BQ #7: Unit V: Derivatives and the Area Problem

The difference quotient is a something we have come across earlier in Math Analysis. However, the question now is, how are we able to derive this? Where does this come from? Presently, with our newfound knowledge of calculus, we know that the difference quotient basically is the first step towards finding our derivative (the slope of all tangent lines). On a side note, we know that in order to find the derivative, we do the difference quotient, then find the limit of it, as h approaches 0. But again, we go back the question of how did we get the difference quotient in the first place? Look at the picture below for better clarification--I will explain below.

http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG

I mentioned before that the derivative is basically the SLOPE of all tangent lines. When you hear slope, you should think of the formula we use to find the slope: (y-y)/(x-x) (rise over run). We are going to use this formula and basically, plug a few more things in to make it the difference quotient. In order for us to the the slope formula, we need to know two points. Look at the graph above. That first point, we will assume that the x value is just "x". Thus, the y value must be "f(x)". So, our first point is (x, f(x)). Our second point ,we move further across the x axis. We will call the distance crossed "h". So the x value would be x+h. So, it only makes sense that our y value is f(x+h). Our second point is (x+h, f(x+h)). Now that we have our two points, we will plug it in our slope formula. So your plugged in slope formula should be [f(x+h)-f(x)]/[x+h-x]. We simplify the bottom because the x's cancel out and you are left with this: [f(x+h)-f(x)]/h. THE DIFFERENCE QUOTIENT. 

Now you know where the difference quotient comes from. However, in calculus, we proceed further, as I mentioned before. If you were trying to find the slope, the derivative, you would find the limit as h approaches 0. Your main question is probably why is h approaching zero. The thing is, we want to be able to find our tangent line (a line that touches the graph at one point). The problem is that we a secant line, where we touch the graph TWICE (as you can see the red line is touching the blue line twice in the picture above). So, in order for us to get one point, we can basically have those two points on top of each other... meaning that they are in the same place. We are able to get them to sit on top of each other by decreasing the "h", the distance between the two points. The smaller "h" is the closer the two points are. Therefore, we have our limit as h approaches 0 because we can't actually have it at 0, but we can get it pretty darn close. 

This is just a hint of what calculus has in store, and I'm still waiting for the rest to be revealed! This is my final blogpost and I hope these posts have helped in some way or another because I know I've learned a lot this year. I've conquered Math Analysis and no lie... I feel like a Mathematician. I'm done! Lights out. *Nox* 

References: 

http://cis.stvincent.edu/carlsond/ma109/DifferenceQuotient_images/IMG0470.JPG