[Music] hey welcome back to AP preal this is section 1.2 we're talking about rates of change and we're going to start out with the average rate of change now this is something you've done before I don't know if you want to believe me or not but let's fill this in average rate of change have you heard of slope before in the past slope you're remember slope back in 8th grade slope is the change in the dependent variable for every change in the X okay or change in y over change in X and we're going to introduce this new symbol here this is Delta and what Delta means is the change in so this little triangle it's Greek letter Delta all right means change in y over change in X sometimes they use that Delta in chemistry it just it's a Shand it's a quick way to write it but you probably have seen Y2 - y1 over X2 - X1 just make sure that the dependent variables are always on top okay it's wise for the dependent variables to be on top that's right the Y's are on top so we're going to start with number one find the average rate of change so we know now that just means the slope between one two and three four the points and here's the function up here so let's plot those two points down so here are the points labeled on our wonderful graph one two and three 4 the average rate of change we want to know between these two points what what is the rate of change you can look at it and you can tell that the rate of change changes a little bit right so we want an average rate of change and to do that we're just going to find the slope so I'm going to do my best there to approximate if we draw a straight line through those two points that'll give us an average rate of change it's a little bit less than the the rate of change here but it's a little bit more it's a little greater than the rate of change up here it's the average rate of change how are we going to show that well we're just going to write out our formula the average rate of change is going to equal and just like we did a long time ago we're going to say Y2 - y1 all over X2 - X1 now some things I'm going to point out and again you probably know this uh the first point we need to designate each point as X1 y1 and X2 Y2 and it really doesn't matter which one you do because if you do it in a different order then essentially what happens is the top and the bottom the negatives all switch and ends up giving you the same answer you can trust me on that but I always like to choose the point with a larger number uh to be X2 all right and then Y2 and then the other point will be X1 y1 okay now uh when we substitute this all in what are we going to get we take the y-v value so it's going to be 4 minus 2 all over 3 minus 1 and you have done this before as I said I don't want to keep reiterating that but we get 2 over two which is one that is the average rate of change now can we take a second you you don't have to write this down but I want to play the what if game what if what if we didn't designate the first point to be X1 y1 and the second one to be x what if we switch that because students always say how do I know which one is which let's pretend like we made this one X1 and y1 now they have to mat these two coordinates have to match they have to be in the same point but what if we just switched it what would that look like mathematically and so when we sub it in our formula here we're going to have the same formula Y2 - y1 over X2 - x one all right what would that give us so in this case we would do what we do three uh no we wouldn't we would do 1 minus 3 on the bottom because the X always goes on the bottom and on the top we look at it we'd have 2 minus 4 if you look and notice we're going to get a -2 over -2 and notice those negatives cancel that's going to give you one so the moral of story is it doesn't matter which point is X1 y1 which point is X2 Y2 you can just plug in the formula that will always work as long as these guys match up so if you notice like two over one that comes from one two here okay and then 3 over four all right enough of the what if you didn't have to write that down I'm just proving it to you instead of a graph let's say we have a table of values you want to calculate the average rate of change on an interval be careful with the output values they always have to be remember output are the Y values the Y values always have to be in the numerator which is in top so here's number two that says what is the average rate of change or the slope on the interval from 13 to 25 so really these two columns are what we're looking at we have time and distance now generally we have to figure out which is our independent which is our dependent so time here is going to be our independent because you can see that D of T okay T is like the independent variable that's the one that has to go in the bottom so remember I'm going to write our formula down Y2 minus y1 all over X2 - X1 and that is the average rate of change so this would equal all right now be careful D pendent has to be on top and remember I like to have bigger numbers first sometimes it's not possible because look this one's bigger on the X and on the Y it's smaller so I'm just going to roll with it so we're going to take the point 2530 the 30 goes on top minus 76 all over 25 -3 now some students might be saying well how do you know which numbers and where to get them if it's easier let's just write this as a coordinate point we'll write it as 1376 and we'll write this one as 2530 and now we can see we're going to do 30 - 76 25 -3 that might make it easier for you if you need to do that all right we're going to work this all out what is 30 - 76 I get 46 for that 25 - 13 is 12 and very very important the word purr is a key indicator that you're dealing with a rate of change miles per gallon students per classroom online gamers per server it tells you which variable is dependent or independent so we have to make sure that we include our units so in the top here are the Y values that was the distance in meters so it's -46 m per every 12 seconds okay or we can work that out if we're going to divide that out in the calculator I get when I put in my calculator -3.8 3 three we're going to go out three places meters per second okay we're going to write that out with our units you can abbreviate if it's standard units there pretty easy right I mean we're just dealing with slope here okay next let's talk about the rate of change at a single point what about the rate of change of a function at a point this tells us about the rate at which the output values would change or the input values to change at that point we can approximate the rate of change by using an average rate of change over really tiny intervals that contain the point to illustrate this what I'm going to do here's our function that we were working with I'm going to go to a website called geogebra you might be familiar with it and they have a graphing calculator you can put in the function which I did right here and what I want to do is I'm just going to put two points if you hit point you can just click on here and it put a point on the function you know here are two points and if I wanted to find the average rate of change then I would find the slope of these two right I would find the slope what I'm going to do is I'm going to put a line how do we do that here's line and I go through a and through b okay so as we move these points closer together suppose I want to know oop I don't want to do that suppose I want to know what the average rate of change is at Point a now if you notice look at the curve it's pretty steep right I mean because we're talking about the slope and if I use point a and point B then my estimate is not really going to be close to what the slope is right at Point a so what we're going to do is we're going to move this a little bit closer to a and notice how this line the slope of it is pretty close to what the slope is at Point a and then after we move it that close we're going to move it this close and then we're going to move it this close and if we get those two points very close together then we can approximate the average rate of change at that single point as they become on top of each other and essentially become the same point that's what we're going to do in this part of the packet so it says estimate the rate of change at x equals 1 okay so to estimate the rate of change at xal 1 I'm going to pick two points that are very close together how about one we'll have xal 1 and xal 2 and I'll plug both of those into our function here and if I do that you can take my word for it but I'm going to get for xal 1 a point of one two and for xal 2 I'll get 2 comma 3.5 now sometimes I show a lot of work here pause the video if you need to to sometime you know to write that down but here's our slope formula average rate of change we work it all out we get 1.5 and that represents the average rate of change between the straight line that would connect these points that is the best job I could do not too bad but estimating the rate of change becomes more more accurate as these points get closer together so instead of using an x value of one and an x value of two let's use an XV value of one and another x value of 1.1 what would that look like well to help us out this is hard calculations we're going to use our calculator so I'm going to go back to the calculator I've already done some stuff but I'm going to quit I'm going to show you this is our first little calculator if we do second and then memory which is above the plus sign and 712 that will help us clear the ram you only have to do this if this isn't your calculator and you're not really quite sure how to you know the setness could be messed up but I just cleared the memory out we're going to plug in this function we're going to make sure we use our parentheses where'd that function go it's all the way over here so use your parentheses and it's - one2 I always put parentheses around fractions it's a great idea here's the X and the squared button I'm just going to use this little carrot that works for exponents you could also use this button right here so so we have - 12 x^2 + 3 go back to the X and we subtract use your parentheses 1 12 all right so here's our function and we put it in y1 that's going to be important later but if I were to go to the table see where the table is here it's in Blue on mine you hit second and then table it gives you a table of values starting at zero one is two we already knew that and then two is 3.5 we knew that so on and so forth the way down you can scroll down you can scroll up as long as your cursor is on that input value right there well that helps us plug numbers in but what I want to do is get these points closer together so I want to use x = 1 and x = 1.1 how can I come up with that 2.95 which is the output for 1.1 now there are ways you can change the settings in the table so I'll show you how to do that right now we go second table set okay this first number tells you where you want to start the table so -2 is I guess okay but uh let's go down to this one here which is the most important it says Delta table now we learned earlier today Delta means changeing we want the table to change in tense because I want to go to 1.1 and you know what I could start this at one that' probably make our life easier so this says start at one and change by tths so if I go back to the table check this out it starts at one and it's changing by 10 the input value so that gives us our two 2.95 easy enough now I could also do this as we work let's work this out work it out work it out oh there we go plug it all in we get a slope of 1.95 that's if we have the two points so close together if we move that all the way down to here uh we get a different slope it's a little more steep than it would be as this point moves away then the slope decreases but as they come closer together the slope increases well that's pretty cool but I'm going to show you one more thing in your calculator because what you don't want to do is have to you know change the table settings let's go ahead and get these points so close together how close real close let's go 1.001 now we could change our table that's not difficult let's do it table set we just go down here to the change in table 0.001 and when we do that we can go up to the table and it gives us a value but let me tell you a little problem with this if we scroll over we notice down here it says the yv value or the output for 1.001 is actually a little bit this is longer this has been uh TR we say it's been rounded to 2.02 right so I'm going to show you another trick not a lot of teachers use this but I love it I'm going to be honest I love it if you go to the home screen which is where you quit and there's nothing there that's fantastic go to where it says vars that stands for variables I want to pull up that y variable so I go over to the right it's a function there there's y1 you remember we plugged the function into y1 so now using function notation you can just plug in an input value and it'll tell you the output it'll just plug that number into the equation so if I want y1 that's right I mean that's where we're at y1 is where our function is so if I want to plug in a 1.001 I'm going to scroll up to y1 I can just grab that again and do 1.001 and it's going to give me the output value then it's very simple I can do some math where I go grab that output we subtract the other output right or you can just type two and you divide it by you got to go the same order you got to be careful but 1.1 minus one that's just 0.1 and that's going to give me an answer of 1. n95 but of course I'm going to show that right so I'm going to show all that out now an important conversation we need to have we get the slope is equal to uh 1.9 995 but we want to go out three places so what do I do well one option is you can round oh well that's not this option that's the second option so the the second option here you can round it so if you're going to round you look at the third okay you got to look at the fourth that's a five or higher so this has to go up when nine goes up it's going to a 10 so you move that over that goes up that goes up that goes up you're going to end up with 2.0000 Z awesome or the other option is you can truncate truncate means you can just cut it off and and we're good to go either option is acceptable we'll take them both so you either put 1.999 or 2. those are two options when you go out to the nearest thousandth when you're changing those two so now it's your turn try it any way you want to you can use your calculator however you want you can use a table you can use a function notation as I showed you but you're going to do number four all by yourself go to the nearest thousand that's what we want this part right here you want to be very very accurate so you're going to have a very very small change in the X okay so my approximation for the rate of change I got uh the slope equal to 0.135 and that's from here's a screenshot I put the function in and then I plugged in ative -2 and a -2. Z1 which be mindful that's on the left right because that's smaller but it doesn't matter and then I go ahead and find the slope and I get uh what was my answer here. 13526 so I can cut that right off I can either round it or truncate it same answer either way all right that's number four one more part left super easy so our very last part and then I promise we're all done talks about positive and negative rates of change or positive and negative slope a positive rate of change means that as one quantity increases or decreases the other quantity does the same another way of thinking about this is as X increases y also increases so think of a graph as one variable increases here the other variable increases on this this side it gives you some type of you know positive it grows it's going uphill from left to right you ever play Mario Brothers means Mario's running uphill uh a negative rate of change indicates that as one quantity increases or as the X increases the other decreases so it goes down like this or Mario would be running downhill left to right so pause the video right now and try to figure out whether uh five and six would be positive or negative based on what they tell you go pause the video okay so for number five did you get positive because as years increase all right let's put the years down here so as years increase then a high school student body we'll just put B for body that also increases so it looks something like this it's positive however number six as Mr Bean weight decreases which means as we go left on the x-axis his running distance increases so it kind of looks like this as we go left it goes up and that's it that's positive and negative rate of change but we know that from positive slope and negative slope that's about it good luck on your Mastery check students and remember it's always nice to be important but it's more important to be nice s