Calculus: A Primer

Introduction

People hear the word calculus and get intimidated. They found math hard in high school and do not pursue it any higher. I feel this is unfortunate and hope to make those intimidated by the prospect of calculus to feel a little less intimidated. Here I will give an overview of the basic idea of calculus so you will have an idea of what calculus is.

First off understand math is like a computer program or a machine. Its easy to use once you understand it. Developing this understanding requires a little work but it is much easier than inventing the machine yourself. This means if you are having problems you do not understand the procedure properly. Its like anything else; difficult to learn, easy to do once you know how.

Next realize that math is continually advancing like any other technology. Using more advanced math can make problems easier. High school math usually covers finding the minimum and maximum of quadratic equations (i.e. ax2 +bx+ c). Using the methods of calculus makes these problems much easier and can be easily applied to higher degree functions unlike the methods taught earlier. If we can understand how to use calculus math becomes easier. Don’t get frustrated cuz its “hard”, grind through and it will become easy.

What is Calculus

Calculus is the branch of mathematics that deals with rates of change. Since the physical world is in a state of constant change calculus proves to be the most used branch of mathematics. It is a core part of any science curriculum and much of engineering is applied calculus. The principles of calculus are derived from the relationship between distance, speed and acceleration. A large part of introductory physics is simply applied calculus, since it is concerned with the relationship between these quantities.

The First Principle

Let us begin with the relationship between speed and distance, concepts we have solid intuitive grasp on. If we want to know how far we will travel in a given time we multiply the time we are travelling by how fast we are going. If we drive 60km/h for 1 hour we travel 60 km, for half an hour 30km. Simple, right?

Now consider the distance covered when we start at a stop. Our speed is no longer constant so we can not simple multiply our time travelling with our speed. Assuming a constant rate acceleration we take our average speed and multiply that by the time traveled, but if our rate of acceleration changes we will want to use calculus.

For steady acceleration we can calculate the area of the triangle to determine or distance [(20+0/2)m/sec*20sec)] for that segment, for steady speed time we will cover [20m/sec*20sec], for the deceleration segment we cover [(20+10)/2m/sec*10sec]

Doing the math we determine we covered 750 m before we begin to accelerate at an increasing rate. We can approximate the curve by splitting it up into small segments. If these segments are small enough we can approximate the curve with a straight line and solve as if it were a steady acceleration. This will only give an approximate answer though. The more triangles we use the more accurate our result. A perfect answer would treat the curve an an infinite number of infinitely small triangles. This is the root of calculus.

The fancy d or delta can be read as “the change of…” as in Q is the point with the coordinates (x+the change of x,y+the change of y). As we bring P and Q closer the triangle becomes smaller. Ideally this means P=Q and they are the same point. However we can’t actually compute the area of a point. Our algebra is insufficient to solve this problem. It can only come up with an approximate answer that will take a lot of work to figure out. Calculus takes the formula and applies the operations to the equations themselves to get around this problem. It does divide by zero, but we won’t write it in and let a unknown represent zero and say its as close to zero as you can possibly imagine without actually being zero. This is written as lim->o. It lets us divide by zero here, but is also used for other things. Don’t worry about it and just know you are expected to write it if you want to break math (i.e divide by zero).

the triangle is little delta, same thing as the fancy d. This is saying we are taking our new point, subtracting away our starting point to determine the change and dividing by the amount we traveled on the x axis. the ‘ in f'(x) means the derivative and represents the rate of change in f. If f is distance f’ is speed. This formula is the first principle of calculus and is the most important to understand although it will never be used in practice. Understanding this is hard, but if you do than your will have an easy time in freshmen math and physics. Its worth the effort.

Here is an example

First they we substitute in for f(x) for what is given. We then expand the equation and sum up like terms. We factor out an h, removing it from the denominator. Once this is done we are allowed to set the remaining h’s to zero and our answer is obtained.

This is to say that if a position is modeled by x3-6x than its speed at any given time x will be given by 3x2-6. We do not need to calculate anything other than simple arithmetic. We plug in the value for x and can determine its position  and speed easily. No need for approximations and lengthy calculations, we just plug in the numbers.

This process is known as differentiation and is easier than its reverse which is known as integration. Learn the first principles of differentiation before looking at integrals.

Calculus can be challenging for many people because it deals with abstract concepts. We perform operations on an equation that we can not punch numbers into. Differentiation should be viewed as a machine that takes distance formula and spits out speed formula. If we feed it a speed formula we will get an acceleration formula. To apply the process we insert our function into the formula and methodically solve for it to obtain our answer.

Once we understand the first principles than we can learn about doing it faster and more efficiently. For instance a simple exponential term will have its current exponent become a coefficient and be subtracted by 1 when we differentiate. This rule can be proven with first principles which we should be able to do, but knowing the rule is enough in practice.

Calculus is the first time abstract thinking is absolutely necessary for many people. It is best viewed as a set of rules applied to symbols. A systematic approach is required. It shows how well a person can learn and follow a procedure and develops their capacity to do so. Empirical sciences use calculus so often it is essential to know if you are pursuing a career in science. If you try to blindly memorize the rules of calculus you will struggle with the class and never truly learn it. If you understand it, it becomes a routine procedure.

It is interesting to note that integration, the opposite process of differentiation is much more difficult and can not be solved exactly for every equation. Some integrals that are needed for science take super computers days to calculate for accurate results. Since they can not be solved exactly approximate methods that need lots of calculation are used instead. Its a mind fuck when you ask a computer a math question and it takes a long time to spit out an answer when one is used to calculators spitting out answers instantly. Integrals are an important part of calculus but are introduced much later, usually in the second semester of a two semester calculus course.

I hope this give you some insight into what to expect from calculus but you will probably need more information to actually learn it. I would recommend checking out MIT’s open course ware if you are really interested in learning calculus on your own. It is the most important math to learn if you are to proceed in math or science based pursuits.