0:02 so now that we have a feel about what
0:05 slope fields are let's try to use them
0:07 in an application a little bit more
0:09 realistic than just drawing slopes
0:10 everywhere and figure out what curve
0:12 goes through it this will give you an
0:14 idea of what goes on in some real-life
0:16 problems using differential equations so
0:19 again let's start off with some sort of
0:21 application let's if it's tough about
0:24 slope fields and we'll get this this
0:25 picture what's really going on with our
0:29 situation so a lot of times in some of
0:30 the simplified word problems you're used
0:33 to we neglect things like wind
0:35 resistance which isn't really realistic
0:37 if you jump out of like let's say a
0:39 helicopter before you jump on a
0:41 helicopter okay so go don't jump up
0:44 because it's spinning but let's say you
0:47 jump out of the helicopter well you're
0:49 gonna fall now that makes sense that if
0:52 you if you jump downward you're gonna
0:54 fall faster and if you just kind of fall
0:58 out you're gonna fall slower but when
1:00 you think about how wind resistance
1:03 works if we just had gravity and and no
1:06 wind resistance yeah we pick up velocity
1:08 until we smash them to the ground or
1:12 pull the parachute but if we have wind
1:15 resistance as our gravity is pulling us
1:18 down at a constant acceleration our
1:20 velocity is increasing yeah but our
1:22 acceleration is not we're falling down
1:25 well we're picking up velocity and as we
1:27 pick up that velocity we encounter more
1:29 air resistance wind resistance we
1:33 encounter this this force upward of the
1:36 wind holding us up or pushing us up so
1:38 gravity's going this way but the air
1:41 resistance is actually a counter force
1:43 pushing us up and it depends on on your
1:46 velocity so as it turns out if we have
1:49 how our velocity is changing that's
1:51 acceleration remember that the way that
1:53 velocity is changing the rate of change
1:54 of our velocity with respect to time
2:03 that's acceleration well in previous
2:04 examples like simplified versions we
2:07 would just do this and we go yeah we're
2:08 accelerating from gravity wind
2:11 resistance but we do have
2:15 resistance so if we look at the way that
2:17 velocity is changing perspective time
2:20 our acceleration it is the acceleration
2:22 of gravity - so that's downward so
2:24 acceleration to gravity let's consider
2:27 that gravity is pulling us down air
2:29 resistance is opposite of that so our
2:32 air resistance is some proportional
2:35 constant like K but it's proportional to
2:37 our velocity so the constant
2:39 proportionality is K that's some sort of
2:42 a number that's based on whatever the
2:46 object is that's falling so we have a
2:49 larger body or something would be
2:51 encountered more air resistance than
2:53 like an arrow or a bullet or something
2:55 and but it is proportional to velocity
2:58 so the faster you are going the more air
3:00 resistance you encountered we know that
3:01 right out because if we're standing
3:03 still you're not feeling any resistance
3:06 but if you were running twenty miles an hour
3:06 hour
3:08 that'd be really fast you would feel a
3:11 lot of wind against you or if you stick
3:14 your hand out of a moving car then
3:17 you're gonna feel that that wind against
3:19 your hand so the and the faster you go
3:21 the more wind that you feel well then
3:23 that's pushing us upwards as a force
3:26 that's acting upwards so in real life
3:28 our acceleration the rate of change of
3:31 velocity respective time is yes gravity
3:33 gravitational constant but - the air
3:35 resistance I hope that makes sense to
3:45 you so G is that gravity and it's the
3:48 constant of acceleration for gravity k
3:53 is the this this idea of you have a
4:27 and that would be objects specific so
4:30 whatever you are throwing out or dumping
4:33 out or jumping out that that K is gonna
4:35 change you know what object you or
4:37 considering their resistance for so
4:40 here's the idea and this is what I
4:41 really want to get get out of this if
4:45 you just consider the gravitational
4:48 acceleration so 32 feet per second
4:50 squared and so every second that you're
4:51 going you're increasing your velocity or
4:56 32 feet per second well if you don't
4:58 consider air resistance yeah we pick up
5:01 velocity until we smash into the ground
5:03 but if we have air resistance something
5:05 pushing up against us at some point
5:15 check this out at some point G will
5:18 equal K times V somewhere along this
5:21 line this is going to equal this look
5:23 that's a constant that's not changing
5:25 we're on earth it's always the same
5:26 gravity yeah it does change as we get
5:29 away from the earth but not not by much
5:32 so that's constant if that's a constant
5:34 and we hold that constant and this
5:37 increases as velocity goes up so so you
5:38 dump something out of the helicopter
5:41 it's gonna pick up speed as it picks up
5:44 speed it gets more and more and more air
5:47 resistance against it so the way our
5:48 velocity is changing our acceleration is
5:52 constant but this is growing as time
5:54 goes on that's growing at some point
5:57 these two things will be equal well if
5:59 you subtract to equal things what do you
6:02 get you get zero so at some point our
6:04 acceleration the way that our velocity
6:06 is changing will be zero it means our
6:09 velocity won't be changing any more let
6:12 me explain that one more time so you
6:14 throw something out your velocity is
6:15 going to change that's acceleration how
6:17 your loss is changing so the rate of
6:18 change of velocity with respect to time
6:21 your acceleration is G minus air
6:22 resistance so right at the beginning you
6:25 just have G but your velocity starts
6:28 going up you're going to gain air
6:30 resistance so that's pushing against
6:32 your acceleration that's pushing against
6:34 your the body and it's going to slow
6:35 your acceleration
6:37 if that's constant and that's growing
6:40 your acceleration diminishes to the
6:42 point where at some point you're getting
6:44 zero acceleration that doesn't mean your
6:46 velocity zero that means though that
6:49 your velocity is not changing so at some
6:51 point you're going to hit a terminal
6:53 velocity or some people call that a
6:56 limiting velocity for speed related type
6:58 problems we're going to hit that and
6:59 we're going to see that in our slope
7:02 field so without any specific techniques
7:04 on how to solve these things we can come
7:06 up with a slope field to describe the
7:09 situation so one last time
7:12 yeah we're jumping out yes the way the
7:13 loss of change spec times called
7:16 acceleration gravity is a constant but
7:18 the the proportionality of your air
7:20 resistance is is it's proportional to
7:22 well your air resistance is proportional
7:24 to plus the faster you go the more wind
7:27 pushes against you at some point
7:30 you're going to reach this where the way
7:33 that air resistance is forcing up is
7:36 equal to how gravity is pulling down
7:39 that's going to that's going to be your
7:42 limiting velocity so when that happens
7:44 it's gonna be your limiting velocity and
7:46 we're gonna check that out so let's say
7:48 let's explore one so let's say that this
7:50 is our situation we're gonna jump out of
7:53 a helicopter and that right there that's
7:55 gravity that's 32 feet per second
7:58 squared we have found somehow the
8:01 constant of variation or how we're
8:04 proportional is this constant of 1.6
8:06 times whatever velocity you're traveling
8:10 that's gonna be how our wind resistance
8:12 is affecting us let's say this is us
8:16 jump down so if we have a first order
8:18 differential equation we know that this
8:20 means slope so first derivative means so
8:24 that means our slope is 32 minus one
8:27 point six V the where's-where's fatigue
8:29 remember that velocity is already a
8:32 function of time so velocity is changing
8:34 with respect to time the T is wrapped up
8:34 in that
8:36 that's our dependent variable but T's in
8:38 there somehow so all we really need is
8:40 velocity what that means is this is
8:43 going to be weird but as our time goes
8:45 on these numbers are not going to change
8:47 there's no place to even plug in T
8:50 ve already has all the t's in it so I
8:54 hope you've got to do is go plug in zero
9:00 if we plug in zero for our V and here's
9:02 all of our T's plug in zero you're gonna
9:12 get 32 and that's not going to change
9:15 our T doesn't affect that our velocity
9:18 is already already has those it's
9:19 already bunch of T it already has those
9:22 T's in effect that plug into five what
9:28 we get if we plug in five we get 24 so
9:46 if we plug in 10 well that's 32 minus 16
9:53 so we need 16 all the way down if we
9:56 plug in 15 32 minus 1 point 6 times 15
10:08 gives us 8 if we plug in 20 we're gonna
10:14 get 0 this is what you're looking for
10:17 and if you continue this this is
10:32 negative 8 and negative 16 you know if
10:34 you were watching for the this slope
10:38 field video the one right before this we
10:40 had diagonals that weren't changing that
10:42 happens a lot when you have like X is
10:44 plus or minus wise this when they're
10:47 your x I did a variable on here you're
10:48 going to get these verticals that aren't
11:04 changing happens and I know that you can
11:07 plug in 40 and do 30 minus 1.6 times 40
11:10 and get these these same exact numbers
11:17 I'm getting so let's make all this makes
11:19 sense before we do our slope field and
11:22 look at what's going to happen on the on
11:25 our graph so we know that in real life
11:27 we get air resistance that means that
11:29 when you jump out you're going to reach
11:31 some sort of limiting velocity at some
11:34 point your air resistance is going to
11:36 meet up with acceleration of gravity if
11:37 you're jumping high enough and don't get
11:39 the ground first or or polar parachute
11:41 or something they're gonna be equal that
11:43 means that you're not going to be
11:45 changing your velocity after that you
11:47 had this zero change as zero
11:49 acceleration means that you will have a
11:52 constant constant velocity at that point
11:56 so this is what that does this says if
11:57 you jump out
12:02 it used fall so your initial velocity is
12:05 zero you're not jumping downwards you're
12:07 in a helicopter right so you're probably
12:09 not gonna jump up because there's a
12:12 blade that would be a negative initial
12:14 velocity so if you jump down with
12:16 negative five you're gonna probably hit
12:18 the propeller you don't want to do that
12:21 so this would be jumping downward so if
12:25 we jump out at zero so just fall well
12:27 initially we're going to be going this
12:31 30 - oh well let me let me say that and
12:33 say this is they will see this on our
12:35 arm picture we're going to be increasing
12:40 our velocity at this 32 feet per second
12:43 if we jump out of five does it make
12:46 sense that if you jump out going
12:48 downward you're going to immediately
12:49 encountered air resistance
12:52 immediately that means that your rate of
12:54 change of velocity is going to be slower
12:58 you've already jumped out at five feet
13:00 per second so the rate at which you're
13:01 changing is a little bit slower now
13:03 let's say you jump out of ten you're
13:05 jumping downward at ten feet per second
13:07 already the reign of changing to be
13:08 slower you've encountered that air
13:11 resistance immediately here you did it
13:12 that's why we're Exelero
13:14 that's why our velocity is changing
13:17 faster here no here no here a little bit
13:20 little bit slower now what about 20 and
13:22 20 if you jump downward at 20 feet per
13:24 second right from the get go out of this
13:27 helicopter your rate of change isn't
13:27 going to be anything at all you've
13:29 encountered just enough air resistance
13:32 to meet up with your gravity so this is
13:34 weird and we're gonna see it on the
13:35 picture but if you jumped out of this
13:37 helicopter at 23 per second you're not
13:40 gonna feel your velocity increase or
13:42 decrease you will have immediately met
13:45 your limiting velocity that's wild
13:48 now let's say that you jump out faster
13:49 than that let's say that you jumped down
13:51 wow I just really wanna get on a
13:53 helicopter and you're jumping down at 25
13:56 feet per second you're actually going to
13:59 slow down so if you're jumping faster
14:01 than your limiting velocity air
14:04 resistance is pushing up more than gravity
14:05 gravity
14:06 he's pulling you down should kind of
14:09 slow down that that's the same principle
14:12 like when you when you shoot if you shot
14:15 a gun downward so it's at some point
14:18 your guns going so fast that it's a it's
14:20 meeting so much air resistance that
14:22 gravity's not gonna overcome that your
14:24 air resistance is pushing up more than
14:25 gravity's pulling that down you have an
14:27 initial velocity there that's so fast
14:29 it's counting so much air resistance
14:31 it's going to slow that bullet down even
14:33 though you shouldn't downward that's
14:35 what's happening here so you jumped out
14:36 at forty feet per second downwards that
14:38 really pushing it you're gonna encounter
14:40 so much air resistance that that's
14:42 pushing you up more than the gravity is
14:44 pulling you down that's weird but that's
14:46 what this is gonna play on us so long
14:48 story made real short we have air
14:50 resistance it's proportional to your
14:51 velocity and we can show the picture of
14:54 it really kind of nicely this point
14:57 where we have a zero change that is
15:00 called your limiting velocity or
15:02 terminal velocity or when we talk about
15:05 population will get like this population
15:07 limiting factor layer carrying capacity
15:09 something that limits how much we can
15:12 grow if you're below that you're going
15:14 to increase if you're above that you're
15:15 gonna decrease until you find this
15:18 leveling spot this limit so let's put
15:20 these slows remember these are all
15:22 slopes how our velocity is changing is
15:25 right here let's put them on our graph
15:26 let's see what happens again we made the
15:29 grid it's a lot easier to use so because
15:32 our time isn't affected well our time is
15:35 affecting our our velocity but it's
15:37 wrapped up in the velocity itself so
15:40 four times zero one two three four we're
15:44 gonna have these these intersections of
15:48 time and velocity so zero zero right
15:52 down here we have a slope of 32 but
15:54 that's really steep I'm not gonna really
15:57 build it draw that as not a vertical
16:04 line that's gonna be darn near a
16:08 vertical line 32 rise over run but also
16:13 at so here's our velocity so 0 5 10 15
16:14 20 years of times 1 through 6
16:18 so at 0 0 we got 32 at
16:21 equals one and velocity zero we also
16:23 have thirty-two at T equals two and
16:26 velocity zero will get 32 so all along
16:29 this velocity of zero we have a slope
16:31 with 32 this is acts like a diagonal
16:34 notice not bad no but this does not
16:45 change darn your vertical lines now as
16:48 we bump it up to five our velocity goes
16:50 to five feet per second then our slope
16:55 starts to two not starts to decrease
16:57 from 32 and we talked about why that was
16:59 so you jump out with an initial velocity
17:01 you immediately encountered air
17:03 resistance which means your velocity
17:05 can't change as drastically as it did if
17:07 you jump down at zero at zero you have
17:09 no air resistance for a little while
17:10 that means that you're going to increase
17:13 velocity faster because you're it's only
17:15 gravity that's affecting you as you
17:16 increase speed you start encountering
17:18 air resistance so the way your velocity
17:21 is changing can't be as drastic is if
17:23 you didn't have air resistance so the
17:24 more air resistance you have the less
17:26 that your acceleration of gravity can
17:29 take over here we get 24 so all along
18:16 at 20 we have zero and that is just as
18:18 that should be striking to be like oh
18:20 well hobby
18:22 obviously if our spokes are really no
18:24 matter where where our soaps are gonna
18:25 increase we're gonna go uh we're gonna
18:28 level out right here so if we and this
18:30 is what is saying to you as we're
18:32 looking our soap field that's this isn't
18:33 this is important because if models
18:35 real-life concepts really really well
18:36 that's what so people to do that's how
18:39 we want to use them if you jump out at
18:41 any velocity up to twenty feet per
18:43 second going downward you're going to
18:46 increase your velocity gravity is going
18:48 to overcome your air resistance until
18:50 you reach that point and then you're
18:53 gonna level out if you jump out faster
18:57 than twenty feet per second down so
19:06 let's see 25 you get negative eight at
19:08 thirty we have negative sixteen so in
19:34 and putting this all together our slope
19:37 field gives us how are our velocity is
19:40 changing with respect to time if we jump
19:46 out or fall out or are putting an
19:49 initial velocity that's slower than 20
19:51 feet per second this is all saying that
19:53 yeah you're going to increase until you
19:55 reach that level if you jump out that
19:57 faster so going down we're jumping
19:59 downward at faster than 20 feet per
20:00 second you're actually going to slow
20:02 down here
20:04 gravity is overcoming your air
20:07 resistance until you hit 20 here air
20:10 resistance is overcoming your gravity so
20:12 this would be bigger than gravity that's
20:15 what you're slowing down what we what's
20:16 gonna happen though is that these two
20:18 things are gonna level out in real life
20:20 your air resistance and your gravity are
20:21 going to come to some sort of
20:23 equilibrium an equilibrium solution
20:26 that's what this is the the solution
20:30 where if you started at 20 so 20 feet
20:31 per second you jump out you're jumping
20:33 down route 23 per second you are not
20:35 gonna feel a change so it's not gonna
20:37 feel like you're increasing speed it's
20:38 not gonna feel like you're decreasing
20:40 speed check out slower than that it's
20:42 gonna feel like you're speeding up
20:44 because gravity's pulling you faster
20:46 than your air resistance when you start
20:50 to a point to that that level of 20 feet
20:52 per second where we even out here if
20:56 you're jumping faster than that so jump
20:58 me down we're faster than that you're
21:00 going to feel like you are slowing down
21:03 your slope is negative here you're gonna
21:06 be decreasing your velocity until you
21:07 reach that equilibrium until you reach
21:10 that limiting velocity in that weird
21:13 it's kind of cool so right here this
21:21 this 20 that right there is our limiting
21:24 velocity no matter what happens whether
21:26 you jump out whether you're pushed out
21:28 jump downward whatever you're going to
21:32 reach that velocity given enough time so
21:33 let's kind of interesting we'll figure
21:36 out how much time you really need again
21:37 you're not going to jump up so
21:38 don't have a negative velocity not gonna
21:40 jump upward because of the helicopter
21:43 idea so our limit is this 20 feet per
21:50 second so let's say that we let's say
21:54 that we jump down at 0 so so we start
21:56 here so remember from our our slope
21:59 fields that we had an initial condition
22:03 and starting spot of 0 0 so at x 0
22:06 we're not traveling any we're just
22:07 jumping down when I choking downward
22:11 let's see how fast we actually come up
22:13 to that that 20 feet per second so
22:19 starting starting here remember that
22:21 slope fields that they've got to find
22:23 this curve that matches these slopes so
22:28 we're starting really steep when we get
22:30 to this level we better have a slope of
22:34 positive 24 so when we hit 5 we gotta
22:37 have a slope of 24 when we hit 10 we've
23:03 this right here looks like it's about
23:07 seventeen and a half or something so we
23:10 reach 17 and a half feet per second
23:12 after only two seconds so we go from 0
23:17 to 17.5 feet per second in 1/2 that's
23:19 like well what is that that's a good
23:23 percentage of 23 per second when we if
23:25 you divide that so do set about 70 and a
23:26 half divided by 20
23:30 that's a percentage of that limiting
23:31 velocity that you reach after two
23:34 seconds so you can find out how long
23:36 it's going to take you to reach a
23:37 certain percentage of 20 now
23:39 theoretically do you ever reach 20
23:43 it's an asymptote so technically no or
23:45 theoretically no but that's the limit so
23:47 you're going to get to that point get
23:48 closer and closer and closer but but
23:50 you're never going to exceed that that
23:53 velocity you can only limit to it wild
24:10 you're gonna slow down until you reach
24:12 your limiting velocity I hope this is
24:14 making sense to you it's very
24:15 interesting the way that works in real
24:17 life because at some point when you
24:19 really do think about it gravity and air
24:22 resistance if you're jumping up faster
24:24 than gravity slow down you're jumping up
24:25 slower than gravity
24:28 you're gonna speed up but that air
24:30 resistance is forcing you backwards so
24:33 when you're when you're when you're self
24:34 when you're jumping up so fast
24:35 air is just this is overcoming gravity
24:37 when jumping out so slow gravity's
24:39 overcoming air resistance but some what
24:40 they mean that's what we're talking
24:42 about with an equal image solution you
24:44 jumped out of twenty that can feel
24:46 change faster slurring that you're gonna
24:48 reach that limiting velocity we're gonna
24:50 do one more example about population
24:52 will talk about limiting population or
24:55 carrying combat capacity so hang on for
24:57 that one I'd like to walk you through
24:59 one more example about population we're
25:01 gonna do this very quickly the idea is
25:03 exactly the same I just want to show you
25:06 how slope can do some funny things in
25:08 the middle of this this problem to show
25:10 you that on the slope field and to talk
25:12 about something called carrying capacity
25:15 so so when we look at this we have the
25:17 way that that a population is changing
25:20 with respect to time now again we don't
25:22 have a T up here but the population
25:25 itself is a function of time and so the
25:27 way that population is changed or you
25:28 think of it the acceleration of
25:30 population that's kind of weird but how
25:33 fast is the population growing or decreasing
25:35 decreasing
25:37 well that's given by this for a certain
25:41 population with TB months now because we
25:46 don't have a team in here our slope is
25:48 still equal this here is the slope of
25:50 the population the way the populations
25:52 change with respect to time is still
26:02 equal to this but there's no T involved
26:04 the P hat is a function of time already
26:06 so it's it's involved in that population
26:09 well if we want to go ahead and do a
26:11 slope field our slope is given by this
26:15 function of P a function of T with our
26:17 dependent variable P in there let's just
26:19 start plugging in initial populations so
26:20 what if we started
26:27 zero well okay 0-0 these are all going
26:27 to be zero
26:29 do you see that it's not gonna change as
26:32 far as what time we have we're gonna
26:34 have zero this whole way down same thing
26:36 happens for all this I'm just going to
26:40 be one number if we start with ten
26:47 our slope would be point two all the way
26:56 down if we start with twenty point three
26:58 three would be our slope start with
27:00 thirty point for our slope start with
27:03 forty point for to move your so let's
27:09 start with fifty point 0.38 and usse
27:11 know something funny kind of happening
27:16 so our slope is increasing and then it
27:18 starts decreasing again that's a very
27:20 interesting than 60 would be point two
27:26 seven 75 would be zero you're looking
27:29 for that and then 80 would be negative
27:33 point one two here's what this is saying
27:35 this would be all the way down for all
27:43 of these remember my goal in teaching
27:45 you this not just how to do it
27:48 understanding what's going on if this is
27:49 the way that the population is changing
27:52 what this says is that if you start with
27:55 these populations here's what is going
27:57 to happen if you start with a population
27:59 of zero are you gonna see any growth
28:03 probably not there's nothing to grow if
28:05 you start with a population of ten
28:07 whatever this population size is you're
28:08 going to immediately start seeing a
28:11 population increase twenty increase
28:13 thirty increase faster for to increase
28:15 faster and that should make sense right
28:17 because if you have more things to
28:19 reproduce the population should be
28:22 growing faster to a point because our
28:25 natural resources aren't limitless which
28:27 means that as we get closer and higher
28:30 and higher our population is growing but
28:32 not as fast because we're reaching some
28:32 sort of
28:36 factor here at some point we hit what's
28:39 called in carrying capacity so you start
28:41 with 75 units of population whatever
28:44 that is it's not gonna grow it's not
28:47 gonna fall it's limited if you start
28:48 with more than that you're gonna
28:51 actually have a die-off if you put say
28:53 you've had a really small pond and you
28:57 just you don't zero fish in no Fisher
28:59 code you go okay that they don't just
29:01 spontaneously come about listen make
29:05 sense if you dump in way too many fish
29:08 like a billion fish in a really small
29:10 pond there's not enough resource just a
29:12 lot of them are going to die so as we
29:14 start having our population go higher
29:15 and higher and higher it makes sense
29:17 that at some point or not you'll be able
29:19 to grow anymore in fact we're going to
29:21 start dying off because there's just too
29:22 much of this population in one spot
29:25 there's a middle ground that because if
29:27 you start low and it's gonna grow you
29:30 start high it's gonna die that's both
29:31 wrong that's kinda fun then at some
29:33 point there's this equilibrium solution
29:37 and that's what that 75 does that's zero
29:37 right there
29:39 look positive slopes negative slopes
29:42 there's got to be a zero that right
29:44 there that's your carrying capacity that
29:45 means that no matter whether you start
29:49 lower than zero or higher you are going
29:50 to level off you're going to limit
29:54 yourself to that 75 popular units of
29:55 population whatever that is and the
29:57 picture is gonna bear that out so I'm
30:00 just really quickly but at zero zero we
30:06 have of course zero and if we had a
30:08 population start at zero we're gonna
30:13 zero all the way no matter what if we
30:16 start up with ten so we're gonna have
30:18 this part of this is slope with 0.2 so
30:26 all the way along get a little point to
30:29 slope and we're going to increase until
30:33 we get to 40 populations start on
30:49 I'm kind of exaggerating my slope right
31:03 now so you see what's going on but then
31:05 once we get to 50
31:08 please start to slow down so the most
31:09 rapid growth that we have is right
31:11 around start with 40 you start with that
31:14 much if you have a lot of things or
31:16 whatever these beans are to reproduce
31:18 and you but you have enough resources to
31:19 make it sustainable
31:21 when we get over that if we start
31:40 slowing back down then even slower and
31:42 probably even so I don't have this
31:44 listed here I'm missing my 70 I don't
31:45 want to put that out there but even
31:55 slower 70s point one of you on them to
32:03 the point where we have zero and above
32:14 so if you're looking at your slope field
32:17 that looks like this this should be
32:19 really telling what's going on if you
32:21 start with a population of zero you are
32:23 not going to see any growth obviously if
32:24 you start with just slightly more than
32:27 that it's going to grow slowly at first
32:29 but then as our time moves on look how
32:32 our curve has to flow through this slope
32:34 field so it seems like well time doesn't
32:37 matter it does matter but it's just like
32:39 you're starting at a different level
32:42 here population wise so if we start at
32:44 zero and we have no population no matter
32:46 what our time is we have zero that that
32:49 is an equilibrium solution but it's not
32:51 a good one I mean it's like it's like
32:53 the trivial solution you know yeah it's
32:55 it's nothing it's like the same you're
32:56 gonna launch a basket where you launch a
32:59 rocket from the ground at time zero it's
32:59 on the ground
33:02 when's the rocket on the ground and time
33:05 zero Thanks I meant like after it's all
33:07 done once it hit the ground so if you
33:09 start with no population obviously
33:10 you're not gonna have when I'm at one
33:12 time but if you start with just higher
33:14 than that you are gonna flow along this
33:16 curve you're a mattress slopes up but
33:19 eventually you're gonna level off if you
33:23 start higher you just gain quicker but
33:25 you're gonna level off if you start
33:27 lower yeah my team may take you a long
33:29 time to finally reach that because you
33:31 don't have as many many reproducing
33:34 items if you start way higher what
33:36 you're actually gonna drop this is like
33:39 what the last example your gravity to
33:40 overcome in your air resistance or your
33:42 air resistance overcoming your gravity
33:45 this is the same thing your level of
33:48 food is overcoming your population which
33:51 means you can grow or your populations
33:53 overcoming your level of fitness called
33:55 food which means your population is
33:57 going to decrease you'll have enough so
33:59 that's the way that these slope fields
34:02 work they're looking for the 0 that 0 is
34:05 typically your carrying capacity or your
34:07 limiting velocity for speed problems or
34:09 your equilibrium solution if you start
34:13 with that so if we started with 75 units
34:21 of population you wouldn't change if you
34:29 make sure you're hitting these slopes as
34:49 we travel along but eventually we are in
34:54 love allowed so how you can figure out
34:56 things like how fast would it take you
35:02 to hit 90 percent of your or 75 percent
35:06 of your your limiting capacity or the
35:08 two thirds or whatever and just look for
35:13 where you reach let's say 60 a level of
35:15 60 that happens when do I hit a level of
35:18 60 let's go if I start at 10 if I start
35:20 at 10 units population well that would
35:23 mean that I did right around the 60
35:26 months or so they said 48 15 somewhere
35:30 48 60 months so the whole idea of soap
35:31 fields one last time as last we're gonna
35:32 really talk about it
35:35 so fields model hopefully some realistic
35:37 examples that you can't
35:38 or maybe aren't able to solve with
35:40 different equations right now or maybe
35:43 even ever what they do is they consider
35:44 a first derivative to be the slope which
35:47 we have they say that's the formula for
35:48 your slope
35:50 no problem I can just plug in some
35:52 points I got it then it's gonna give you
35:54 a slope at every point the whole picture
35:58 is your general solution so every
36:00 possible curve would happen fit these
36:03 slopes so this is your general solution
36:05 it tells us a lot about situations
36:07 though if you just look at the slopes it
36:08 goes about that's your that's your
36:11 limiting factor right there so if you
36:13 start with the population 75 you ain't
36:14 gonna grow if you jump out of the
36:16 helicopter 20 feet per second year I'm
36:18 not gonna fall any faster so they tell
36:20 us a lot about a situation just by
36:23 looking at it to get a specific curve
36:25 you need a starting point so you could
36:26 say I'm going through this point and
36:27 then you can find a particular solution
36:30 that's how all initial conditions work
36:31 any way this condition gives us a
36:34 particular solution a lot of times we
36:37 can see what's our trends in in gaining
36:39 or losing and how that would really
36:42 affect our real life situation so I'm
36:44 I'm hopeful that this is making sense to
36:47 you I'm hoping that you see the the
36:49 niceness about this that you can
36:51 approximate solutions without actually
36:53 having to do anything with the problem
36:54 that's pretty cool
36:56 computers do this really well and what's
36:58 nice about it is remember that whole
37:00 talk about how we have this trade-off
37:04 between perfectly modeling the situation
37:07 and doing the math behind it well if
37:10 you've got computers you can kind of do both
37:11 both
37:13 you can perfectly model situation and
37:15 then do a slope field and it will show
37:16 you a lot of trends in it I'll show you
37:18 limiting capacities are limiting
37:21 limiting factors or carrying capacities
37:24 or limiting velocities and include more
37:26 variables even though United will solve
37:28 it and that's that's pretty neat so I
37:30 hope this makes sense I hope that right
37:33 now you're able to take a slope from a
37:35 difference of the equation make up a
37:37 table to get yourself a slope field and
37:40 interpret what's going on look at how
37:44 slowly we go we start low or how quickly
37:46 we decline if we start high look and
37:49 think about how that would make sense in
37:51 a real life event with low populations
37:54 we'd grow slowly but as we get more and
37:56 more population we start living
37:58 ourselves because of resources to a
37:59 point where we're going to level off
38:02 just enough resources to support us but
38:03 not grow any more you start too high
38:05 there's not enough resources to support
38:07 and we would decline population that's
38:09 what so feels about is interpreting
38:11 real-life situations so next time what
38:12 we're going to talk about is how to
38:15 determine when we're when we're gonna
38:18 have solutions or not so so fields one
38:20 way to deal with it when we don't have a
38:21 solution but now we're going to start
38:24 determining when we do have solutions
38:27 and when we can expect a unique one so
38:28 we'll talk about that and then after
38:30 that we start dealing with solving
