today's lesson 4.4 rates of change and polynomial functions so the first thing you need to remember is that if you have a polynomial function the rate of change is going to be different as you go around the curve of the function so think about a polynomial function let me just sketch you one something over here let's say have something just that looks like this okay so it's probably a cubic function and I can see that the rate of change if I drew a series of tangent lines to this graph would be different as I go around the curve right there free tangent line has a different slope so the rate of change is going to be different as we move around the curve so we talked about two things previously in Chapter two on rates of change and that is the average rate of change being the slope of a secant remember a secant line is just any line that joins two points on your graph so if I took these two points and I do a line that would be a secant line the instantaneous rate of change is the slope of a tangent now these are concepts that are critical for your understanding of calculus so all these little lessons at the end of every chapter when they do rates of change and there will be a rate of change section at the end of each chapter as you go through this textbook and that is because they're warming you up for calculus so instead of going over this I'm going to do a few examples from the textbook that will help to illustrate what they're trying to get across in this section so it's really just a repeat of what you did in chapter 2 only using polynomial functions so here's an example they give you it's obviously a quadratic you know where the vertex is 2 and minus 2 you know it's concave up it's very stretched by a factor of 3 the question says find the average rate of change for X between 2 and so let's begin initially by making just a quick sketch of it so that we can discuss a little bit about rates of change to this for this graph as we go around so I'm going to make it a quick little sketch here so I have 1 2 and minus 2 minus 2 what's the y-intercept can we plug in X is 0 and we would have 12 minus 2 is 10 so it's going to be way up here somewhere it's not even going to be on my graph ok so it's going to go something like this right there we go so it says find the average rate of change so remember the average rate of change all you have to do is pick two points and the ass between 2 & 6 so I need to know what are the Y coordinates so that's first thing find the Y coordinates because I'm doing slope that's it you did it in grade 9 you can do it now so what's F at 2 I need effort to and I'm going to need F dot 6 so I can have two sets of coordinates which I will use to find the slope so f at 2 2 minus 2 is 0 oh great 0 squared 0 so 2 and minus 2 so I'm going to do 2 minus 2 and it's a good idea to write these points up because then you know how to do the slope F at 6 6 minus 2 is 4 squared is 16 times 3 is 48 minus 2 is 46 so I have 6 and 46 so now that I have these two points I can find the average rate of change so I'm going to write average rate of change is going to be equal to the slope between these two points so you can do it in any order you want I'm going to do 46 minus -2 don't forget the minus minus ax so you'll get numbers over 6 minus 2 46 minus minus 2 is 48 divided by 4 equals 12 so that says between 2 & 6 so this was 2 here and 6 2 3 4 5 6 so somewhere up here I have a steep line a slope of 12 okay so that's how you find average rate of change so don't forget all you have to do here is to find the slope between two points find the Y coordinates and you're off to the races okay another one that's just a little bit different okay now the other thing sorry before I turn the page is that you should understand what the difference is in the ends the rates of change as we go around this curve so if I were to draw a series of tangent lines to this graph you would see that this tangent line is going to be negative it's going to be very steep but as I approach this vertex the tangent lines are going to get less steep until the slope is zero at the vertex here okay so anytime you have a change in direction or a maximum or a minimum value you will have zero slope that's also a very important concept when you're doing calculus to find places of zero slope now as soon as we go around the other side here you can see the slope is I don't know maybe it's three here and then as you go up the slope is going to get steeper okay so important points to understand is the change these are positive slopes on this side of my graph and negative slopes on this side and zero slope here okay so important for you to understand how the slopes go okay it's going to flip over and we're going to do an example from number 9 and let's see if I make sure I get this so you can see it so it says f of X equals 3x squared minus 4x minus 1 estimate the slope when x equals 1 okay now this time again we have an equation so remember when you have an equation to find the the slope so Frank going to estimate the slope they always say asked me because until you're using calculus you don't know the exact slope calculus is really easy and sometimes I show my students little tricks to use calculus to check their answers because you're not supposed to know that yet and maybe I'll do that in a minute here okay I want to estimate the slope where X is equal to one okay so that means I need to find the slope so I'm going to say the slope of the well it's going to be a tangent so I'm going to just say slope of tangent or instantaneous rate of change right you're estimating it so the slope of the tangent maybe even es-tu for estimate it's going to be that's a gain it's just an equation so it's F at so X plus 8 minus F at x over 8 now remember this from chapter 2 what we're doing is we're making the H really really small so the slope when X is equal to 1 is going to be equal to the function at 1 and we're going to take a small interval so one point zero zero one minus F at 1 over one point zero zero one minus one all right so this is my H what I've added I've added in point zero zero one so if you do that entire calculation and I did it just before I started so I'm just going to pull up some numbers here because it's it's a calculator work right you can't do this in your head and it comes up to well it comes up to approximately two and I'm going to show you how to do the calculus on this because it's so much easier so it's approximately - I think you end up with like two point zero zero one or some so don't forget your approximately equal to sign so that's just plugging that into this equation here right FF 1 is 3 minus 5 is minus 2 so minus minus 2 you're adding it to that and dividing by point zero zero one okay so in calculus all you have to do is I'll show you something really quickly here so we take derivatives and the derivative just means the rule for the derivative of x squared is you bring this number in front and you subtract one so the derivative of x squared is 2x so that's your derivative so for this one the derivative would be 6x because I'm multiplying by two reduce the exponent by one so I had 6x minus 4 so f prime X is going to be 6x minus 4 and then the prime this gives you the slope of any function so this would be 6 minus 4 is 2 so I know the slope is 2 that's just a little site for you in case you want a little quick way to check okay so find the point of tangency so I need to know what's the y-value when x is 1 so f at 1 is equal to 3 minus 4 minus 1 so 3 minus 4 minus 1 is minus 2 so that means the coordinates are 1 and minus 2 okay so once I have the coordinates the last part of the question says here to find the equation of the tangent at x equals 1 now to remember how to find the equation of a line so you're going to use y equals MX plus B and you have a point so my X is equal to 1 by Y is equal to minus 2 and my slope is equal to 2 so those are all the things you need to find the B value I remember doing that over and over at nauseam okay plug it all in so I have minus 2 is equal to 2 times 1 plus B so that means B is going to be equal to negative 4 and therefore the tangent line is going to be y is equal to 2x minus 4 okay so that's how you find the equation of a tangent line it's not all that hard right you can do it and again if you want to try it a little bit of calculus makes your life easy okay so hope that helps you out that's handy chapter 4 please subscribe I get a thousand subscribers I'll do calculus and vectors for you that's my bride bye for now