00:00

in this episode we’ll practice approximating the zeros or root of a function

using the python programming language don’t worry everything is explained

let’s do some math [Music] all right welcome back everybody um

before we get started with the uh python part

uh let’s quickly review what a zero of a function is so the zeros of a function

are those inputs whose output is zero so we can look at this

you know and just say we we have a generic graph here a function right here

so here’s a zero here’s a zero and here’s a zero so like this is a graph of

a function and this would be say root number one let’s call it root one um

[Applause] and let’s say here this would be root number two

00:01

and this would be root number three and this function just keeps going down

and so this is just decreasing decreasing here

and a little bit increasing but anyways we get to a root and we get to another

root and we get to another root so we call those uh x values those inputs

where the heights the or the outputs are zero

so those are roots and we can go with a formal definition here if f is a

function then the x values for which you plug in and get out 0

are called the zeros of f or sometimes we call them the roots of f

all right and so in the last episode we did some examples where we looked at

lots of graphs and we talked about finding the zeros

so in this uh episode we’re going to use python to help us find these uh

approximating these roots here so before we move on though i’d like to

00:02

mention that this episode is part of the series functions and their graphs

step-by-step tutorials for beginners especially if you’re in a pre-calculus

course this is a good video to help you understand uh zeros and help help you

define them now you can find the zeros using a graphing calculator

but if you don’t have the money or you know you just don’t want to buy a

graphing calculator then you can do it if you have some kind of computer or if

you have a phone or something like that so check out the link below in the

description and you can start with your own python

notebook and you can follow along just click on the link below and then i

think you click on something like new document excuse me

and then you’ll be able to do everything

that we’re going to do here so let’s get started let’s go to the code now

so let’s get this out of the way here all right

00:03

so let’s uh zoom in here and make this bigger here and

let’s get our setup going here so i’m in the python notebook

and these are the packages that i’m going to import to make things quick and

easy so we’re going to import matplot library and numpy and senpai

and this right here for this video right here this is different uh we haven’t

looked at this um yet and so we’re going to import the fsolve from this package

right here and this will allow us to do a lot of work really quick all right

now because we’re going to be making some graphs

i talked about this in previous episodes but this is the way i like to customize

my axes so i call this function pc for precalculus axes

and so yeah just use that if you don’t want to customize your axes you don’t

have to but i use this function i use this definition right here

all right so let’s execute both of those i’m going to click in the cell and hit

shift enter and then click in the cell right here and hit shift enter

00:04

and then well let’s go look at the functions that we’re going to use for

this episode here in fact actually this uh function right

here is the exact same one we used in the last episode where we talked about

finding the or approximating the relative extrema so i’m going to plot

the function first so we’re going to input a function

and i got some default values here so the default for the input of x minimum

is minus 10 and x maximum is 10 and the increment is

0.1 so these are just some friendly default values to help you get started

but the point is is that we’re going to define a figure and some axes

we’re going to customize the axes we’re going to give it a whole bunch of input

inputs sorry give it a bunch of inputs we’re going to go from -10 to 10

in whatever increments we want and then we’re going to calculate the

outputs so we’re going to take the each of the inputs

00:05

and calculate them so we need to have a function uh set into this function right

here then we’re going to plot it and then

we’re going to show the plot now once we

get a good plot going then we’ll be able to try to find the zeros of of that

and the way to do that is by using this function right here

which you have to give it a function and you have to give it a guess now to find

that guess is what we’re going to use this function for so we’re going to plot

the function we’re going to look at it and we’re going to make some kind of

educated guess and then the the meaty part the reason why this is

only one line long right here is because we’re using the fsolve function

and that takes in a function and a guess and actually gives you a whole bunch of

values right here so i’m only going to choose the first or the zeroth

uh entry in the array and that will be the root and so you

want to use this function right here when you’re certain that there’s only

one zero in that around that guess right there so i’ll

00:06

talk about that by looking at some examples right here

all right so let’s execute those two right there and get those defined

and so there we go so one two three four we’ve executed four cells so far in our

notebook and now let’s go look at some examples here

now these are going to actually be the exact same uh functions that we worked

with in the previous episode but that that episode talked about

finding relative extrema so now we’re finding the um zeros so find the zeros

all right so find the zeros of the function right so i’m going to define f1

as a function and i’m using f1 because this is the

first example and i’m just using generic f right so there’s the name of my

function f1 so it’s a minus two x squared so there’s

how we do an x squared and then plus one

00:07

so i’m going to define my function let’s hit shift enter

and now what i’m going to do with that function is i’m going to plot it before

i start trying to find a 0 for it i’m going to plot it and i’m just going to

use the standard window minus 10 to 10 and i think it has a zero through there

in fact maybe it has two it’s hard to tell um let’s find a better window

so i’m going to try minus one to one and yeah looks like we’re getting two

zeros right there zero right there and zero right there

so i’m going to try to find those two zeros right there so let’s first find

the zero right here the one on the left now you want to zoom out more

perhaps before you start trying to find zeros to convince yourself that those

are the only two zeros or we zoomed out really far um

00:08

and it just looks like it’s just increasing and then it’s just decreasing

and i say really far what do i mean by that i mean we really zoomed out we’re

getting minus 20 000 down here minus 20 000 is pretty large compared to these

numbers over here compared to the way this is structured

uh minus 20 000 significantly larger that it convinces me

that the only zeros we’re going to have are in here

so i’m going to go back to my my window here um now that i looked at it

uh zoomed out i’m much more confident that these are the only two here

now that looks pretty close to negative 0.75 so that’s what uh is going to be my

guess so i called my function here 0 function i’m going to use this with f1

so here we go we’re going to use the function called 0 not 0s and then f1 and

then my guess is going to be 0 negative 0.75 there we go

00:09

so there’s my 0 right there so that’s the actual x value

that we’re getting right there now at this point i would like to just

mention the difference between this episode and the last

the last episode we were trying to find um or we were finding the relative

extrema the relative extrema are y values output values

and so in that video or that episode we returned y values the heights where you

actually had the relative extrema in this episode we’re returning x values

because the roots are the actual x values right there so this is the actual

root right here the x value where the function height is zero

all right so now we can go try to find this one right here and what am i going

to use for a guess i could use like 0.6 0.8 0.4

00:10

but for this guess right here to find this one i don’t want to choose minus 1

1 is not a good guess for this root minus 1 is an okay guess for this root

but if i give it a guess of minus 1 it’s probably going to give me that one back

there so let’s just try that real quick so let’s try 0 again and

let’s get a new cell escape b and this time for my guess i’m going to

choose minus 1 which is a little bit farther away than minus 0.75

so if i tried -1 it should give me that one right there right

so that’s a bad guess for this one over here so this one over here i don’t know

exactly where it’s at but i’m going to guess 0.75 all right so let’s try that

00:11

there we go um and let’s take away the negative sign

all right there we go and now coincidentally we’re getting the same

number here but positive because this graph is symmetric by the way it’s

symmetric with respect to the y-axis so you could imagine that

whatever negative that was is where over here would be the same but positive

and so that that right there is is just confirming that all right so there’s uh

two roots for this function right here we found the two zeros

at least approximately all right and so now let’s go on to

another one right here um so let’s um again we’re going to find the zeros

or the roots let’s call it roots just for fun this time

and now let’s type in the definition so definition i’m going to call this f2

00:12

and then we’re going to go with 2 times and then so i’m going to return

so return 2 times and then x to the third

so we’ve got an exponent there and then minus two times x and n plus one

all right so there’s our function there and since i accidentally clicked on that

let’s change that real quick find the roots

all right so now let’s go here and make a new cell i use escape b for that

all right so now what we’re going to do is we’re going to plot this right here

so let’s go up here and copy that plot function here and

i’m going to use what windows should we use so let’s plot f2

and i’ll just use the default window there and so i’m going to use here um

about uh you know we’re looking for roots here this is just going to keep

00:13

going up and there’s not this is not going to

curve any it’s just going to it’s just going to be increasing here so i’m going

to say minus 2.5 to 2.5 all right and so that we’re definitely

getting a root here and are we getting one here or not let’s

wait on this one let’s find this one right here first so let’s just uh copy

that right there and get a new window or new cell sorry and then paste that in

and change our function we’re on the second example now

so for my guess i’m going to choose so for my root right here

um i’m just going to choose minus 1 minus 1 is close enough it’s not going

to get confused with that one [Music] so -1 here for my guess

00:14

there we go so it gave me minus 1.19 so that right there is going to be the

the zero right there is minus 1.19 and now let’s try to find this one over

here if there is even one right there at

all so to find out if there’s one or not there

i’m going to change the window here to a zero um and it seems like i got

um oops missing a comma all right so it doesn’t look like it

touches there let’s just uh double check our efforts there

yeah definitely doesn’t look like now you see how it’s a little bit uh

line segmenty like it’s a little bit segmented like that that’s because my

increments are pretty small right now or pretty large um

so you know that’s the default value right there i’ll throw in some other

zeros we can make it a little bit more curvier but that’s not going to change

the fact that there’s no zeros there it’s just not going to cross or touch

00:15

the x-axis there all right so we found one root for this right here

uh on example two and it was this one right here

all right so now let’s look at uh this one right here example three

x to the third minus three x squared minus nine plus six let’s execute that and

i’m going to borrow this copy and paste again

and come down here and get a new cell and maybe a second one just in case i

need one all right and now let’s plot f3 using the defaults

and there we go so it looks like it’s going to have some roots maybe one maybe

two maybe three let’s zoom in a little bit just to um

you know get a better better better picture better guesses

00:16

so how about minus three to um i don’t know five maybe

there we go yeah that’s a good window yeah so there we can see our three zeros

right there very good um and so now let’s just go up here and

copy and paste this and i’m gonna use it once and so i need to f3

and i’m going to get another one and another one

all right so there we go so we’re going to find three of them so what’s a good

guess for this first one here so the first one i’m just going to guess of -2

boom we got the root and now this one right here i’m just going to guess a 0.5

so i just guessed a 0.5 in here and it says 0.577

00:17

and then for this one over here i’ll just guess a 4.5

all right and there it is it came up with 4.656 so there we go

4.656 right there for that zero right there all right so um there’s uh the third

example we got two more to go to right here real quick find the roots

and our function here is x to the fourth minus four x to the third plus six

you know try to find the roots of this by hand

factoring or you know exactly how you’re going to do that how are you going to

solve this equation right here can you do that

so you’re not going to try factor by grouping or fact all right

so let’s execute and declare f4 so we got f4 defined

00:18

and now let’s go up here and borrow this and then get a new cell

and maybe we need a couple of them and i’m going to get a zero in there but

first i’m going to get a plot going all right so here we go i’m going to plot f4

and i just guess that the same window might work and it doesn’t look too shabby

it looks right here like there’s a zero and it looks right here there’s a zero

now i know this keeps going up because i’ve looked at this function in the last

episode we looked at it we changed the domain

or the x min and x max we changed the window a lot

same thing on this right here we know this is just going to keep decreasing

if you’re not clear on that change the window

so you can view it up here and you can view it up here more

but we’re going to try to find these roots right now these zeros

so i’m going to execute this twice here but they’re going to be for the function

00:19

f4 and for this guess right here for this root right here

i’m just going to guess 1.5 and then we actually get 1.3

and then for this one right here i’m just going to guess a 4

or since i already have in here typed up 4.5 i’ll just use that

and there we go 3.898 okay so there we go there’s the two roots of that

and then let’s do example five now find the roots

anything interesting couldn’t come up in this one all right

now actually um i like this example here because what this tells me

is that i don’t need a computer i can find them really fast

because this is already factored so x is 0 and x is 2.

00:20

those are going to be the zeros right there so we don’t need to go do anything

else and so uh we define the function here f5

and let’s get some cells going here and we plot

f5 and we said they’re going to happen between 0 sorry 0 and 2

so i’ll look at minus 1 to 3 and then we can see the 0 there and we

can see the 0 at 2. um and then we can just say 0 of f5 and

my guess would just be zero and you get out of zero and here my guess is

00:21

say oops parentheses f5 here let’s say my guess is say i don’t

know 1.8 or something just to be fun so it basically just gives you two

so you can see how it’s an approximation we know the exact values too

simply because in this example we can actually solve for it

and the other examples we couldn’t um i just realized

hope i just did it on this one that that’s mistyped there and

i did it there again too i wonder if i just did it in the last episode

all the way through but uh sorry about that um in any case um

let’s look at two more examples before we go here let’s look at example 6

and find the roots of the function and got it typed up already so x to the

00:22

3rd minus three x plus two all that is the numerator and then x

squared plus two x plus one all that is the denominator so let’s plot that

um let’s go up here to get the plot and we’re going to plot f6

and we got the divide by zero which we talked about in the last um example we

can just ignore that there’s a zero uh somewhere in here and

i mean it’s dividing by zero in here it’s it’s there’s an isotope that’s the

problem um so the question is does this have any

zeros right here so does it cross the x-axis anywhere here

so let’s zoom out here and see if we can figure this out here so it

looks like it is decreasing and then it starts to

00:23

increase but does it ever actually cross so right here um it certainly crosses

um looks somewhere around negative 2.5 but what about here so let’s zoom in

around between let’s say negative 0.5 and 2. um

still can’t tell if it crosses or not let’s try positive 0.5 right there

oh there that looks like it almost touches at one let’s go to 1.2 and

let’s go to 0.8 and let’s make the increments better so that

00:24

it doesn’t look weird looks like it’s hitting zero right there all right

well i moved it let’s see what happens um when we find the zero

and let’s go between a zero point or let’s say the guess is one and

we’re getting a two let’s see here when i guess at the root of one

the zero gives me out the zero function gives me out a two um let’s see here

what if we move over to a 2.2 so it’s just giving me out gibberish

00:25

there’s no there’s nothing at two here um what happens if we actually go up here

and say what is f of six at one you see we actually get the zero out right there

um oh i see what’s happening um there we go all right that makes sense

f of six is my guess of one yeah we get the zero one there okay

all right that makes perfect sense all right so now let’s go um

so there’s one zero at one and now what about for the other side so

now let’s go back to the other side what was happening over there

00:26

um let’s see we need to go further and let’s say we go to negative one here

and how about changing that to a one so let’s see here

we’re looking somewhere around here um where does it cross maybe we need to go

a little further right and let’s just go to positive one here

positive five maybe so i can’t seem to remember what window where it looked good

00:27

for us so let’s just come back out and go to -10 to 10 and minus 2.5 2.5 and

yeah so why did i think there was a zero right there so what if we went to minus

25 all right so i think it crosses here but um so let’s try to zoom in now

i don’t seem to be having good luck on the left side there

00:28

so let’s say here this is i don’t need all this stuff over here so

let’s say negative 0.05 and where is a good guess over here so we can say the

uh here it is so let’s try the guess for over here let’s just guess negative 1.4

and it looks looking like it’s happening at negative two

so let’s go this out to negative three and negative one

and it’s hard to tell that it’s crossing right there at negative two [Music] um

00:29

but that’s what’s happening so this is decreasing right here

and then it’s just decreasing long term so what if we zoom out really far and

it doesn’t seem like it’s decreasing long term here but it is 300 all right so

we’re getting 0 at negative 2 here really close in here

and then after that it’s just decreasing but it’s doing it very slowly here

let me try this here just in case something uh got messed up here

00:30

let’s go and save this and then restart and run the kernel again here right

all right so let’s move on now to this last one here let’s get here

back in here let me move out of the way over here

all right so i’m going to define this function right here f7

for this function right here x over square root

and then we have square root of x squared minus 1 here

and then now let’s go over here and um get a new cell and then

we’re going to plot this f7 right here and let’s just use the standard window

here and we talked about last time about the domain of this function right here

00:31

from from 1 and on and -1 and on here so the domain is right there

uh open these are open right here so minus infinity to minus 1

union 1 to positive infinity here so that’s the domain of this right here so

we’re not going to see anything defined in here

and so we’re going to really break this up into two pieces here let’s look at it

the first one let’s plot the right half first so let’s say one two

five let’s say for example and then we get a nice curve right there

so the question is does this ever curve down and make a zero out of it so let’s

zoom out a little bit let’s try 50 here [Music]

and yeah so it’s just leveling off right there

um so we’re not going to have any zeros over here

00:32

and then let’s try the opposite let’s try -50 over here and a minus one here

and we’re going to get the same behavior over here

it’s just leveling off and we’re never going to get any zeros here so here’s a

function um [Music] you know here’s a here’s a function

right here that doesn’t have any zeros if you go to try to solve this

algebraically and say all right let me try to find the zeros here

so then you’ll say oh the x has to be zero which is true for the numerator has

to be zero but if you try zero in the denominator

it’s not defined x equals zero is not in the domain so this function right here

has no zeros all right so there we go so there’s

seven examples we dived in pretty good and um hope you got a better um feeling

for how you can use python and um get these zeros

so i look forward to seeing you in the next episode have a great day

00:33

if you enjoyed this video please like and subscribe to my channel and click

the bell icon to get new video updates