You must have landed on this page as somehow you are interested to know what calculus is and how to learn it. Calculus is a very complicated concept to understand. Merely attending the lectures on calculus is not enough if you really want to learn calculus. There is no direct answer to the question of what the best way to learn Calculus is. It varies from individual to individual to individual. But there are specific methods which are very helpful and time-tested for learning calculus. We will discuss them in the article.
What is Calculus?
But before starting the process of learning calculus, we must know properly what calculus is used for and what exactly the calculus is. Calculus is a branch of mathematics which focuses specifically on the derivatives, integrals, limits, functions and infinite series. Actually, it is an extensive study of how things change. The calculus, in a way, provides the basic framework with the help of which you can model a system that is changing. You can construct a quantitative model of change which is relatively simple to understand, and you can even make or deduce predictions of the models that are continually changing.
What is the use of Calculus in real-life situations?
Most of us at school level we do not understand the use of calculus in real-life situations, and as a result, we get demotivated. So it is essential to understand that there is extensive use of calculus in real-life situations, else why would we study such a difficult to understand the concept. Actually, the study of Calculus endows the learner with the ability to understand and establish the effects of the changing condition on the subject/system of study. The extensive research and learning of the calculus can enable you even to control the system to work in accordance with you. You can make the system to do what you want it to do. Isn’t it exciting? Do you know Calculus has an excellent role to play in the industrial revolution? It was because its application to engineering/physics found to play a significant role in the industrial development and the development of the modern science.
A Gentle Introduction to learning calculus:
So when we say that we are learning Calculus, what exactly are we learning? I mean, do you know what does the study of Calculus incorporates? Actually, the study of Calculus starts with a very basic framework. It starts by describing the old notions of speed, position, and acceleration. After that, comes the study of the single variable calculus. It usually makes a study of an object moving along a fixed path. Then comes the multivariable calculus, which deals with the movement of an object on a surface (non-linear) or space.
As you know, that if there is some motion, there will be an origin of that motion. The movement can then be marked at relevant points at some definite time or some definite distance. When we know the distance and the time of movement, there comes the ‘Function.’ A Function is a set of positions and time that describes any motion.
This Function has a vital part to play in studying a system. It is used to define the quantities of interest in any system. Applying calculus to the system using the functions produce the results. The major course starts with numbers and functions. After that the study of Calculus encompasses the following topics:
Finding instantaneous changes of the varied functions. (Differentiation).
Using derivatives for finding the solution to some kinds of problems
Going back from the derivative of the function (Integration)
How to integrate different types of functions?
The application of the integration to the varied geometric problems
Finite and infinite series coverage
What is Calculus used for?
Now we know in general what calculus is and what it has in its stride. But it is not all. The problem comes when we start to understand calculus. Talking about or discussing the Calculus is not of much help. We must actually have a firsthand experience of solving the Calculus related problems to understand its application. The college level class of the Calculus can only teach you all about Calculus. But here, in this article, I am trying to give you a push to start learning Calculus. Here, I am discussing some important concepts related to the Calculus, to provide you with an idea about the Calculus learning. Let us divide the learning process into three parts:
Basics of Calculus
(A) Basics of Calculus:
Before starting the learning of Calculus, you must become familiar with specific concepts and topics in detail. Some of the concepts that you must clarify in your head are:
Understand how Calculus is a study of how things are changing.
Understand what actually a Function is: In short, a Function is a rule that governs how the numbers relate to each other. Functions are extensively being used to make graphs. There is a detailed study of the function, which you must learn in order to understand the Calculus well.
Understand the concept of infinity: Infinity is when a process never stops, and you repeat it again and again. The concept of infinity is of great value in Calculus.
After infinity, make a good study of the concept of limits: When any process/value is not infinite, it means that it is something near the infinity, and this is the limit of it. In calculus, the limit is generally set to 0. With the help of it, you can solve various mathematical problems like how many times one has to divide 2 by 4 to get zero?
Revise and clear any doubts related to mathematical concepts like trigonometry, algebra, etc.
If possible, purchase a graphing calculator
Finding how something is changing at some exact moment or instantaneously is the heart of Calculus. The derivative is just ‘how fast something is changing.’ If you have studied Calculus before, you must have read the term ‘derivative of speed.’ Do you know what the derivative of the speed is, it is ‘acceleration.’ Acceleration tells that how fast or slow something is moving. It means acceleration tells about the change of speed taking place. Position and time here play a major part. While learning the derivatives, you will encounter and learn the following:
The rate of change is the slope of the line:
This is one of the major deductions of the calculus. If there are two points, a and b, the rate of change between these two points is equal to the slope of the line connecting them. If there is a line equation, b = 2a, here 2 is the slope of the line. This is to say that whenever the value of ‘a’ changes, the value of ‘b’ will also change by 2. We can also say it in other words that the value of the line is changing by 2. For e.g. when a = 3, b = 6; when a = 6, b = 12. So the deduction from the equation is that the slope of any line is a change in the value of ‘b’ divided by any change in ‘a.’ Moreover, it also establishes the fact that bigger is the slope, the steeper is the line.
Finding the slope of a curved line:
This is a bit complicated. For example, take the complex equation with curves b=3a2, the slope of this is harder to find. However, you can find the rate of change, and it is not impossible. Draw a line and find the slope. In this case, if (a,b) = (1,1) and (a1,b1) = (3,4) . Then the slope would be (b2-b1)/(a2-a1). In this case, it is (4-1)/(3-1) = 3/2= 1.5. This means that the rate of change between a =1 and a=3 is 1.5.
Finding the accurate rate of change by marking your points closer.
Finding the instantaneous rate of change use infinitely small lines.
Finding derivatives depending on the equation
Finding the derivative equation to get an idea of the rate of change
(C ) Integrals:
Is it possible for you to measure the amount of water is preserved in some deformed or not properly shaped lake by using simple calculations? No, it is not possible for sure. Here you need to make use of the Calculus. Calculus enables us to measure even the most complex shapes. In the case of the lake, the best way to find the amount of water present in the water by measuring the edges and the associated changes at the edges.
Integral here plays a major role. Integration involves making of a geographic model and then studying the volume. In short, integration helps to find out complex area and volumes. While learning to use the integration technique, you will come to know the following things:
Integration finds out the area of complex figures and finds out the volume as well.
Integration helps to find the underneath area of a graph.
You will learn that you cannot integrate the entire function if it goes on forever. Instead, you need to choose an area.
For effectively finding out the area with the help of integration, you need to know how to find the area of a rectangle. As most of the times, to find out an area, integration adds on many small rectangles.
There is a specific way of reading and writing the integrals for mathematical use.
To integrate different functions, different formulas are used. You will have to learn various different formulas to deal with all kinds of functions.
Integration reverses differentiation. For e.g., if you know the acceleration, you can integrate it to know the speed.
Integration can also find the volume of the 3D objects.
Is all of this discussion sounding scary? It is not so. The learning of the Calculus demands some time and effort, but once mastered, you will be able to tackle a lot of things alone. There are many ways to learn calculus like taking a tuition, learning calculus online or on your own.
How to learn Calculus on your own or online?
The best way to learn Calculus on your own is to divide the entire calculus into parts and then practice each and every part continuously and with dedication. We have already discussed the major areas of calculus. Apart from all the things discussed above, you must concentrate on three more specifically:
Convergence Test for series
Vector products (cross and dot)
Trigonometric Substitution in Integration
If you get into some confusion, you can take the help of online resources. You can even learn calculus online on your own. There are a number of videos, youtube channels and lessons of calculus available online. Some of them are even free. Some websites also give you the facility to personalize your teaching experience. You can mold the course according to your wish. If you are clear with your basics, you can directly start with the higher level. Queryfloor, etc. are some good sites where you can get good help. Such sites offer one-to-one lessons as are given in face to face tutoring, what more can you expect?
Though traditional calculus courses are helpful at large, to learn quickly, practicing with computer spreadsheets is a better idea. Now-a-day some sites provide its user with the applets which further lessens the effort. Have you heard of the programs/software Mathematica or MAPLE? These are excellent programs to help you with the practice of the Calculus. Try the wonder of such software to ease the process of Calculus learning for you.
In conclusion: So this is it. We have provided the basic guide. The easiest way to learn calculus is to develop a positive attitude towards it, start learning it step by step and then practice more and more. Solve more and more calculus questions and answers. There is no shortcut to it. Remember, Calculus is a very important tool which is employed in various areas of knowledge ranging from economics, statistics, to engineering, chemistry, physics and ultimately mathematics. It has helped to make many real-world inventions.