The Case with Differential Calculus :
Do you like to watch the skateboarding competitions?Where do you find the thrill, in the skating on the flat ground or on the curved part?
Just for a moment…. rewind the skateboarding video that you are watching alongside as you read this article…. and observe the motion of the head of the skateboarder? Do you see that for very short time intervals, the head moves approximately as a straight line?
Did you know that simple mathematics can help you calculate the tilt of this ‘line’ (something we call the slope)? If not, then differential calculus is the way to go. Finding the slope is one of the basic tasks that differential calculus deals with. That is not all, however. Differential Calculus is a much more diverse and beautiful science which explains a lot about the world. In fact, all the major researchers in the world use calculus extensively. Simply speaking, calculus is the most logical and useful part of mathematics, specially in JEE syllabus.
Come to think of it, we as IITians are taught about it extensively. Much of engineering requires a strong command over Differential Calculus.
Let Us Explore the ‘Thriller’ that Calculus is…. (I am sure that MJ fans would understand) 🙂
Chapter – Wise Analysis :
- Functions : Fundamentally the most important chapter of calculus. While having a strong command over tis chapter does not ensure that you would have a strong command over calculus. However, if you do not have a good hold of this chapter, then for sure you cannot excel in calculus. Practice graph making a lot and try reducing most questions to graphs, even in the later chapters. This is one of the most time saving tricks for solving maths problems. Practice functional equations a lot because they are tougher, and can only be attempted successfully if you have practiced them a lot earlier. You should also be knowing about the different types of functions that are there and certain properties, especially those of inverse functions.
- Limits : The chapter that makes most sweat initially. However, this chapter has to be approached from a certain mind perspective. Either strengthen your grasp over L’Hopital Rule or over Taylor Expansions. Both these methods are useful to solve most the questions. Also certain forms should be practiced a lot, like 1 to the power of infinity, etc. Graphs are mandatory while solving questions involving points of discontinuity/non differentiability. Also cover critical points & inflection points well. They are in the syllabus and it is better you know of them. One extremely important theorem is the Sandwich Theorem. Make sure you practice it well.
- Differentiation : Quite an easy chapter. Just a knowledge of First Principles Method and of certain Standard Derivatives is enough to help you solve most questions. Also, use the different rules like chain rule, etc pretty well. Certain subtle things like differentiation of inverse functions are important and should be focussed upon.
- Application of Derivatives : The most important chapter here, having an immense weightage in every exam. The Mean Value Theorems are of paramount importance. Practice them well. Also, make sure tha the concepts of Critical Points/Inflection Points/Maxima&Minima are clear in your head. The questions of this chapter as usually framed indirectly so be sure to practice a lot to develop your thinking skills.
Books (Buying links included for your comfort) :
- McGraw Hill : Comprehensive Mathematics for JEE Advanced should be sufficient. If you want further practice, pick up questions from papers/coaching revision packages.
- Amit Agarwal : Differential Calculus is praised a lot by many of my friends here at IITB. Hence, added it to the list.
Tips and Tricks : Infinite Practice is needed. Additionally, practice questions to develop your thinking skills and not for the sake of problem solving or homework completion. Feel the chapters and then proceed.
All the best,