Introduction to the Average Rate of Change (Must Know for Calculus)

Video Series: Functions and Their Graphs (Step-by-Step Tutorials for Precalculus)

(D4M) — Here is the video transcript for this video.

00:00
in this episode you’ll learn what the average rate of change is and why it’s
important let’s do some math [Music] all right well everyone welcome back um
this episode we’re going to talk about the average rate of change of a function
and so we’re going to need a function and we’re going to need a subset of the
domain which is going to be a closed interval and so the average rate of change
between x1 and x2 is so sometimes we’ll use this delta y over delta x
but this right here explicitly the formula f of x2 minus f of x1
all divided by x2 minus x1 provided that
these aren’t the same so we’re not going to be dividing by zero
now before we continue i would like to just mention that
this episode is part of the series functions and their graphs step-by-step

00:01
tutorials for beginners in the previous episode we talked about
the difference quotient and i briefly mentioned what the average rate of
change is and so now we’re going to go more in depth in terms of what is the
average rate of change so we’re going to look at a really good
example here first so we’re going to see um something that looks like this here
we’re going to have this function coming through here
and it’s going to come up and come down and then it’s going to go down and then
come up and so this is going to be the function right here f of x equals
x to the third minus 3x and this is the point right here 1 minus 2
and this is the point right here minus 1 2
and it does go through the origin right here zero zero so there’s some

00:02
points on there it’s nice and curvy it’s a nice cubic so [Music] there we go
we also have a point down here um minus 2 2. all right and so [Music]
there we go so we got a 1 and then somewhere over here we have a 2.
and all right so now we’re going to ask the question
what is the average rate of change so let’s write that here average
rate of change of this function right here f on the interval here
and let’s go from -2 to 0. so i’m looking at -2 right here
to zero what’s the average rate of change of this function right here so it’s
coming up and it’s coming down so what is the average rate of change here

00:03
so the average rate of change will be given by this right here this will be my
x1 this will be my x2 so i’ll use here x2 which is f of f of 0
minus f of minus 2 all divided by 0 minus to minus 2
and so now what happens if we substitute in zero into the function we get out
zero so this part right here is zero so zero minus now what happens when we
substitute in minus two everywhere here we’re gonna get out of minus two and
and then we have zero plus two zero plus two or just two
so we’re getting two over two so we’re just getting one so that’s the average
value from minus two to zero the average value is just a one there [Music]
there’s a height of 2 right there all right so the average value is 1.
so now let’s look at the average rate of change of f on a different interval

00:04
so average rate of change of f and this time let’s look at the interval
right here from zero to one so from zero to one right here we’re
looking at this part right here so here we’ll have the x1 and x2 again
so f of x1 minus f of x0 f of x2 minus f of x1 and then 1 minus 0.
so when we substitute in 1 here we’re going to get out minus 2 and then 0
and then 1 minus 0 is 1 and so we’re going to get out minus 2.
so now what do these average rate of change represent here
so if i’m looking at the second one for example right here the average rate of
change from 0 to 1 is this point to this point right here

00:05
and if we look at the line through those two points there
and we ask what is the slope of this line the slope is minus two
and if we look at the line going through these two points here so -2
the first part right here is -2 so this point right here and 0 right here and so
now we’re looking at and asking what is the slope of this line
so the slope of this line right here will be one
and those are the average rates of change of f over the interval
so just depends on what the function is doing how the function is shaped and
what interval you choose all right and so there’s a good first
example there but we’re going to do more so let’s
look at what we’re going to do next all right so in our first example here

00:06
we’re going to look at this function right here quadratic and we’re going to
be looking on 0 3. so here we go here’s the average rate of
change up here so it’s going to be f of 3 minus f of 0 over 3 minus 0.
so what happens we substitute in 3 so we’re going to get 9 minus 12 plus seven
minus what happens when we substitute in zero so that’s just seven
and then all over three and so what is this right here that’s a um
the sevens are going to cancel right so we’re looking at 9 minus 12 which is
minus 3 over 3 so i’m going to say -1 here all right so there’s number one
number two we’re going to have so now we’re using
x1 x2 so now we’re going to use f of 2 minus f of 1 minus sorry f of minus 1

00:07
and then all over 2 minus minus 1. now i need to plug in 2 into here
so i’m going to get 10 minus 4 so in other words i get 6
6 minus there’s the minus sign right there
and now what happens when you substitute in minus 1 i’m going to get minus 5
minus 1 so minus 6 and then all over 2 plus 1 is 3
and so i’m going to get 4 out right that’s 12 over 3 is 4.
all right and so next one um x to the third plus three x
um from minus one to one and so let’s see here we’re gonna go to
f of one minus f of minus one all over um one minus minus one
and so let’s see here what happens when we use a one here we’re gonna get one

00:08
plus 3 so we’re going to get 4 minus and then what happens if we substitute
in a minus 1 here we’re going to get minus 1 minus 3 in other words minus 4
and then 1 plus one is two and so we’re going to get eight over two
again we get four and then for part four we’re going to say we have f of five
because we’re looking on one to five so f of five minus f of one all over five
minus one and so what happens when we substitute
in five here we get we’re going to get one over 25
minus f of one which is just one and this is just a one but i’m going to
say it’s 25 over 25 just to know what’s coming up ahead
and then the denominator is four so this will be minus 24 over 25

00:09
and then instead of dividing by four i’ll just say multiply by
um you can’t see that right so let’s get smaller
so this will be minus 24 over 25 and then instead of dividing by four let’s
multiply by one fourth and then here the 24 and four so it
looks like we’re getting minus six over twenty five all right
so there’s the average rate of change for this function on this interval
is minus 6 over 25. and basically what it means is what’s the slope of the line
slope of the secant line it’s going to pass through this curve at two points at
the endpoints of this interval right here and the slope will be this right here
all right so now let’s try this one right here see if we can squeeze it in
down here so this will be f of four minus f of 0 all over 4 minus 0.
so what happens when we substitute in 4 we’re going to get square root of 20

00:10
minus square root of 5 times 0 so that’s 0 over 4
and so let’s say square root of 20 is 2 square roots of 5 over 4
so in other words i’m getting square root of 5 over 2.
all right there we go there’s five examples right there
and now let’s see how to do this on so this was one two three four
and part five right there so now let’s see how to do this um using python using
a computer and if you’ve never used python before it’s pretty easy to do it’s um
a link is below in the description for you to get started on your own notebook
um yeah and so let’s see how this works here so let’s go to
that right here so let me erase all this stuff right here for us real quick

00:11
all right so getting it done by computer is going to be actually very quick and
easy so let’s zoom in here so we can see um
this a little bit better all right so here’s the setup we’re going to need so
we’re going to need this uh set up right
here so we’re going to need you’re going to need to type in all this now the
reason why i’m going to do this here is because i’m actually going to not just
actually find the average value the numeric value but i actually want to
look at the plot so i can kind of see because remember the average rate of
change is the slope of the secant through the endpoints and so i want to
actually see the plot there to see what’s going on there just so i can see that
and we’re going to use some symbols here and
now if you haven’t seen in the other previous episodes then well i like to
customize my axes i don’t like the axes that python uses by default if you like
the default axes then you don’t need to use this right here
all right and i’m going to plot the functions so i’m going to use a plot
function which is going to take a function in and i have a default window

00:12
and a default increment um minus 10 10 and 0 1 and so i’m going
to plot these functions right here so this is all the setup so you just
need to type this up as it is make sure you watch out for spaces and tabs and
all that stuff all right and so let me minimize that and now let’s go to
let’s make sure i executed all those cells real quick
all right and now let’s go to the average rate of change
so here’s the average rate of change it’s just a definition
it’s just a function i mean so the average rate of change i’m going to take
in a function and an x1 and an x2 just like we did in our examples
so i’m going to plot the function and i’m going to use the function and
the x1 x2 to plot it and then i’m going to return the average rate of change
right here it’s just f of x2 minus f of x 1 all divided by x 2 minus x 1.

00:13
so execute that cell to get that in and now we can look at our examples right
so here’s our definition of the average rate of change it’s the delta y over the
delta x we need to make sure and have a subset of the domain
and so we’re going to subtract the y’s subtract the x’s
and so here we go let’s look at some examples here
here’s example one that we did so we got x squared minus four x plus
seven and then on zero three so for my function i’m going to define
here the x squared minus four x plus 7. and then to find the average rate of
change i’m just going to use execute this function right here
make sure and execute this one right here first
and then execute the average rate of change function which we just built
and it plots the function right there and it gives us the output there’s the
average rate of change on zero to three so if we were to look between zero and

00:14
three so if we were to look at these two points right here this is the endpoints
the zero and the three and if i were to find the slope through here
slope of this line is minus one and that’s the average rate of change
all right good so now let’s go on to the
next one which looks like garbly [ __ ] for some reason
so let’s just click on this and execute it again
all right so here’s our second example that we did a minute ago you don’t need
to type that in that’s just to refresh our memory on what it is but we
definitely do need to type in and you need to make this definition
here so i’m calling this f2 so it’s going to be 5x and then minus x squared
and so let’s define that function right there and now i’m going to find the
average rate of change of this function on this interval right here so minus one
to two so i’m going to execute that cell right there
and then we’re going to get the graph right here
and so we’re at minus one so we’re right here at this point right here

00:15
and at 2 we get a hide out and so now you have to ask what is the
slope of the line through these points here
what is the slope of the line through those endpoints right there so the slope
is 4 and that’s the average rate of change of f over this interval right here
all right let’s do again for this one right here
and so yeah i just typed up the example for us right here there it is x to the
third plus three x and now we’re looking at minus one to minus one to one
so i’m gonna define the function x to the third plus three times x
so let’s execute that definition and then now let’s find the average rate
of change of this function on this interval right here minus one to one
so let’s execute that and there we go there’s the curve right there
and here’s the endpoints minus one and all the way to one and so we get
these two points on the graph right here and right here

00:16
and if we were to look at the slope of the line
we would get four and that’s the average rate of change of this function on this
interval right here all right so now let’s do number four
what was number four again let’s click in here and see
refresh our memory all right number four is one over x squared on one to five
hopefully you can see that um and so the function f4 will be one over x squared
execute that um and then the average rate of change of f4 on one to five
just type that in right there and it’s going to give us this graph right here
and so we’re at this point right here and this point right here
and so if we were to find the slope of that line through the slope of
the secant line through those two points right there given

00:17
by the endpoints the one and the five the slope would be negative 0.24 excuse me
and now approximately all right and so now on these last one right here
square root of 5x on 0 4 and so for this function right here
we’re going to use the mp square root so the mp was the numpy
package that we’re using and so we’re going to use square root of 5 times x
so let’s execute that definition and now we have f5 defined
and now we can go find the average rate of change from 0 to 4 and execute that
and there it goes so there’s our where’s that coming from um any case

00:18
uh in any case um square uh f504 all right so there we have we have zero zero
and then at four we have the square root of 20.
and now if we find the slope of the line through those two points right there
we’re going to get 1.118 approximately and then that’s going to be the slope of
the line through those two points and that is going to be the average rate of
change all right and so that takes care of those five examples which we did
um right here a minute ago so let’s see here we can get back to the
examples right here so we did these five examples right here we did them by hand
and we did them using a programming language all right so now i want to
remind you uh just one last time about the average rate of change and the

00:19
difference quotient so i talked about these at the end of the last episode
um but i want to do that again here so the difference quotient and the average
value of a function they’re very very similar
but they are you know different they are different so the average value of a
function we’re going to have a function right here f of x
and we’re going to have a closed interval and so when we sketch something like
this we can say here’s our function here’s our a here’s our b
and we can talk about the average value which is going to be the slope of the
secant line so this slope will be f of b minus f of a all over b minus a
or if you want to use an x one and an x two like we did then you’re certainly
welcome to do that so then you would just have f of x two

00:20
minus f of x1 over x2 minus x1 now the difference quotient we have a function
and we have an h and the way that we’re going to
interpret this right here or both of these is in calculus one
where we’re going to turn these these definitions into processes using
limits so in calculus one you’ll start off by studying limits first usually
and then we’ll turn these two definitions into processes
and these secant lines will become tangent lines so for right now we’ll
have something that looks like this and then we’ll have an x right here
just an arbitrary any x and then we can have any h and i usually
think like to think about h as a um positive distance but it doesn’t have to
be can be any real number x plus h and so let’s actually move it out a little

00:21
bit further x plus h and so h is this distance right here
and so then we get some height right here and so the difference quotient right
here will be the slope of this line through those two uh points
right there f of x plus h minus f of x and then here we get x plus h minus x
and the 8 x’s add up to 0 right there so you really just get an h down there
so there’s the difference quotient right there
so they’re both centered around the idea of finding the slope of the secant line
and what you want to do is try to squeeze this interval down smaller and smaller
so that you actually end up getting a secant line
and that will be used to define what a derivative is either approach but
usually this one right here that will be used to define and so in
physics you’ll have something like the average
uh speed of something or you could talk about the instantaneous

00:22
or you could have the vl the average velocity or the instantaneous velocity
so you’ll see applications lots of applications of the derivative when you
get to calculus so for right now we have the difference
between the average rate of change and the difference quotient right here um
and so i hope that you enjoyed this video and i look forward to reading your
comments in the um in the you know i look forward to reading your comments below
and i’ll see you in the next episode bye-bye if you enjoyed this video please
like and subscribe to my channel and click the bell icon to get new video
updates

About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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