Probably one of the scariest courses a person can take in high school is calculus. Some people have taken the class, passed (barely sometimes), and gone on to college or whatever without even knowing what calculus is all about and just how useful it is. Well, calculus is definitely one of the (if not the) most interesting areas of mathematics as we will see in my next tutorial. This article will not have any math in it. First, a little about why calculus was invented. The credit for calculus is mostly given to, I believe, Leibniz and Newton. Even though they both contributed a lot to calculus Leibniz by himself thought of the notation of calculus we use today. I read somewhere that he would sit and think for days at a time trying to come up with an appropriate notation for a concept he had come up with. I am really bad with the history of mathematics so the only one of those two men I know anything about is Newton. Sir Isaac Newton was more of a physics guy than a math guy. He came upon many physics problems which the mathematics of the time could not solve. For instance: finding the velocity of a particle at a given time. It sounds easy but think of what it is really asking. If you were given the position of a particle at a certain time and the particle's position at some time later you could easily figure its velocity by taking the difference of position and dividing it by the difference of time. That is called average velocity. When you have two time frames it is so easy it is almost trivial to find the velocity. But, what if you were only given one time frame? Finding the velocity at a certain time (an instant) is called instantaneous velocity, which happens to be the line tangent to a position versus time graph. This concept is what differential calculus is entirely based on. But, that was not enough for Newton. Another challenge of his was finding the area under a curve. Why would anybody want to do that? Let's say you had a graph where the independent variable (x-axis) was time and the dependent variable (y-axis) was velocity. What might you get by multiplying a velocity by a time? Distance traveled right? If you were to drive twenty meters a second for ten seconds you would have traveled two-hundred meters right? So the area underneath a velocity versus time graph represents distance traveled. We had actually done that with the previous example. But most of the time things are not so easy. In the above example there was no acceleration. When there is acceleration and especially varying acceleration the velocity versus time graph can become a complex curve. Finding the area underneath that curve is what another branch of calculus is based on: integral calculus (something I do not plan on writing about any time soon). So, Newton needed calculus for two reasons: to find the slope of a point (instead of two points) and to find the area under a curve. As you can see from the title of this article we are only going to be talking about differential calculus. We will be getting into the nitty-gritty stuff pretty soon so I will not get too specific. Well, in differential calculus (I am going to just say calculus from now on) we are going to be constantly searching for a ratio. It is a very magical ratio indeed for it is the slope of a line tangent to a point on a graph. For now you can think of a tangent line as a line which only "touches" a graph at a point. The most popular form of the ratio is (dy/dx). That is not the only form though. You could have (dx/dt) or (dz/dy) or anything else. The variables that the "d" is attached to represents the independent and dependent variable of a function. But, if you were to take away the d's from the ratio you would be left with (y/x). In mathematics what does that ratio represent? A slope correct? Rise over run? Delta-y over delta-x? In any case, when you attach the "d" to "y" and "x" (or any variable) you are simply saying "a little bit of y" or "a little bit of x". This will become obvious later, but calculus is only concerned with very, very, very small portions of "y" divided by small portions of "x". But, the two must always stay proportional. For example: x = 50,000
y = 10,000,000
y/x = 200
-- you could also think of it like this
dx = 5
dy = 1000
dy/dx = 200
As you can see the (dy/dx) equals the (y/x) but separately the
x, y, dx, and dy's are not equal. But,
remember that you would never be able to calculate how much dx or
dy is equal to because it is so extremely small. In fact
they are so small they hardly even exist. This is exactly how small
we want them to be. Which leads us to an important observation to be made.
What would happen if we were to square or cube the dx or dy
term. If the term is already indefinitely small squaring it or taking it
to any power greater than one would just make it smaller. Thus,
whenever we encounter a term like that we can simply disregard it.
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