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MAT-152 Intermediate Value Theorem

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hello everybody Welcome to the intermediate value theorem there's actually attached to the end of chapter 2.2 but it's such an important awesome topic that I thought it deserved its own video what we're going to discuss is how we can use this theorem to approximate zeros of a polynomial and Ross are going to formally Define it so let's get started an intermediate value serum here it is let's get you the formal cast speak definition for any polynomial which is a continuous function like we talked about before for any continuous and therefore for any pound Nair for continuous function what we can say is that there's going to be some a value sunseed value and some P value and I also means there's going to be an FFA value do I should have made that a tire that's okay NFL see now you undersell so going to be enough of a p-value USA Christopher colors does today can you go to sfb Harry kalas a little bit easier to see and then we can call you the blue one or we'll just leave that see Value Place okay and here's what the intern play value set a and be the real numbers such that a is less than be okay so far there is over here B's all the way over there so is definitely smaller than me if F fancy asked if the function is a polynomial function search LA SFA is not equal to SSD and it certainly is not this value is not this value whose values not equal each other then we can say that f takes on every value between us and be it's kind of like an odor sort of the theorem basically it's saying that for a continuous function to function that doesn't do any of those jumps or doesn't do any sorts of holes or anything like that it just flows really nicely the whole time I would never pick up my pen if I drew it for any function that's continuous especially polynomials Anna polynomials are continuous then basically between one value of x and a bigger value on X between an A and A B you will have a y value for every single one of those X Trooper what does this graph very clearly there's no holes no jobs no nothing so for every single EX value in here there's going to be a y value basically this just defines a prettier picture of what it even means to be continuous right here is how that's useful why this is useful his wife if FFA and SMB are opposite signs there must b80 a route to our polynomial between them between a NBA monster such a revelation we're going to put an! Next to it how cool and here's a little picture as to why and I say if we had our y-axis and we have our x-axis we know that we have all of how did values up here and we have all of our negative y values down here so if that any points if at any point let's say I don't know at 1 or graph is positive let's see what happens at 2 play me fine but I'm not sure maybe over here at 3 our graph is now negative one because there has to be enough value for everything between here and here because cast to be a y value for everything between here and here are good line are graph would be a line between this point and this point it might do some funky stuff at might like go up and down I don't know but we know for sure do I have to go from here to here without stopping that means that it must cross the x-axis at some point let's stay right here I'm not sure and it would have a zero this a NSB value let's just a little arrows at the end of this between this and this be there would have to be a zero same thing if say started out on the negative side very hears negative for why here's positive for y I'm not sure maybe we had a point at negative line let's say negative to negative one- to and then we found over here when x equals 3 do we had a positive value of one again because there has to be a y value for everything between this a point we chose and this be Point Liscio's because my line is never going to stop I never break in every jump I never do anything there has to be is zero a point where my a crosses or heard me where my graph crosses the x-axis to get from A to B it could look like that for example I'm not sure the point is that there would be a point where from this negative value today's positive value on the why we would have to cross the x-axis at some point to get there let's do an example of this so for the function we have little fancy f equals X cubed plus x squared plus 1 we're asks the question does a 0 exist on the interval negative to negative 1 and we could look at a table of right now just for the sake of time and the fact that I know the numbers work we're going to substitute in -2 and -1 um we're going to see what happens to the signs of the Exotics remember this is kind of like our right now and this is kind of like Arby we want to see if we're going to get opposite signs for aim for isolette substitute those if we take off of -2 we evaluate our function negative to negative 2 cubed + -2 Square calamus pattu saree Plus we get -8 + 4 + 1 we get -3 if we evaluate our function at negative one we get negative 1 Q + -1 squared plus one so we get negative one plus one plus one we get one oh my goodness oh my goodness look at that we have a value for a and we have a positive value for RB that means at some point somewhere between -2 and -1 or x-axis there is point where a graph crosses the x-axis how interesting okay balloon graph cubed plus x squared plus one holy smokes we totally free rare Point let's do that with graphic let's call this negative negative 2 and we'll call the sky out here one and we'll call this guy right here one will go down 1/2 -3 he was a few things we already know -3 no there's a point negative one one wow how cool and actually what we can do to kind of help ourselves but the rest of the graph is we can take a look at our calculator and we can put that in there so we can do y equals x cubed plus x squared plus one when we Graphics my window range so I'm going to quit out of that I'm going to go to zoom and I'm going to go to zoom I'm going to go to zoom fit fixing I want to get close and I send over there I want to look right here right here so I'm going to do and more how cool it looks like you know what let's change my window just to fit what I'm trying to do pardon me for being so picky here but let's take a look what's make this go from 3 two positive two I must go from -4 to positive to try that girl that's alright so we can see that from -2 - -1 there is in fact a zero looks like it's going right between so when we continue to make our graph here we go up a little bit more not really and then you go through one and we can go back up awesome and what we can say disabled aircraft versus f of x equals X cubed plus x squared plus what we can say here is that FCF has a zero in the in -2 - -1 what's your other example of this let's use here i v t intermediate enter the zero of f of x equals X cubed minus x squared plus one okay my similar but a little bit different I'm over going to do here is we're going to take a look at our table to approximate this one to see what intervals this might be happening on so we're going to take our function we're going to put it into our grapher we're going to say x cubed minus x squared plus one kind of similar to the last equation and we're going to have to press grass where can I press seconds window to get our table gone we want to take a look let's see it about -2 and we're going to change by once and then we're going to go second graph to take a look okay so it looks like negative and then negative and then positive o look at it happen there it looks like we were negative negative and Amigo positive positive positive but right here is there something interesting happens from the x value -1 to 0 we were negative in her why and then we got positive and our went from negative one to positive one so we know using the intermediate value theorem what has to be a zero in between -1 and 0 so what we can do now is we can break up that interval even more we can take a closer look between -1 and 0 where's how we can do that we can press second window we can take a look from -1 this time you want to look from -1 to 0 and we want to change not by ones anymore we already did that change my something smaller let's change my 1/10 so let's change my .1 and let's see what's happening at point nine point 8.7 and so on I work at a second graph okay they're taking a closer look Frederick still goes negative at -1 at point nine it's negative at point eight it's negative okay here we go so it looks like there's definitely a zero specifically not just between -1 and 0 but between negative 8 and -.7 because we go from negative to a positive and as we can keep on doing this we could start our table at -.8 we could change by .1 and we could take a look to see what is really happening there and then exactly what point we get a zero you can keep on doing that to get more accurate however we're going to call it good here and we're going to stay that there is a zero that exists in the interval -.8 - -7 and here's what else we're going to do our calculator can get even more accurate we could keep on approximating a butt find exactly where that 0 is what we could do is we could press graph and our previous window still works for this what can you can see that's right between -.8 and -.7 and what we're going to do now is we're going to press seconds count we're going to go down to zero and we're going to pick a point to the left of our zero we're going to take a plane to the right of our zero and ask Matt guest for us and there we have it it found the zero at exactly .755 round to 3 places Emerald have our final answer of the zero of the function occurs a taxi 0.0 / -0.755 pretty cool stuff next let's talk about an exam was Finding intervals and then I'll totally leave you alone I bet they were given the function f x equals x to the fourth plus 2x cubed minus 3x squared minus 4X Plus and we're asked the question in what intervals are there zeros Oregon to use art able to help us with this one so what we're going to do is we're going to put it in there grapher again add we're going to do our function x to the fourth + 2x -3 x squared minus 4X plus 2 and again we're not going to go to graph right away we're going to go to table sets and we're going to take a look at when are values begin at -4 and we're not interested in small changes like we were last time we're not interested in 10 so what let's just go buy ones look at everyone and now we're going to crack it cable okay add let me take a look at these values what we can see is that there appears to be a zero between -3 and -2 and I know that because our y values start positive and negative 3 and then they get Negative at -2 so we can stay so far there are zeros in the in -3 - -2 and it's also important to understand how many zeros were going to be looking for write the degree of our tell me all is for and value is for that means there can be at most a little note to ourselves in here at most for zeros at most for intervals so we know once we get 2 for Wii stop alright so from -3 to -2 there's definitely a zero in there positive positive okay if we take a look from 0 to 1 we go positive to negative to so there's definitely a zero in there it's a from the interval 0 someone if we continue down our table we can see it went turn positive and then it was positive 98 positive and negative I didn't catch that it wouldn't friend what is 2 - positive 14 most definitely is 0 also between 1 and 2 and I'm realizing I missed crazy I miss one if we take a look again back between there we go we talked about going from -3 to -2 we look for positive 14 - -2 that was definitely a zero but then chica from negative to negative one we had a negative and a positive negative to positive means they're also there's also a zero there tween -2 + -1 and again I know I'm good to go I can stop now because there's four intervals here that means there are four zeros and our degree was for that the most this polynomials going to have that's all I have for you will see in class

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MAT-152 Intermediate Value Theorem

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