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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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