You know, I have always been fascinated by calculus. It is one of the last class some people will take, and others won’t take it at all. This make the subject all the more mysterious and fascinating. Especially after knowing that Isaac Newton and Gottfried Liebniz invented it independently. Well, after reading a book about it, I can say that it is as fascinating as I thought it would be, and not as difficult as I thought it would be. Basically, it is this: if you suck at algebra, you will suck in calculus (some calculations are difficult!), and if you are good at algebra, you can be good at this with a few practices.
So, what is calculus, and why is it important? To understand calculus, you have to understand the slope of a curve. It can be any curve, a linear one or a curved one. The slope of a curve at any point on it is a line tangent of it. What does that mean? Well, it is like this:
As you see, the line only touches one point on its curve. As for the linear equation, the tangent of the line is itself. Basically, in the equation mx+b, the slope is mx, which can be found by dividing the coordinates of of 2 points on a line (y2-y1/x2-x1). That one is easy, but there is a problem. How the hell do you find the slope of a point in a curve?! If you choose two points on the curve, it won’t be the slope of a point anymore, it will only be an approximation! The closer the points on the line are, the more accurate it is, but it is not an exact value! You bring it closer and closer together, and it is still to no avail. Great, you need to bring both points infinitely close to each other to know the slope. You start to think the problem is impossible…
Then it strikes you. Of course, you have to bring both points infinitely close! At that point, you introduce the concept of a limit. It means that you go infinitely close to a point, but NEVER EVER, arrive to that point. For example, limit–>x=0 means that you go really closely to 0, but never to 0. If you tell me it is exactly 0, I will list you in my annals of complete idiots! (kidding! jeez, can’t a guy joke around?) But seriously, don’t do it. Since the slope of two points are the difference between them, so you start forming an equation applicable to all curve, which is called the derivative. A derivative is the difference between y divided by difference of x, as shown in this picture:
P, in this case, is y, and q is x. This leads you to the equation (the derivative): dy/dx, d is a triangle symbol called delta, which means the difference. The x part remains the same, but the y has to be more descriptive, since y is the product of a function machine:
insert number in x—>f(x)(it can be x or x^2 or 2x/3+1, etc)—->result is y
So, the dy part changes to f(x+dx)-f(x). That is because the second point is certain distance away from the first point, which is dx, so you can tell f(x+dx) is the second point. So, that makes the derivative: limit–>dx=0 f(x+dx)-f(x)/dx. The limit means the difference between the two points are infinitely close together, but never together. (note the words between two points, it does NOT mean that the points are going to x=0)
Simplify to: x+dx-x/dx
x-x is 0, so dx/dx=1. As you see, the slope of a linear equation x is 1, and it is consistent with mx, when m is 1. Now, let’s try with a quadratic equation, shall we?
limit–>dx=0 f(x+dx)^2+f(x)^2/dx =(apply FOIL!!!(First Outside Inside Last))
2x*dx+dx^2/dx = 2x+dx
Which means as dx approaches 0, the slope is 2x. This is it! We did it! We found the slope of a cuve! Congratulations, you passed introductory calculus! Of course, this is not everything calculus has to offer. Doing everything with just the derivative is too much, so there are many rules wich simplifies derivation. The rules are all made with the equation above, so as you can see, the derivative is the basis of everything in calculus.
Now, what is the importance of calculus? Calculus is especially important to physics because many terms in physics are expressed in the change of a point on a line. For example, position is f(x), which can be in case of gravity, say, f(x)=-1/2gx^2+ax+b. Velocity is the change in position, which is -gx+a. Acceleration is the change in velocity, which is g (9.8 in case of Earth). As you can see, calculus is very important in physics. Another importance is that one can find the area under a curve with calculus, isn’t that cool!! With it, you can find the area of an ellipse, or the volume of a, say… a perfectly parabolic mountain. As you can see, calculus is versatile, and I hope you continue to investigate more about the subject. Too bad I can’t have a course in calculus at my High School, since I started in a lower level stuff.