A Fun Little Informal Calculus Introduction

[Kvasir 1121]

Spoiler

Short clarification: ~ means approximated to here.

 

Hello all, I'm bored so I'm writing this up because I'm neglecting doing my algebra homework and I'm sick. Anyway, I wanted to show off some small application or at least example of calculus. 

 

Now, I know calculus sometimes makes people feel bored or things like that, and I think that is because of a boring look at what you are really doing. Now, I'll just show off this interesting little example that will kind of be cool to those who have knowledge up to quadratics (or who can remember them.) Here we go.

 

Calculus, in very simple terms, is the study of change. So, if I were to throw an apple at somebody, being able to study how that apple changed in distance and height over time could be done with calculus given a correct modeling equation (i.e something like x^2 if you somehow threw it from a negative distance.) We will use an example like this, actually. 

 

The Algebraic Part

 

Consider the graph of -0.1x²  + 2, if you wanted to solve this, you would use the quadratic formula, or since we do not have two X's, solve thusly: 

-0.1x²+2 = 0

-0.1x²=-2

x²=20

x ~  ±4.5

Now, that was rather straightforward, and you end with a graph like this. 

 

Notice, that the graph does go below its root, as in the X's we found (yes too, pay mind to the +/- I denoted)

 

Okay, this is all well and good, and something you might notice quickly when doing basic algebraic quadratics is the standard form that they come in:

ax²+bx+c

Let us break this down very simply to any that might be confused on something like this, the a is just a number separate from b or c, this does not mean they cannot be equal to each other, b is likewise the same. They are both being multiplied by x, the unknown, and in this form, we are simply outlining how you would fill out a quadratic equation. You do usually know your a, b, and c, and they can be any number. The c is a constant, meaning that it is just any number.

 

But there is something we notice when throwing this apple, after breaking down the equation and looking at the graph, there will at some point be a maximum point where it turns and then begins to descend, as you can see the graph is constantly decreasing. You know this because it is never inflecting in upon itself as if you were to crush an aluminum can in your hand, an example of a constantly increasing function would be 

 

2^x

 

 

The concept of increasing and decreasing forms of graphs might seem trivial for now, but it would do you good to understand the difference as it shows the form of the function. 

 

Carrying on, back to this maximum point, this point is important because it is when we knew it has reached a maximum y. For quadratics, you are given the equation

 

x = (-b)/(2a)

 

The observant reader will note that we are still referencing the all-encroaching form of ax² + bx +c constantly. This equation will give you x at which you are at a maximum. Meaning at which point in time are you at a maximum, so in the case of the first graph, we are at a maximum at let's say 0 seconds. You can subsequently find out your vertex by plugging zero into x for our equation. So y = -0.1(0)² + 2, is of course 2. 

 

Now that this example is out of the way, we can get to the meat of what we are doing; a simple example of calculus. All of this is just some preparatory knowledge that you would need or at least have to semi-understand.

 

The Calculus Part

 

Now, the only thing we will need to know of calculus that at first seems like a kind of complicated idea becomes simpler the more you think about it. The idea of a derivative, we will not worry about limits as that is not what we need to know about for this part, if this was an actual lesson in calculus I would start there but I am here to entice not to instruct for now. 

 

The idea of a derivative, we will not worry about limits as that is not what we need to know about for this part, if this was an actual lesson in calculus I would start there but I am here to entice not to instruct for now. I will now introduce the idea of a derivative, just be patient and think with your head about this. I will first give the unthinking answer and then we will delve deeper into the concepts. I am first going to give you the definition Wikipedia gives me: The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. This is all well and good, no? No, it is not all well and good. Of course, you could dissect the meaning by reading it flat out but one thing you must realize and I hope you take away from this document is that there is no learning if you just read the dictionary definition of something. You must think about its underlying concepts and what it actually means, and then we can start writing frivolous and rigid definitions that are more accurate. When you learn math, seek for the soul of it, not for the high-lexiconic verbiage that most scholars prefer to use in formal writing. If we were formal mathematicians, yes that would be useful if reading a mathematical proof, but I have gotten off track.

 

I will help you understand this definition now, but first, we must introduce the idea of a rate of change. This is fairly simple, let me give an example: Let us say I am running across a field, and I run five meters in a whopping two seconds, and let us say I upheld this speed to the decimal somehow, we can express this relationship as r= d/t, meaning that my rate is equal to distance divided by time. So I can say that I was going at 2.5 m/s, or two point five meters per second. Now, in the real world, I cannot run at a constant sustained speed to the decimal, as I am a human and there is such thing as gravity and differences. So it seems I am out of luck if I wanted to find my exact speed at some moment. Now, let's say I wanted to find my rate of change at some moment. This is a strange thing to think about, as it is a paradoxical statement. I want to find the rate of change in a moment, of course, this is impossible. But there is something we can do that we call Instantaneous Rate of Change.

 

The name of this paradoxical in nature, fitting the rate of change at an instant moment does not make any sense, though the actual meaning is something slightly different. I will demonstrate the difference with a very simple geometric example. I will introduce some vocabulary:

 

Secant: This is a trigonometry term, but also a geometry term (though those two are one in the same) I use this word here to define a straight line that is crossing a certain curve at two points. 

Curve: This is very unnecessary to define as I hope you have gathered what a curve is by this point, but is a result of a function we can say, that either increases or decreases at different concaving points through time. It has form in geometry. This is a shoddy definition but it is the best I can come up with for now.

Tangent: A line that only touches one point on a curve, I can get into the trigonometric link in another post if anybody wishes. 

 

Let us have a curve, let us say -x^2+5

 

 

 

And let us form an approximated secant line that cuts through the points (strictly above the x-intercept line in the positive y's.) (~ -1.6, ~ 2.5) let us call this point D, and then our other point, E which is (1, 4) In between these two points on the curve there is point C defined at (-1, 4)

 

Let us call this secant line segment DE. 

 

 

Now, let us say this line DE begins to 'shrink' in distance, the points D and E getting closer and closer to C. We can put A tangent liner on C. 

 

 

In the way I graphed this, one is able to see where I am headed with how this will go down. Let us say that the two points D and E now begin to approach C, their line DE beginning to change as well. Now, we cannot have them be equal to each other since C is at a single point, and if we think of our secant line as an approximation of a function, let us say in point-intercept form ( https://en.wikipedia.org/wiki/Linear_equation#Point–slope_form ), this approximation is narrowing. Let us say we get to a point where both points get very very very close to this C, but it never touches. It never equals, but it approximates to where there is no mathematical difference. This may be confusing at first, but it just means that the two points get infinitely close to C and our line DE basically becomes the tangent line of C.

 

This may have been a bit confusing, but the thing is to take away that our approximation of a certain point on a graph for its rate is not actually at the point, but just extremely and when I say extremely I mean very much extremely close, to where there is no difference. If this concept still makes perfect sense to you, you need to back and re-read and analyze the graphs, as this is a concept paradoxical and amazing still. The true power of it comes from its absurdity and its elegance. This may have seemed left field, but when we are finding a derivative what we are really finding is the formula to find the rate of change at any point on a curve. 

 

There is a certain equation that was created by Sir Isaac Newton, it goes as follows: f(x) lim h->0 (f(x+h) - f(x))/(h) = the derivative of f(x)

 

 This seems a bit confusing, I can sense, but what it is saying is basically what we have just done. The limit, which we can just improperly define for now as when something is approaching another number ad infinitum. So let us just say this, the limit of f(x) as h is approaching zero for the function (f(x+h) - f(x))/(h) will be the derivative, aka the formula for the instantaneous rate of change at any moment on the curve. Now, we could go about and plug things into this and figure out our derivatives, but it would be boring. But if you wish to, I challenge you to find the derivative of x^2 using this formula, but for now, we will be moving on.

 

This takes a long time as you get into the higher exponents and learn more rules (i.e the derivative of a constant is always 0), but there is one thing we can see that is very useful a pattern. We love patterns because it makes our lives easier. This pattern is called the POWER RULE, it is cooler than it sounds! 

 

The pattern goes like this, let us say I have x^2, and I find the derivative of it. My exponent will go in front of my x, as a coefficient, and my exponent will be subtracted by one. 

 

https://i.gyazo.com/56bd9afc1226c4b4905404deef195196.mp4 

 

Now that we have all of this knowledge, we can get started with some examples, since it is easier to write down I will be using newtons notation, despite it being in my opinion inferior for understanding, later on, it should suffice in our short tutorial. 

 

Let y=x^2, 

Thus follows y'=2x. 

 

We pronounce the ' as 'prime', it has nothing to do with prime numbers. But this is just a fancy way we can say that we are finding the derivative of something. Let's try it out with some others things now.

 

Let f(x) = x^2 + 2x + 3

Thus follows f'(x) = 2x+2

 

This is interesting, and you can do this yourself, but every time you have a constant, it has no x, so it automatically becomes zero.

 

Now, I will leave you all with one last thing that I think is very interesting and should tie our algebraic connections before together into one nicely backed package. I mentioned before the useful base formula for quadratics and its vertex form. Now, I told you how you could derive the base formula on your own, but not how you could get the vertex formula. If you really wished, you could spend some time thinking about it, but I have a better way to figure it out, using our brand new shiny tools. 

 

Let us create our expression: ax^2+bx+c

 

Now, let us find the derivative of this, and remember what this exactly means, we are finding the slope of a point at any given position on the curve. Let us go for that now. 

 

y'=2ax+b

 

Now, you may notice something, we have this new expression, and it is basically a formula for any point on the curve... Now, look at it some more and think, what if I wanted to find when the movement is at zero. This brings us back to the concept of increasing and decreasing and what a vertex really is. It really is the point at which the graph stops moving at a certain point, this is a strange idea to some but basically, all it means is that at that point its tangent line is a horizontal line. So what we can do is try to find the point at which any position on the graph creates a horizontal tangent, aka our vertex.

 

2ax+b = 0

2ax=-b

x=(-b)/2a

 

We seem to have gone full circle now, showing our vertex formula in a simple manner just by the things we have learned today. There is actually one last thing I will leave you with, a physics connection and a prelude or exposition to something I will write possibly, later on, the idea of derivatives of derivatives, higher order derivatives we can call them, and a brief physics connection. I'll just leave you a little chart.

 

Position Curve (i.e x^2, x^3 + 5x + 3, etc) - A model for certain action.

The First Derivative  - The velocity of the position curve at any certain point.

The Second Derivative - The acceleration of the velocity at any point.  The rate of change of velocity.

The Third Derivative - The jerk, or lurch, the rate of change of acceleration. (This one is rare and very cool, but we can use the example of you starting up a very nice car and it suddenly accelerating and your head kind of bumping back on the seat's head. Read this Wikipedia article on it: https://en.wikipedia.org/wiki/Jerk_(physics)) 

 

Well, this is it. I hope you all enjoyed it and learned something if you stuck with reading it thus far. Thank you, have a nice night.

 

[Arygon 135]

Yes. I am mentally ill too.

[GODHawkEye 106]

you're putting off your algebra homework but are tryna teach us calculus

 

ill admit your info is pretty accurate but i have a very hard time believing you wrote any of it lol

 

plagiarizing is nearly as bad as pvp, dont do it kiddo

[Gone 869]

 

You could probably tutor people tbh

[Kvasir 1121]

3 minutes ago, GODHawkEye said:

ill admit your info is pretty accurate but i have a very hard time believing you wrote any of it lol

 

I spent like 2 hours writing this pls don't discredit me. But I actually do know how to do this stuff, and yes I wrote it :(

[Dromui 755]

This ****** goes ahead to say he plagiarized someone else's work.

[Starfelt 1437]

I don't like math :/

[1 1393]

the **** it's accurate and you wrote that instead of doing ALGEBRA homework? the ****

 

i like you

[GODHawkEye 106]

33 minutes ago, Skale said:

This ****** goes ahead to say he plagiarized someone else's work.

0

Upvote

@ me noob

[mmat 7460]

[Salvo 1852]

minus one that's three quick maths

[OzYmandi 647]

the ****

the ****

the **** is in the air

 

the **** 

the ****

there's math **** everywhere

[Archipelego 2188]

he most likely did it himself, which is ******* sadder than plagiarism

[TheIchorDruid 1986]

Pretty impressed, for someone who struggles with the advance mathematics of calculus. This helped to remind me of the basics, good work!

[Endovelicus 613]

no.