Isoclines: A Way to Analyze Slope Fields

all right we're going to look a little bit at slope field so we've already talked about what slope fields are what i want to emphasize in this video is isoclines and so if you just kind of think about this word and the different pieces of this word iso means same and klein's means slope so what we're looking at here is the locations in our slope field where we're going to have the same slope iso clines and so i've just written down a very straightforward first order differential equation here and so when you look at this what i want to emphasize here is not necessarily doing this by hand although i'm going to start doing this by hand but how to think about this in kind of a big picture way so that you can get a sense of what the slope field looks like without necessarily having to get out that computer to do the graph we will look at the graph on the computer at the end of this video as well just to kind of compare what we've done by hand with what we see on the computer all right so i've got my differential equation here solved for the derivative and what i want to look for here are the different places in the graph where my slope my d y d x which is what your slope is is going to be constant and you might notice from looking at the right hand side of this that for different x y values you're going to end up with negative output values for this dydx equals equation positive and 0. and so that's always a good way to really look at anything is thinking about when the values are positive negative and 0. so i'm going to start with 0. so this would be when our slope is 0 and then what we want to think about here is where are the locations of those points where the slope would be 0. so that's going to be coming from setting this right hand of the equation equal to whatever this constant is this d y d x so here we're going to have x squared plus y squared minus 2 equals whatever this constant is all right so in this case x squared plus y squared minus 2 equals 0. if that's a little bit easier to think about what that looks like if i add the 2 to both sides of the equation so i get x squared plus y squared equals 2 when i put in that 0 for con for c all right so when i think about this this is a circle of radius square root of two centered at the origin so that tells me that my slope field if i think about a circle of radius square root of 2 centered at the origin that all along that circle i'm going to have slope 0. and so you can visualize what that looks like slope 0 is where you're going to have horizontal tangent lines to the curve so what i did here was i graphed that circle of radius square root of 2 centered at the origin just a rough sketch but really what i want to emphasize there is that along there is where we're going to have slope 0 or flat curves horizontals tangents that are flat and then we can also think about where the slope is positive or negative and you can think about that in a more global sense or you can just choose a positive number and choose a negative number that's what i'm going to go ahead and do here so just a nice convenient positive number 1 so when i set x squared plus y squared minus 2 equals 1 maybe add the 2 to both sides x squared plus y squared equals three so that's going to be a circle of radius square root of three so just a little bit bigger than the circle i just drew there i'll draw this one in a slightly different color here but the point here is not the circle but that along that circle i've got slope one so my tangent lines are going to have slope one so it should be pretty easy for you to visualize what slope one looks like those are where rise and run are equal or your tangent lines are going to form 45 degree angles with the coordinate axes so you can visualize that and sketch in some of those tangent lines slope one all along this circle and again the point here is not to get a beautiful sketch the point here is to get an overall sense of where you've got positive slope negative slope and zero slope so you can get kind of a big picture visualization of what these slope fields look like all right and then let's do a negative slope all right so when i have x squared plus y squared minus 2 equals negative 1 and i add 2 to both sides i get 1 so i have a circle of radius 1 centered at the origin it's going to be right in here and along this circle i'm going to have slope negative 1. so make sure you think about that correctly it's just basically going to be this tilted the opposite direction so 45 degree angles with the coordinate axes but in the negative slope direction and you can certainly do lots of others but the main idea here is breaking this down into positive slope negative slope zero slope so you have kind of this big picture idea of what the slope field looks like and so the curves here there are some questions in your homework the curves along which my slope is constant these curves here are called isocline so i've sketched a slope field along three different isoclines for these three different values for the slope so there are a couple questions in your homework that use that word isocline so just want to make sure that you're clear about that it's the curve along which the slope is a particular constant value let's look at the overall slope field on the computer using d field so here i've pulled up just the default window setting on d field and remember that there are a few things that you need to be careful about changing when you use d field so the first is that the box in the left side here is it's hard to see the prime but there's a little prime next to that and the default settings have the the differential equation given in the form x as a function of t x prime is a function of x and t so my example had y as a function of x so i'm going to change this x to a y and remember there's a prime there so this is my y prime d y d x this box turned red because i now introduced a variable that it doesn't know how to handle that so i need to make sure i change my independent variable to x and then my equation that was here in the default was in terms of x and t i need to put it in terms of x's and y's like the equation i'm working with so i type x squared plus y squared minus 2. so there are some default parameters here which we didn't we're not going to use so i don't need to change those and then the default settings on the window are not necessarily what you might want so i'm going to go ahead and change those a little bit here i'm just going to go negative 3 to 3 in both the x and y direction and graph the phase plane alright and so you can see in there where i've got my slope zero my curves my circle along here where i have slope zero it's a little bit hard to see where that circle is but if you look at the scale there you can see that and you've got slope negative here in the center of that circle negative slopes and then positive slopes outside that you can see these curves uh click anywhere here remember to see these numerical approximations for these solution curves all right so the main idea again with isoclines is just to look for where you have constant slope and if they don't tell you what constant slopes to look at positive negative and zero are just a good place to start in general

Channel: Brenda Edmonds

Views: 2,040

Rating: 4.9111109 out of 5

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Id: YC39cHuwmvs

Length: 8min 13sec (493 seconds)

Published: Mon Aug 24 2020