again to add some meaning to this we'll write this as a fraction.
So the change in y is one hundred dollars,
and the change of x is one computer.
So we can see that the average rate of change in cost has decreased
dramatically on the interval from x equals four hundred to six hundred.
So if the average rate of change
in cost is one hundred dollars per computer
on the interval
from x equals four hundred to x equals six hundred.
And let's go ahead and take a look at this graphically as well.
We just found the slope of the secant line passing through these two points.
And you can see this line is not as steep as
the green line, therefore, the average rate of change
is less on this interval.
I hope you found this helpful.