Calculus - Awesome Mathematics

This is going to be a long one. Better grab something to eat.

Calculus has remained my favourite topic since JEE. It's the best, the most logical and the most beautiful topic for me. It has it's own beauty. It's a mixture of everything - co-ordinate geometry, algebra, vectors and what not. It's because of my interest in calculus that I performed good in our calculus course. It's extremely important from the point of engineering.
Coming back to JEE, 
Here is a chapter wise discussion of JEE calculus.

Functions : forms the framework of calculus. Understand this concept well otherwise studying calculus will be a tough time for you. Common questions are :

• finding domain/range of a given function : students commit silly mistake here
• given a domain and range, finding the number of one-one, onto, etc functions
• plotting graph and finding number of intersection points of 2 or more functions

study this chapter well, especially graph plotting. It's going to help you a lot.
Practice a number of problems of finding domain/range. There are tricks like :

• no negative number inside a square root.
• no division by zero
• no negative number zero inside log
• base of log is not equal to 1 etc etc.

They expect you to know these basics. A good practice is to cross check your answer from the options available.

Limits : Most important part. Entire calculus is based on finding limit of a function. L'Hopital's rule is extremely important. Apply it as much as you can. You might also give a try to series expansion but I avoided it because most questions of series expansion can be solved if you are good at L'Hopital's rule. So, I never needed it. However, it is advisable to remember the exponential and logarithmic series.
Questions from this chapter are usually straight forward and simple. You just have to figure out the zero by zero form or the infinity by infinity form and apply L'Hopital's rule.
The variety of questions is large.

Continuity : another important topic. This chapter can be solved if you are accurate and quick at drawing graphs of functions. Continuous functions have many interesting and important properties like "intermediate value property", invertibility etc. Understand them and apply them.

Differentiation : One of the easiest chapter. Doesn't need much effort. Learn the formulas of differentiation - quotient rule, product rule, chain rule, implicit differentiation, differentiation of exponential and logarithmic functions, higher order derivatives. Practice enough questions.

Application of derivatives : Most-most-most important chapter because maximum number of questions are asked from this topic. Rolle's theorem, Lagrange's mean value theorem, concept of maxima-minima, increasing and decreasing functions, inflexion points, etc are really really important. Practice as many questions as you can from these topics. Be prepared to see maximum questions in JEE from this topic. If you are weak at it, then better do something otherwise you will face trouble. This chapter basically is the application of all that you studied in the previous chapters and JEE is all about applying concepts.

Integration : Relatively easy chapter. You won't find many questions in JEE from indefinite integrals. There are a number of formulas to be remembered from this chapter. The formulas are extremely important because you will sometimes need them in definite integration.

Definite integrals : most important topic of integral calculus. There will be a good number of questions from this chapter. Most questions will be based on the properties of definite integrals rather than just applying fundamental theorem of calculus to a definite integral.
Practice enough questions from this topic.

Differential equations : A relatively simple and interesting chapter. Need a good amount of practice and some memorizing skills. Learn the known methods of solving the standard differential equations and you will enjoy this topic.

Books : Arihant differential calculus, Arihant integral calculus (Both by Amit M. Agarwal) are the best books for this topic. They will cover almost everything (except one topic - inflexion points which you may cover from somewhere else). Solve all the subjective and objective problems of the book. You may choose to ignore the extremely long ones because they are meant for olympiads etc.

Revision : Revise from the standard revision packages in the market. FIITJEE GMP or Resonance rank booster or the one provided by your coaching will be sufficient.

Final word : Calculus needs a lot of cramming. There are a number of standard methods and techniques that you are expected to remember. You won't derive the formulas in exam. Just understand them at home and then learn them by heart.
Practice a number of problems especially from application of derivatives and definite integrals. These are the favourite topics and definitely there will be questions from them. You can't afford to lose marks in calculus because it is one of the easiest topic.
Enjoy calculus.