Slope fields provide a visual method to understand the behavior of solutions to differential equations, particularly in real-world applications like falling objects with air resistance and population dynamics, by illustrating the rate of change at various points.
Mind Map
클릭해서 펼치기
클릭해서 인터랙티브 마인드맵 전체 보기
so now that we have a feel about what
slope fields are let's try to use them
in an application a little bit more
realistic than just drawing slopes
everywhere and figure out what curve
goes through it this will give you an
idea of what goes on in some real-life
problems using differential equations so
again let's start off with some sort of
application let's if it's tough about
slope fields and we'll get this this
picture what's really going on with our
situation so a lot of times in some of
the simplified word problems you're used
to we neglect things like wind
resistance which isn't really realistic
if you jump out of like let's say a
helicopter before you jump on a
helicopter okay so go don't jump up
because it's spinning but let's say you
jump out of the helicopter well you're
gonna fall now that makes sense that if
you if you jump downward you're gonna
fall faster and if you just kind of fall
out you're gonna fall slower but when
you think about how wind resistance
works if we just had gravity and and no
wind resistance yeah we pick up velocity
until we smash them to the ground or
pull the parachute but if we have wind
resistance as our gravity is pulling us
down at a constant acceleration our
velocity is increasing yeah but our
acceleration is not we're falling down
well we're picking up velocity and as we
pick up that velocity we encounter more
air resistance wind resistance we
encounter this this force upward of the
wind holding us up or pushing us up so
gravity's going this way but the air
resistance is actually a counter force
pushing us up and it depends on on your
velocity so as it turns out if we have
how our velocity is changing that's
acceleration remember that the way that
velocity is changing the rate of change
of our velocity with respect to time
that's acceleration well in previous
examples like simplified versions we
would just do this and we go yeah we're
accelerating from gravity wind
resistance but we do have
resistance so if we look at the way that
velocity is changing perspective time
our acceleration it is the acceleration
of gravity - so that's downward so
acceleration to gravity let's consider
that gravity is pulling us down air
resistance is opposite of that so our
air resistance is some proportional
constant like K but it's proportional to
our velocity so the constant
proportionality is K that's some sort of
a number that's based on whatever the
object is that's falling so we have a
larger body or something would be
encountered more air resistance than
like an arrow or a bullet or something
and but it is proportional to velocity
so the faster you are going the more air
resistance you encountered we know that
right out because if we're standing
still you're not feeling any resistance
but if you were running twenty miles an hour
hour
that'd be really fast you would feel a
lot of wind against you or if you stick
your hand out of a moving car then
you're gonna feel that that wind against
your hand so the and the faster you go
the more wind that you feel well then
that's pushing us upwards as a force
that's acting upwards so in real life
our acceleration the rate of change of
velocity respective time is yes gravity
gravitational constant but - the air
resistance I hope that makes sense to
you so G is that gravity and it's the
constant of acceleration for gravity k
is the this this idea of you have a
and that would be objects specific so
whatever you are throwing out or dumping
out or jumping out that that K is gonna
change you know what object you or
considering their resistance for so
here's the idea and this is what I
really want to get get out of this if
you just consider the gravitational
acceleration so 32 feet per second
squared and so every second that you're
going you're increasing your velocity or
32 feet per second well if you don't
consider air resistance yeah we pick up
velocity until we smash into the ground
but if we have air resistance something
pushing up against us at some point
check this out at some point G will
equal K times V somewhere along this
line this is going to equal this look
that's a constant that's not changing
we're on earth it's always the same
gravity yeah it does change as we get
away from the earth but not not by much
so that's constant if that's a constant
and we hold that constant and this
increases as velocity goes up so so you
dump something out of the helicopter
it's gonna pick up speed as it picks up
speed it gets more and more and more air
resistance against it so the way our
velocity is changing our acceleration is
constant but this is growing as time
goes on that's growing at some point
these two things will be equal well if
you subtract to equal things what do you
get you get zero so at some point our
acceleration the way that our velocity
is changing will be zero it means our
velocity won't be changing any more let
me explain that one more time so you
throw something out your velocity is
going to change that's acceleration how
your loss is changing so the rate of
change of velocity with respect to time
your acceleration is G minus air
resistance so right at the beginning you
just have G but your velocity starts
going up you're going to gain air
resistance so that's pushing against
your acceleration that's pushing against
your the body and it's going to slow
your acceleration
if that's constant and that's growing
your acceleration diminishes to the
point where at some point you're getting
zero acceleration that doesn't mean your
velocity zero that means though that
your velocity is not changing so at some
point you're going to hit a terminal
velocity or some people call that a
limiting velocity for speed related type
problems we're going to hit that and
we're going to see that in our slope
field so without any specific techniques
on how to solve these things we can come
up with a slope field to describe the
situation so one last time
yeah we're jumping out yes the way the
loss of change spec times called
acceleration gravity is a constant but
the the proportionality of your air
resistance is is it's proportional to
well your air resistance is proportional
to plus the faster you go the more wind
pushes against you at some point
you're going to reach this where the way
that air resistance is forcing up is
equal to how gravity is pulling down
that's going to that's going to be your
limiting velocity so when that happens
it's gonna be your limiting velocity and
we're gonna check that out so let's say
let's explore one so let's say that this
is our situation we're gonna jump out of
a helicopter and that right there that's
gravity that's 32 feet per second
squared we have found somehow the
constant of variation or how we're
proportional is this constant of 1.6
times whatever velocity you're traveling
that's gonna be how our wind resistance
is affecting us let's say this is us
jump down so if we have a first order
differential equation we know that this
means slope so first derivative means so
that means our slope is 32 minus one
point six V the where's-where's fatigue
remember that velocity is already a
function of time so velocity is changing
with respect to time the T is wrapped up
in that
that's our dependent variable but T's in
there somehow so all we really need is
velocity what that means is this is
going to be weird but as our time goes
on these numbers are not going to change
there's no place to even plug in T
ve already has all the t's in it so I
hope you've got to do is go plug in zero
if we plug in zero for our V and here's
all of our T's plug in zero you're gonna
get 32 and that's not going to change
our T doesn't affect that our velocity
is already already has those it's
already bunch of T it already has those
T's in effect that plug into five what
we get if we plug in five we get 24 so
if we plug in 10 well that's 32 minus 16
so we need 16 all the way down if we
plug in 15 32 minus 1 point 6 times 15
gives us 8 if we plug in 20 we're gonna
get 0 this is what you're looking for
and if you continue this this is
negative 8 and negative 16 you know if
you were watching for the this slope
field video the one right before this we
had diagonals that weren't changing that
happens a lot when you have like X is
plus or minus wise this when they're
your x I did a variable on here you're
going to get these verticals that aren't
changing happens and I know that you can
plug in 40 and do 30 minus 1.6 times 40
and get these these same exact numbers
I'm getting so let's make all this makes
sense before we do our slope field and
look at what's going to happen on the on
our graph so we know that in real life
we get air resistance that means that
when you jump out you're going to reach
some sort of limiting velocity at some
point your air resistance is going to
meet up with acceleration of gravity if
you're jumping high enough and don't get
the ground first or or polar parachute
or something they're gonna be equal that
means that you're not going to be
changing your velocity after that you
had this zero change as zero
acceleration means that you will have a
constant constant velocity at that point
so this is what that does this says if
you jump out
it used fall so your initial velocity is
zero you're not jumping downwards you're
in a helicopter right so you're probably
not gonna jump up because there's a
blade that would be a negative initial
velocity so if you jump down with
negative five you're gonna probably hit
the propeller you don't want to do that
so this would be jumping downward so if
we jump out at zero so just fall well
initially we're going to be going this
30 - oh well let me let me say that and
say this is they will see this on our
arm picture we're going to be increasing
our velocity at this 32 feet per second
if we jump out of five does it make
sense that if you jump out going
downward you're going to immediately
encountered air resistance
immediately that means that your rate of
change of velocity is going to be slower
you've already jumped out at five feet
per second so the rate at which you're
changing is a little bit slower now
let's say you jump out of ten you're
jumping downward at ten feet per second
already the reign of changing to be
slower you've encountered that air
resistance immediately here you did it
that's why we're Exelero
that's why our velocity is changing
faster here no here no here a little bit
little bit slower now what about 20 and
20 if you jump downward at 20 feet per
second right from the get go out of this
helicopter your rate of change isn't
going to be anything at all you've
encountered just enough air resistance
to meet up with your gravity so this is
weird and we're gonna see it on the
picture but if you jumped out of this
helicopter at 23 per second you're not
gonna feel your velocity increase or
decrease you will have immediately met
your limiting velocity that's wild
now let's say that you jump out faster
than that let's say that you jumped down
wow I just really wanna get on a
helicopter and you're jumping down at 25
feet per second you're actually going to
slow down so if you're jumping faster
than your limiting velocity air
resistance is pushing up more than gravity
gravity
he's pulling you down should kind of
slow down that that's the same principle
like when you when you shoot if you shot
a gun downward so it's at some point
your guns going so fast that it's a it's
meeting so much air resistance that
gravity's not gonna overcome that your
air resistance is pushing up more than
gravity's pulling that down you have an
initial velocity there that's so fast
it's counting so much air resistance
it's going to slow that bullet down even
though you shouldn't downward that's
what's happening here so you jumped out
at forty feet per second downwards that
really pushing it you're gonna encounter
so much air resistance that that's
pushing you up more than the gravity is
pulling you down that's weird but that's
what this is gonna play on us so long
story made real short we have air
resistance it's proportional to your
velocity and we can show the picture of
it really kind of nicely this point
where we have a zero change that is
called your limiting velocity or
terminal velocity or when we talk about
population will get like this population
limiting factor layer carrying capacity
something that limits how much we can
grow if you're below that you're going
to increase if you're above that you're
gonna decrease until you find this
leveling spot this limit so let's put
these slows remember these are all
slopes how our velocity is changing is
right here let's put them on our graph
let's see what happens again we made the
grid it's a lot easier to use so because
our time isn't affected well our time is
affecting our our velocity but it's
wrapped up in the velocity itself so
four times zero one two three four we're
gonna have these these intersections of
time and velocity so zero zero right
down here we have a slope of 32 but
that's really steep I'm not gonna really
build it draw that as not a vertical
line that's gonna be darn near a
vertical line 32 rise over run but also
at so here's our velocity so 0 5 10 15
20 years of times 1 through 6
so at 0 0 we got 32 at
equals one and velocity zero we also
have thirty-two at T equals two and
velocity zero will get 32 so all along
this velocity of zero we have a slope
with 32 this is acts like a diagonal
notice not bad no but this does not
change darn your vertical lines now as
we bump it up to five our velocity goes
to five feet per second then our slope
starts to two not starts to decrease
from 32 and we talked about why that was
so you jump out with an initial velocity
you immediately encountered air
resistance which means your velocity
can't change as drastically as it did if
you jump down at zero at zero you have
no air resistance for a little while
that means that you're going to increase
velocity faster because you're it's only
gravity that's affecting you as you
increase speed you start encountering
air resistance so the way your velocity
is changing can't be as drastic is if
you didn't have air resistance so the
more air resistance you have the less
that your acceleration of gravity can
take over here we get 24 so all along
at 20 we have zero and that is just as
that should be striking to be like oh
well hobby
obviously if our spokes are really no
matter where where our soaps are gonna
increase we're gonna go uh we're gonna
level out right here so if we and this
is what is saying to you as we're
looking our soap field that's this isn't
this is important because if models
real-life concepts really really well
that's what so people to do that's how
we want to use them if you jump out at
any velocity up to twenty feet per
second going downward you're going to
increase your velocity gravity is going
to overcome your air resistance until
you reach that point and then you're
gonna level out if you jump out faster
than twenty feet per second down so
let's see 25 you get negative eight at
thirty we have negative sixteen so in
and putting this all together our slope
field gives us how are our velocity is
changing with respect to time if we jump
out or fall out or are putting an
initial velocity that's slower than 20
feet per second this is all saying that
yeah you're going to increase until you
reach that level if you jump out that
faster so going down we're jumping
downward at faster than 20 feet per
second you're actually going to slow
down here
gravity is overcoming your air
resistance until you hit 20 here air
resistance is overcoming your gravity so
this would be bigger than gravity that's
what you're slowing down what we what's
gonna happen though is that these two
things are gonna level out in real life
your air resistance and your gravity are
going to come to some sort of
equilibrium an equilibrium solution
that's what this is the the solution
where if you started at 20 so 20 feet
per second you jump out you're jumping
down route 23 per second you are not
gonna feel a change so it's not gonna
feel like you're increasing speed it's
not gonna feel like you're decreasing
speed check out slower than that it's
gonna feel like you're speeding up
because gravity's pulling you faster
than your air resistance when you start
to a point to that that level of 20 feet
per second where we even out here if
you're jumping faster than that so jump
me down we're faster than that you're
going to feel like you are slowing down
your slope is negative here you're gonna
be decreasing your velocity until you
reach that equilibrium until you reach
that limiting velocity in that weird
it's kind of cool so right here this
this 20 that right there is our limiting
velocity no matter what happens whether
you jump out whether you're pushed out
jump downward whatever you're going to
reach that velocity given enough time so
let's kind of interesting we'll figure
out how much time you really need again
you're not going to jump up so
don't have a negative velocity not gonna
jump upward because of the helicopter
idea so our limit is this 20 feet per
second so let's say that we let's say
that we jump down at 0 so so we start
here so remember from our our slope
fields that we had an initial condition
and starting spot of 0 0 so at x 0
we're not traveling any we're just
jumping down when I choking downward
let's see how fast we actually come up
to that that 20 feet per second so
starting starting here remember that
slope fields that they've got to find
this curve that matches these slopes so
we're starting really steep when we get
to this level we better have a slope of
positive 24 so when we hit 5 we gotta
have a slope of 24 when we hit 10 we've
this right here looks like it's about
seventeen and a half or something so we
reach 17 and a half feet per second
after only two seconds so we go from 0
to 17.5 feet per second in 1/2 that's
like well what is that that's a good
percentage of 23 per second when we if
you divide that so do set about 70 and a
half divided by 20
that's a percentage of that limiting
velocity that you reach after two
seconds so you can find out how long
it's going to take you to reach a
certain percentage of 20 now
theoretically do you ever reach 20
it's an asymptote so technically no or
theoretically no but that's the limit so
you're going to get to that point get
closer and closer and closer but but
you're never going to exceed that that
velocity you can only limit to it wild
you're gonna slow down until you reach
your limiting velocity I hope this is
making sense to you it's very
interesting the way that works in real
life because at some point when you
really do think about it gravity and air
resistance if you're jumping up faster
than gravity slow down you're jumping up
slower than gravity
you're gonna speed up but that air
resistance is forcing you backwards so
when you're when you're when you're self
when you're jumping up so fast
air is just this is overcoming gravity
when jumping out so slow gravity's
overcoming air resistance but some what
they mean that's what we're talking
about with an equal image solution you
jumped out of twenty that can feel
change faster slurring that you're gonna
reach that limiting velocity we're gonna
do one more example about population
will talk about limiting population or
carrying combat capacity so hang on for
that one I'd like to walk you through
one more example about population we're
gonna do this very quickly the idea is
exactly the same I just want to show you
how slope can do some funny things in
the middle of this this problem to show
you that on the slope field and to talk
about something called carrying capacity
so so when we look at this we have the
way that that a population is changing
with respect to time now again we don't
have a T up here but the population
itself is a function of time and so the
way that population is changed or you
think of it the acceleration of
population that's kind of weird but how
fast is the population growing or decreasing
decreasing
well that's given by this for a certain
population with TB months now because we
don't have a team in here our slope is
still equal this here is the slope of
the population the way the populations
change with respect to time is still
equal to this but there's no T involved
the P hat is a function of time already
so it's it's involved in that population
well if we want to go ahead and do a
slope field our slope is given by this
function of P a function of T with our
dependent variable P in there let's just
start plugging in initial populations so
what if we started
zero well okay 0-0 these are all going
to be zero
do you see that it's not gonna change as
far as what time we have we're gonna
have zero this whole way down same thing
happens for all this I'm just going to
be one number if we start with ten
our slope would be point two all the way
down if we start with twenty point three
three would be our slope start with
thirty point for our slope start with
forty point for to move your so let's
start with fifty point 0.38 and usse
know something funny kind of happening
so our slope is increasing and then it
starts decreasing again that's a very
interesting than 60 would be point two
seven 75 would be zero you're looking
for that and then 80 would be negative
point one two here's what this is saying
this would be all the way down for all
of these remember my goal in teaching
you this not just how to do it
understanding what's going on if this is
the way that the population is changing
what this says is that if you start with
these populations here's what is going
to happen if you start with a population
of zero are you gonna see any growth
probably not there's nothing to grow if
you start with a population of ten
whatever this population size is you're
going to immediately start seeing a
population increase twenty increase
thirty increase faster for to increase
faster and that should make sense right
because if you have more things to
reproduce the population should be
growing faster to a point because our
natural resources aren't limitless which
means that as we get closer and higher
and higher our population is growing but
not as fast because we're reaching some
sort of
factor here at some point we hit what's
called in carrying capacity so you start
with 75 units of population whatever
that is it's not gonna grow it's not
gonna fall it's limited if you start
with more than that you're gonna
actually have a die-off if you put say
you've had a really small pond and you
just you don't zero fish in no Fisher
code you go okay that they don't just
spontaneously come about listen make
sense if you dump in way too many fish
like a billion fish in a really small
pond there's not enough resource just a
lot of them are going to die so as we
start having our population go higher
and higher and higher it makes sense
that at some point or not you'll be able
to grow anymore in fact we're going to
start dying off because there's just too
much of this population in one spot
there's a middle ground that because if
you start low and it's gonna grow you
start high it's gonna die that's both
wrong that's kinda fun then at some
point there's this equilibrium solution
and that's what that 75 does that's zero
right there
look positive slopes negative slopes
there's got to be a zero that right
there that's your carrying capacity that
means that no matter whether you start
lower than zero or higher you are going
to level off you're going to limit
yourself to that 75 popular units of
population whatever that is and the
picture is gonna bear that out so I'm
just really quickly but at zero zero we
have of course zero and if we had a
population start at zero we're gonna
zero all the way no matter what if we
start up with ten so we're gonna have
this part of this is slope with 0.2 so
all the way along get a little point to
slope and we're going to increase until
we get to 40 populations start on
I'm kind of exaggerating my slope right
now so you see what's going on but then
once we get to 50
please start to slow down so the most
rapid growth that we have is right
around start with 40 you start with that
much if you have a lot of things or
whatever these beans are to reproduce
and you but you have enough resources to
make it sustainable
when we get over that if we start
slowing back down then even slower and
probably even so I don't have this
listed here I'm missing my 70 I don't
want to put that out there but even
slower 70s point one of you on them to
the point where we have zero and above
so if you're looking at your slope field
that looks like this this should be
really telling what's going on if you
start with a population of zero you are
not going to see any growth obviously if
you start with just slightly more than
that it's going to grow slowly at first
but then as our time moves on look how
our curve has to flow through this slope
field so it seems like well time doesn't
matter it does matter but it's just like
you're starting at a different level
here population wise so if we start at
zero and we have no population no matter
what our time is we have zero that that
is an equilibrium solution but it's not
a good one I mean it's like it's like
the trivial solution you know yeah it's
it's nothing it's like the same you're
gonna launch a basket where you launch a
rocket from the ground at time zero it's
on the ground
when's the rocket on the ground and time
zero Thanks I meant like after it's all
done once it hit the ground so if you
start with no population obviously
you're not gonna have when I'm at one
time but if you start with just higher
than that you are gonna flow along this
curve you're a mattress slopes up but
eventually you're gonna level off if you
start higher you just gain quicker but
you're gonna level off if you start
lower yeah my team may take you a long
time to finally reach that because you
don't have as many many reproducing
items if you start way higher what
you're actually gonna drop this is like
what the last example your gravity to
overcome in your air resistance or your
air resistance overcoming your gravity
this is the same thing your level of
food is overcoming your population which
means you can grow or your populations
overcoming your level of fitness called
food which means your population is
going to decrease you'll have enough so
that's the way that these slope fields
work they're looking for the 0 that 0 is
typically your carrying capacity or your
limiting velocity for speed problems or
your equilibrium solution if you start
with that so if we started with 75 units
of population you wouldn't change if you
make sure you're hitting these slopes as
we travel along but eventually we are in
love allowed so how you can figure out
things like how fast would it take you
to hit 90 percent of your or 75 percent
of your your limiting capacity or the
two thirds or whatever and just look for
where you reach let's say 60 a level of
60 that happens when do I hit a level of
60 let's go if I start at 10 if I start
at 10 units population well that would
mean that I did right around the 60
months or so they said 48 15 somewhere
48 60 months so the whole idea of soap
fields one last time as last we're gonna
really talk about it
so fields model hopefully some realistic
examples that you can't
or maybe aren't able to solve with
different equations right now or maybe
even ever what they do is they consider
a first derivative to be the slope which
we have they say that's the formula for
your slope
no problem I can just plug in some
points I got it then it's gonna give you
a slope at every point the whole picture
is your general solution so every
possible curve would happen fit these
slopes so this is your general solution
it tells us a lot about situations
though if you just look at the slopes it
goes about that's your that's your
limiting factor right there so if you
start with the population 75 you ain't
gonna grow if you jump out of the
helicopter 20 feet per second year I'm
not gonna fall any faster so they tell
us a lot about a situation just by
looking at it to get a specific curve
you need a starting point so you could
say I'm going through this point and
then you can find a particular solution
that's how all initial conditions work
any way this condition gives us a
particular solution a lot of times we
can see what's our trends in in gaining
or losing and how that would really
affect our real life situation so I'm
I'm hopeful that this is making sense to
you I'm hoping that you see the the
niceness about this that you can
approximate solutions without actually
having to do anything with the problem
that's pretty cool
computers do this really well and what's
nice about it is remember that whole
talk about how we have this trade-off
between perfectly modeling the situation
and doing the math behind it well if
you've got computers you can kind of do both
both
you can perfectly model situation and
then do a slope field and it will show
you a lot of trends in it I'll show you
limiting capacities are limiting
limiting factors or carrying capacities
or limiting velocities and include more
variables even though United will solve
it and that's that's pretty neat so I
hope this makes sense I hope that right
now you're able to take a slope from a
difference of the equation make up a
table to get yourself a slope field and
interpret what's going on look at how
slowly we go we start low or how quickly
we decline if we start high look and
think about how that would make sense in
a real life event with low populations
we'd grow slowly but as we get more and
more population we start living
ourselves because of resources to a
point where we're going to level off
just enough resources to support us but
not grow any more you start too high
there's not enough resources to support
and we would decline population that's
what so feels about is interpreting
real-life situations so next time what
we're going to talk about is how to
determine when we're when we're gonna
have solutions or not so so fields one
way to deal with it when we don't have a
solution but now we're going to start
determining when we do have solutions
and when we can expect a unique one so
we'll talk about that and then after
that we start dealing with solving
텍스트나 타임스탬프를 클릭하면 동영상의 해당 장면으로 바로 이동합니다
공유:
대부분의 자막은 5초 이내에 준비됩니다
원클릭 복사125개 이상의 언어내용 검색타임스탬프로 이동
YouTube URL 붙여넣기
YouTube 동영상 링크를 입력하면 전체 자막을 가져옵니다
자막 추출 양식
대부분의 자막은 5초 이내에 준비됩니다
Chrome 확장 프로그램 설치
YouTube를 떠나지 않고 자막을 즉시 가져오세요. Chrome 확장 프로그램을 설치하면 동영상 시청 페이지에서 바로 자막에 원클릭으로 접근할 수 있습니다.
