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# Worked example: average rate of change from equation

Video transcript

y equals 1/8 x to the
third minus x squared. Over which interval does y
of x have an average rate of change of 1/2? So let's go interval by
interval and calculate the average rate of change. So first let's think about
this interval right over here. x is between negative 2 and 2. So negative 2 is less than
x, which is less than 2. So let's just think about what
is the value of our function when x is equal to negative 2? So y of negative 2
is equal to 1/8 times negative 2 to the third power
minus negative 2 squared, which is equal to--
let's see, negative 2 the third power is negative 8. Negative 8 divided
by 8 is negative 1. Negative 2 squared
is positive 4, but then you're going to have to
subtract that, so it's minus 4. So this is equal to negative 5. And y of 2 is equal
to 1/8 times 2 to the third power
minus 2 squared. And that's going to be equal
to 1/8 times 8 is 1 minus 4, which is equal to negative 3. So if you want to find your
average rate of change, you want to figure out how much
does the value of your function change, and divide that by
how much your x has changed. So we could make a table here. x, y. When x is negative
2, y is negative 5. When x is positive
2, y is negative 3. So how much did your y change? Well, your y increased by 2,
and your x increased by 4. And you could get these
numbers-- you could just look at it that y increased
from that point to that point, x increased from that
point to that point. Or you could say hey, negative 3
minus negative 5 is positive 2. That's the difference between
negative 3 and negative 5. If you said 2 minus
negative 4, well that gives you, once again, the
distance, or the difference. It would give you positive 4. But if you look at
it here, it's clear. When y increased
by 2-- Or we could say when x increased
by 4, y increased by 2. So our average rate of
change over this interval is going to be
average rate of change of y with respect to x is going
to be equal to, well, when x changed by 4, by positive
4, y changed by positive 2. So it's equal to 1/2. So it does look like the
average rate of change over this interval
right over here is 1/2. So we got lucky
in this situation, our first choice-- and
this is a multiple choice, so it's not a multiselect here. So our first one
met our criteria. So we know that
that's the answer. But let me do one more
of these other ones to show you why that
is not the answer. So let's find the
average rate of change between that point
and that point. So let's do another-- let
me do it in another color. So I'll do this one in purple. So 0 is less than x,
which is less than 4. And I'll just do the
table right over here. x and y. So when x is 0, what is y? Well, it's going to be
1/8 times 0 minus 0, and y is just going to be 0. When x is 4, what is y? Well y is going
to be-- let's see, it's going to be 1/8--
trying to do this in my head. 1/8 times 4 to the third. 4 to the third is 64. 1/8 of 64 is 8. It's going to be 8 minus
4 squared, which is 16. 8 minus 16 is negative 8. So in this one, x
is increasing by 4, x has increased by 4,
what happened to y? y has decreased by 8. So the average rate of change
of y with respect to x here is y changed-- so I could
write my change in y-- this Greek letter delta
just literally is shorthand for change in. My change in y is negative
8 when my change in x is 4. So the average rate of
change here is negative 2. It's negative because as x
increased, your y decreased. And on average, for every 1 that
x increased, y decreased by 2. That's where you get
your negative 2 here. So this clearly, the average
rate of change is not 1/2. So that confirms that
that is not the answer. And we know that
these other two, and I encourage you to try them
out, will also give you an average rate of
change of something other than positive 1/2.