Slope of a line secant to a curve

so let's review the idea of slope which you might remember from your algebra classes the slope is just the rate of change of a line or the rate of change of Y with respect to X as we go along a line and you could also view it as a as a measure of the inclination of a line so the more inclined the line is the more positive of a slope it would have so this right over here this has a positive slope it's increasing as x increases and if this had an even higher inclination like this if it increased even more as x increases then it would even have a higher slope so this right over here is some line so that's some line and just as a reminder we can figure out the slope between two points two points define a line and between those two points we can find the rate of change of of Y with respect to X so let's put two points on here so let's say that this point right over here this this x value is X sub well this is pronounced X naught or you just X sub 0 is just X naught and when and when X is x naught for this line y is y naught y naught so this is the point X naught comma y naught and let's say we have another point all the way over here and let's say that this x value this x value right over here is X sub 1 and the Y value over here is y sub 1 so there's a point X sub 1 Y sub 1 so just as a review the slope of this line and a line by definition has a constant slope between any two points that you pick the slope of this line which is often denoted by the letter M is your change of rate of change of Y with respect to X or another way of thinking about it for a given change in X for a given change in X how much are you changing Y or change in Y divided by change in X just as a reminder this triangle that's the Greek letter Delta it's shorthand for change in whatever so change in Y over change in X so let's think about what this is going to be for this example right over here well let's think about change in X first so we are moving from X naught to X one so our change in X so this is our change in X right over here we're starting at X naught and going to X 1 that is our change in X I'll put that in that pink color that is our change in X and what is it equal to well if we're ending here we started here let's just let's just do ending point minus starting point so it is X 1 minus X minus X naught and that way doing it this way I would have made sure that I have a positive value I'm just assuming that X 1 is larger than X naught now what is my change in Y well once again ending point ending Y value minus starting Y value y 1 y 1 minus y 0 now you might be saying hey could I have done Y is y sub y naught minus y 1 over X naught minus X 1 absolutely you could have done that then you would have just gotten the negative of each of these values on the numerator the denominator but they would have cancelled out the important thing is that you're consistent if you're subtracting your starting value from your ending value in the numerator you have to subtract your starting value from your ending value in the denominator as well so this right here you probably remember from tape from from algebra class the definition of slope is the rate of change of our of Y with respect to X or it's the rate of change of our vertical axis I should say with respect to our horizontal axis or a change in Y or change in our vertical axis or change in a horizontal axis now I'm going to introduce a little bit of a conundrum so let me draw another axis right over here scroll over a little bit just we have some space to work with so that was for a line and a line by definition has a constant slope if you calculate this between any two points it's going to be cut any two points on the line it's going to be constant for that line but what happens when we start dealing with curves when we start dealing with non line or non linear curves so let's imagine a curve that looks that looks something like this that looks something like this so what is what is the rate of change of Y with respect to X of this curve well let's look at at different points and we can at least try to approximate what it might be at any moment so let's say that this is one point on a curve let's call that x1 and then this is y1 and let's say that this is another point on a curve right over here x2 and let's call this let's call this y2 so this is the point x1 y1 this is point x2 y2 so we don't have the tools yet and this is what's exciting about calculus we will soon have the tools to figure out what is the rate of change of Y with respect to X at exactly this point but we don't have that tool yet but using just the tools from algebra we can at least start to think about what is the average rate of change over the interval from X 1 to X 2 well what's the average rate of change well that's just how much did my Y change how much did my Y change so that's my change in Y for this change in X for this change in X and so we would calculate it the same way y2 minus y1 over x2 minus x1 so our change in Y over this interval is equal to y2 minus y1 and our change in X our change in X is going to be equal to x2 minus x1 x2 minus x1 so just like that we were able to figure out the rate of change between these two points or another way of thinking about it is this is the average rate of change for the curve between x equals x1 and X is equal to x2 this is the average rate of change of Y with respect to x over this interval but what if we have also figured out here well we figured out the slope of the line that connects these two points so we figured out the slope of the line the slope of the line that connects that connects these two points and what do we call a line that intersects a curve in exactly two places well we figured out we call that a secant line so this right over here is a secant line so the big idea here is we're extending the idea of slope we said okay we already knew how to find the slope of a line a curve we don't have the tools yet but calculus is about to give it to us but let's just use our algebraic tools we can at least figure out the average rate of change of a curve or of a function over an interval that is the same exact thing as the slope of a secant line now just as a little bit of foreshadowing where is this all going how will we eventually get the tools so that we can figure out the instantaneous rate of change not just the average well just imagine what happens is if this point right over here got closer and closer to this point then the secant line is going to better and better and better approximate the instantaneous rate of change right over here or you could even think of it as the slope of the tangent line