today's lecture is on Section 1-3 rates of change key topics for the lesson is just what do we mean by a rate of change um trying to remind you of some key terms uh related to it like slope maybe something you've heard about before average rate of change which is a pretty specific computation how to calculate that using a graph or using a table or using function notation and then lastly how does that relate to where a function is increasing or decreasing we will use integral notation again when we get to that part so this is the basis of calculus is this concept of rate of change and so uh making sure we know how to do the computation is helpful here in an algebra context so here we have an example function P of T measures the position of the car at time t on a drive from Pittsburgh to Shippensburg so we have times is our independent or our X Type variable and the position how many miles you've traveled is the Y value or the dependent value in this case the p-value so between one and two we would be looking at there trying to find the average rate of change so we're looking at those two times in between those two times you traveled 60 miles and a total amount of time of one hour has passed and so we would say your average rate of change is 60 miles per hour it's also known as the average velocity now during the second hour is it greater than or smaller than so let's look we did 175 miles at hour three and we'd only gone 117 at our two and again we traveled one hour's time and so all together we have 15 minus seven we've gone 58 miles in that one hour so it's 58 miles per hour it is a little slower in that second hour so that is an average rate of change it is a slope between two points so we fix two points we look at where we are at the beginning and at the end and we're finding a slope which is in this case the difference in the mileage over the difference in the time the formula you'll often see is this one right here F of B minus F of a you'll see oftentimes I I abbreviate it as Arc for average rate of change but it's F of B what's the function value at B and the function value at a you're going to subtract function values or Y values in the numerator over where did you start and where did you finish B minus a those are the X values and so it's the slope of two points the point at a in the point of B so here's an example here's a picture where we can see it graphically and you can see the dashed line actually shows the line connecting between when X is negative one and when X is two so our average rate of change formula what we're going to do is we'll have 2 minus negative 1 so we'll take the difference of the x's on the bottom and then on the top we'll have F of 2 which is three that's the function value and we'll have f of negative 1 which is zero that's the function value read off the graph and so we have three over three which is one so the slope of that dashed line is one and hopefully you can see that because I go over and up one over and up one so that should fit with what we expect so here is a table we're going to do the same thing between eight and four we're going to do F of a minus F of four over eight minus 4 so we're going to read from the table F of the 8 is 16 and F of four is six so we have 10 over 4 otherwise known as five over two otherwise known as two and a half so that would be our average rate of change between four and eight if we're doing it between four and ten four and ten we do the function value at 10 minus the function value of 4 over 10 minus four 26 minus 6 over 6. 20 over 6 which is 10 over 3. that would be the average rate of change given a table here we have a graph but we also have a function so we're going to do F of one minus F of zero over one minus zero 1 squared is one zero squared is zero so we actually get one and that probably shouldn't surprise you that that slope between 0 and 1 is what let's go between one and two I'm hoping that you see that between one and two the function gets a lot steeper so we're expecting a steeper number in this case F of two minus F of one over two minus one two squared is four one squared is one so we get three which is what we expected something much steeper what about between zero and two that's the slope there F of two minus F of zero over two minus zero two squared is four zero squared is zero so we get two so the slope of that word is two the last thing we're going to do is an example where we have a variable still left in F of 1 plus t minus F of one over one plus T minus one this will give us an expression involving t so this is one plus T Square minus 1 squared which is one all over one plus T minus one which is just a little T 1 plus t squared is one plus two t plus t squared if they're not sure where that came from you're foiling out one plus T times one plus T that's where you get that expression 2T plus t squared all over t and we can actually simplify that so that's two because the t's cancel plus t so we get the expression two plus two so that's a harder one to do you may want to practice ones with letters as well as once with numbers both of them will actually be skills you need to know for calculus so what I would suggest you do next is pause this video and try the first half questions one through five on the section 130 worksheet the next part of the lesson deals with increasing and decreasing if you're trying to read this the function that we're looking at here we read a graph as we look at it from left to right so I'm going to kind of highlight it so you can see so I'm going up up this would be an increasing part of the graph then I stop increasing at that point and then the graph is decreasing all the way down to about there and then it's increasing after that how is this related to the average rate of change the average rate of change in the red pieces is always going to be a positive number indicating it's going up and the average rates are changing the blue zones is going to be negative expressing that it's going down and so if we wanted to express this in terms of intervals we're increasing all the way up to let's call this like negative 1.2 and then we'll call that 1.8 onward is increasing and then we're decreasing in this middle Zone between like negative 1.2 and 1.8 and we're always going to use open intervals because we're going to say it's increasing and decreasing and there's this sort of instantaneous pause at negative 1.2 and 1.8 where it's not increasing or decreasing really at that point so here are some tables and we want to look and see are they increasing or decreasing so I'm going to look at this first one that's definitely going up that's definitely going up that's going up that's going up that's putting up so that one is increasing all the way what about this next one oh in this case this is going down down between there and there still going down we're still going down we're still going down so in this case we are decreasing all along the way okay this one we're going down decreasing decreasing them decreasing decreasing decreasing so this is an all decreasing function and this is increasing then we go on up increasing increasing of increasing and increasing so this looks like it is increasing all along the way so again you just kind of look between two points what's happening is it going up or down and now I would you're done with the video and you should be able to complete section 1.3 and check it in the answer key