What is Calculus
Calculus is a division of mathematics that deals with variables and how they get altered by looking at them through the means of infinitely small pieces called infinitesimals. Calculus, as it is known today, was invented in the 17th century by British scientist Isaac Newton (1642 to 1726) and German scientist Gottfried Leibnitz (1646 to 1716). They had independently come up with the principles of calculus in the fields of geometry and symbolic mathematics, respectively. Cracking calculus can be only done by practice.
While these two separate discoveries are now crucial to understanding calculus as it is practised today, they were initially not isolated incidents. It’s interesting to know that even Archimedes (287 to 212 B.C.) in Ancient Greece and Bhāskara II (A.D. 1114 to 1185) in medieval India had developed and toyed with the ideas of calculus long before the 17th century. Unfortunately, the ground-breaking nature of these discoveries didn’t come to the forefront then or got lost in other new and difficult-to-understand ideas.
The word “calculus” has a humble origin, and is derived from related words such as “calculation” and “calculate”. However, note that these words are all derived from a Latin (or perhaps even older) word meaning “pebble.” In the primeval world, calculi were actually stone beads that used to keep a track of livestock and grain reserves. Today, calculi are small stones that get formed in the gallbladder, kidneys or other parts of the body.
Two halves of calculus
The process of calculus has two halves. The first half is known as differential calculus, which is about examining individual infinitesimals and what happens within that infinitely small piece. The second half is called integral calculus that deals with adding an infinite number of infinitesimals together. Integrals and derivatives are the opposites of each other, and that is basically what is referred to as the Fundamental Theorem of Calculus.
Differential Calculus: It is the study of the rates at which quantities change
Integral Calculus: It is about the accumulation of quantities and areas within the curves.
To make it simpler, it can be said that differential calculus cuts something into very tiny pieces to find the rate of change. And Integral calculus collects tiny pieces to find the quantity. Just the way division and multiplication are opposite to each other, Integral and Differential are inverse to each other.
Integration is a crucial concept and is mostly used for these two purposes:
1. To calculate f from f′. If a function f is differentiable in the interval of consideration, then f′ is defined in that interval. Differential calculus can be used to calculate derivatives of a function. But we can “undo” that with the help of integral calculus.
2. To calculate the area under a curve.
Aryabhatta had made use of differential calculus to find the motion of the moon. Likewise, modern mathematics utilises differential equations and functions to find out maxima and minima of curves that are commonly used.
Here are some other applications:
- Automatic air conditioners- temperature control.
- Cruise control in cars
- Water mixers
- Industrial control systems rockets, ships etc.
In math, a limit can be defined as a value that a function approaches as the input attains some value. Limits are a very crucial term in calculus and mathematical analysis and are quite often used to define integrals, derivatives, and continuity.
The limit of a sequence can be also generalized in the concept of the limit of a topological net and related to the limit and direct limit in theory category.
The basic parts of a differential equation can be classified as functions and their derivatives. The functions stand for physical quantities and derivative are about the rate of change and their relationship is represented by the differential equation.
The myriad applications of differential equations in engineering include:
- Heat conduction analysis
- Understanding the motion of waves in physics
- Modeling the chemical reactions in chemistry
- Monitoring the cancer growth in medical science
Isaac Newton had introduced three kinds of differential equations-
x1 *∂y/∂x1+x2 * ∂y/∂x2=y
Here are the two primary ways of solving a differential equation:
- Separation of variables
- integrating factor
Separation of the variable is achieved when the differential equation is written in the form of dy/dx= f(y)g(x) where f is the function of y only and g is the function of x only. Considering the initial condition, we can rewrite this problem as 1/f(y)dy= g(x)dx and then put them together from both the sides.
The integrating factor method is applied when the differential equation is in the form of dy/dx+p(x)y=q(x) where p and q are both the functions of x only.
First order differential equation is of the form y’+ P(x)y = Q(x) where p and q are both functions of x. Therefore, they are called the first order differential equation as they consist of functions and the first derivative of y. A higher order differential equation is an equation that contains derivatives of an unidentified function that can be either a partial or ordinary derivative. It can be represented in any order.
Stochastic differential equation – that contains one or more terms that are stochastic and the solution it provides is also stochastic.
The secret behind scoring well in Mathematics in JEE is mastering the skills and cracking Calculus. Calculus forms the core of Mathematics and has a wide range of applications. The syllabus covered in JEE is quite vast which covers every aspect of calculus. The aspirants must have an in-depth knowledge of the concerned topics as most of the questions are blended from various sections. Cracking calculus could be a big boost in your preparation for IIT JEE.
“Nothing takes place in the world whose meaning is not that of some maximum or minimum.”
― Leonhard Euler
So there are some things that need to be kept in mind while tackling problems in calculus. We have listed them below taking care every spectrum of possibility. Here are some tips to help you in cracking calculus in JEE:
Focus on fundamentals of Calculus
It is quintessential to have strong fundamentals in order to tackle big problems. Straight away jumping to an advanced level problem is not a good idea.
Channelize your efforts
Devote ample amount of time to each section of Calculus. Consider an example, differentiation is relatively easier and hence can be devoted less time. On the other hand, integration requires a considerably longer period of time.
The most challenging part
Integration is the most challenging part of the calculus, given the fact that it requires the use of specific algorithms. Remembering each algorithm is quite necessary to solve an integration problem. Revision of these algorithms along with practice can make things less complicated. Also to increase confidence, you can solve past year JEE Advanced questions.
Clarity in concepts
Develop a clear thinking process so that the moment you see a question you know the correct theory or formula to be used.
Have proper revision plans
Keep revising the things that have been done. Calculus is a topic which must be refreshed at frequent intervals else the concepts might get wiped off. Maintaining a timetable of the topics for revision will definitely help in a long run.
Well, Calculus needs a lot of cramming. There are a reasonable number of standard methods and techniques that you are expected to remember. You won’t be able to derive the formulas in the exam. Just understand them while practising and then learn them by heart.
An important verdict
Expect your exams to be challenging. If they are challenging, you will be prepared. If they are not challenging, you can expect to have an easy time getting a very high score!
The key to cracking calculus is to start with understanding the roots, i.e., limits and continuity.
Hope this might have helped you in the smallest way possible. If you keep check-listing the above points, you will surely gain an edge over others in any competitive examination and cracking calculus will be a natural thing for you. All the best for your examination! 😀