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# Secant line with arbitrary difference

## Video transcript

a secant line intersects the curve y equals the natural log of X at two points with x coordinates two and two plus h what is the slope of the secant line well they're giving us two points on this line it might not be immediately obvious but they're giving us the points when X is equal to 2 when X is equal to 2 what is y well why they tell us y is equal to the natural log of x so in this case it is going to be the natural log of 2 and when X is equal to 2 plus H what is y well Y is always going to be the natural log of whatever X is so it's going to be the natural log of 2 plus h and so these are two points that sit on the secant line this happens to be where the secant line intersects our curve but these are two points on the line and if you know two points on a line you will then be able to figure out the slope of that line and we can just remind ourselves that a slope is just change in Y over change in X and so what is this going to be well if we view the second one as our end point our change in Y going from Ln of two to Ln of 2 plus h so our change in Y is going to be our end point so natural log of two plus h minus our starting point or our end y-value minus our starting y-value natural log of 2 and then our change in X our change in X is going to be our ending our ending x-value two plus h minus our starting x-value minus 2 and of course these twos cancel out and if we look here it looks like we have a choice that directly matches what we just wrote this right over here natural log of two plus h minus natural log of 2 over H now if you want to visualize this any a little bit more we could we could draw we could draw a little bit I'm going to clear this out so I have space to draw the graph just so you can really visualize that this is a secant line so let me draw my y-axis and let me draw my x-axis and y equals the natural log of X is going to look so let me underline that that is going to look something like this I'm not viously hand drawing it so not not a great drawing right over here and so when when we have so we have the point 2 comma natural log of 2 which would be let's say it's over so this is if this is 2 then this right over here is the natural log of 2 so that's the point 2 comma natural log of 2 and then we have some other we just know it in the abstract 2 + H so it's 2 plus something so let's say that is 2 + H and so this is going to be the point where we sit on the graph that's going to be 2 plus h comma the natural log of 2 plus h and the exercise that we just did is finding the slope of the line that connects these two so the line will look something like will look something like that and the way that we did this is we figured out okay well what is our change in Y so our change in so let's see we are going from y equals natural log of 2 to y equals natural log of 2 plus h so our change in Y our change in Y is our natural log of 2 plus h minus natural log of 2 minus natural log of 2 and our change in X well we're going from 2 to 2 plus H we're going from 2 to 2 plus h so our change in X we just increased by H we're going from 2 to 2 plus h so our change in X is equal to H so the slope of the secant line the slope of the secant line is secant it intersects our graph in two points is going to be change in Y over change in X which is once again exactly what we have over there