hi everyone welcome to calculus one we're going to be doing a ton of work this quarter with slopes and rates of change that's basically the big idea for the entire course so for example we could be interested in how fast a car is moving or how fast a price is changing and we want to really dig into how we talk about that in particular we're headed towards the idea of the derivative that's what calc 1 is all about and the derivative is supposed to tell you how fast something is changing at a very specific instant in time this can be surprisingly tricky mathematically speaking it's fairly straightforward if you pick a range of times so from one hour to three hours how much did the car travel and can we figure out uh speed based on that but that's actually an average rate of change that's not an instantaneous rate of change an instantaneous rate of change would be us being able to say one hour into our trip the car was moving this fast and that's actually not even how a speedometer on a car Works a speedometer on a car would take a very small uh time period and it would calculate how much distance you traveled during that very small time period and it would divide it by how much time had passed and that would give you an average speed so even though it's giving you an approximation of your speed at a given instant it's doing it using an average so this is basically the big idea we're going to be trying to tackle to get to the idea of a derivative how do we calculate an instantaneous speed or an instantaneous rate of change so this section is actually really just about getting us started digging back up some pre-calculus ideas and really framing them in the terms of thinking about calculus so really big ones for us are going to be rate of change that I've just been talking about and function notation understanding your function notation is extremely important and also mixed in with that is going to be a fair amount of messy algebra you've taken your algebra classes you've got all the tools but we're really going to work those skills in this class sometimes the hardest part of calculus for people is just getting really good at all of that algebra so hopefully by the time you get out of this class you're going to feel like an algebra expert which will be excellent okay let's jump into it with this formula and graph that I've given you here just as an intro uh average rate of change I've been talking about it um and I have it defined right here so the average rate of change of a function f from x equals a to x equals B can be found using this formula right here and notice it has kind of a funny I don't know name right here um that's really just an abbreviation aroc is average rate of change and then the a b is just interval notation it's just saying the average rate of change from A to B so just what we were saying here x equals a to x equals B and then I'm hoping maybe the right hand side is looking a little familiar to people technically all that is is your slope formula it's written with function notation but that's really what it is it's a change in y over change in X and the reason it gets this fancy name is because slope implies a constant rate of change which is only true for perfectly straight lines when we're looking at a graph like the one I have here this is most definitely curved so we can't talk about a perfect slope but we can estimate how steep the curve is by finding average rates of change so let's just pick this F of B minus F of A over B minus a apart a little bit so I'm going to mark this off let's say this is a and this is B and they could absolutely be in the opposite order it could be B to the left of a it doesn't matter the formula will work either way just like slope works either way but I'll put it in this order because that's how people tend to like to think about it okay so picking apart the top of this fraction F of B minus F of a whenever you see F of something you should be thinking y value that's the output of your function and specifically F of B is the Y value you get when X is equal to B and F of a is the Y value you get when X is equal to a so we've already got this labeled x equals B and x equals a we know how far from the y-axis that is to get the Y value we're going to run straight up until we hit the curve that's hitting the function and the function pairs an x value with a Y value so that point right there is b f of B so x value is b y value is f of B and to be really literal with this and you need to be really literal you need to be able to visualize this that means that the distance from the x-axis up to the point is f of B so we're taking this y value F of B and then we're subtracting off the Y value when x equals a so we're going to repeat this process but we're going to go back to a and we're going to say okay well we hit the curve a lot sooner we have a smaller y value and we hit at the point A F of a remember F of something is always your y value so what that really means is that this vertical distance is my f of a but what we're being asked to do here is to actually subtract F of a from F of B so I'm going to try and carry this over this is a little rough okay so it's a right here I'm supposed to subtract this distance down here from that entire length F of B and what I'm left with is just this top bit right here the length of that is f of B minus F of a so we have a vertical distance a change in y F of B minus F of a and a really calculusy way to say that change in y is Delta y so this is my Delta y change in y you don't have to use that but you'll see it as we continue um otherwise known as rise change in y is what we would call rise in a slope we have to do the same thing for X's they tend to be a little bit easier to visualize so if we look down here and we say Okay x equals B we already marked off where that tick mark is but what that really means is that this entire length right here is B units long we have to go B units over from the y-axis to get to our point and similarly let's see if I have another good color to use we'll use purple again this length right here is a units long and we're supposed to subtract the B the a from the B so what we're really looking for here is what's left over and what's left over is this little bit right here that's the B minus a the difference between a and b that is otherwise known as run or Delta X if you want to get towards some calculus notation okay so we're taking this orange rise and we're dividing by this blue run and what that gives us is the slope of a line and it's the slope of the line between the two points that we labeled a f of a and b f of B so we can draw this line we'll see if I can do it that's pretty good it's supposed to go right through those two points we can draw this line and its slope is equal to the aroc the average rate of change this line does have a very particular name when you take two points on a curve and you connect them and you make a line that is called a secant line so one way to visualize uh average rate of change is that it is the slope of the secant line okay so that's a pretty good start on some of the ideas that we're going to be working on in this section we want to be able to calculate average rate of change given a graph or a formula or a table and we want to be able to visualize that as the slope of a secant line all right we'll jump into examples next