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Can you give me some good tips on how to approach this section of differentiation and integration? I have a hard time figuring what standard results/equations should I use for a particular question?

Can you give me some good tips on how to approach this section of differentiation and integration? I have a hard time figuring what standard results/equations should I use for a particular question?
 

Grade:12th Pass

2 Answers

Aditya Gupta
2081 Points
4 years ago
Understand the Definition
So what does differentiation of a function implies? d/dx ? Thats correct but how did we arrive to that expression? How do we define a derivative of something and what does that mean? This is one point which generally most of the students tend to ignore but in exams like JEE there will be many questions that will require the use of definition and your standard formulae won’t work. So make sure you are clear with the concept of every topic (integration,differentiation, limits,etc). e.g. differentiation in simple words involves calculating slope of two infinitesimally close points.
Remember standard Formulae
Lot of questions (not all) especially the ones having trigonometric terms in them, require manipulation and substitution in the function onto which an operation has to be performed so that the function gets simplified or the steps required to arrive at the solution becomes easy. For this you should know that particular formulae to be used, the question can be solved without any formula too but it will be time-consuming and lots of steps will be involved.  You definitely don’t want to regret losing marks in a simple question just because you were too lazy to learn the formulae. So make sure you have a list prepared.
Knowing the nature of the functions
Functions is one of the most important topic under calculus. In almost all the questions you will be given a function and the set of operations to be performed on it. For this it is important to know the nature of the function as in whether the function is increasing or decreasing, odd or even, its point of maxima and minima,etc. This will also help in plotting the graph of the function. It is only after you understand the function that you will get an insight of the problem and will help you solve the problem efficiently.
Use graphs whenever possible
This might sound unconventional but use of graphs in the problem will be something that you should start practising from now on. The graph of the function need not be completely accurate but a rough idea of what the function will behave like. Once you have plotted the graph you can get many information from it without doing any calculations. Graphs will help a lot in problems of application of derivative and area under the curve.
Integration
Integration according to many is the toughest part of calculus as there are large number of algorithms for obtaining solution to the problem depending on the type of function. So one needs to practise as many problems in integration as possible because only then will you get the knowledge of the method to use for a particular problem. Also one should know the methods like use of partial fractions, integration by parts and integration by reduction methods.
Application of derivatives/integrals
This includes problems involving finding maxima-minima, area under the curve and slope of complex curves. These topics together hold a significant weightage in JEE both mains and advance every student planning to give these exams must be thorough with the types of problems and the methods to solve the same.
Keep Practising
This is the most important trick that you need to remember in order to ace any topic especially calculus. Once you have gained an insight for every concept, the next step should be to practice as many problems as possible. Take a good reference book, mark some questions (make sure that you practise questions of each type and difficulty) and start solving them. This will not only boost your confidence but also it will help you in remembering the formulae and methods of problem solving.
Understand the Definition
So what does differentiation of a function implies? d/dx ? Thats correct but how did we arrive to that expression? How do we define a derivative of something and what does that mean? This is one point which generally most of the students tend to ignore but in exams like JEE there will be many questions that will require the use of definition and your standard formulae won’t work. So make sure you are clear with the concept of every topic (integration,differentiation, limits,etc). e.g. differentiation in simple words involves calculating slope of two infinitesimally close points.
Remember standard Formulae
Lot of questions (not all) especially the ones having trigonometric terms in them, require manipulation and substitution in the function onto which an operation has to be performed so that the function gets simplified or the steps required to arrive at the solution becomes easy. For this you should know that particular formulae to be used, the question can be solved without any formula too but it will be time-consuming and lots of steps will be involved.  You definitely don’t want to regret losing marks in a simple question just because you were too lazy to learn the formulae. So make sure you have a list prepared.
Knowing the nature of the functions
Functions is one of the most important topic under calculus. In almost all the questions you will be given a function and the set of operations to be performed on it. For this it is important to know the nature of the function as in whether the function is increasing or decreasing, odd or even, its point of maxima and minima,etc. This will also help in plotting the graph of the function. It is only after you understand the function that you will get an insight of the problem and will help you solve the problem efficiently.
Use graphs whenever possible
This might sound unconventional but use of graphs in the problem will be something that you should start practising from now on. The graph of the function need not be completely accurate but a rough idea of what the function will behave like. Once you have plotted the graph you can get many information from it without doing any calculations. Graphs will help a lot in problems of application of derivative and area under the curve.
Integration
Integration according to many is the toughest part of calculus as there are large number of algorithms for obtaining solution to the problem depending on the type of function. So one needs to practise as many problems in integration as possible because only then will you get the knowledge of the method to use for a particular problem. Also one should know the methods like use of partial fractions, integration by parts and integration by reduction methods.
Application of derivatives/integrals
This includes problems involving finding maxima-minima, area under the curve and slope of complex curves. These topics together hold a significant weightage in JEE both mains and advance every student planning to give these exams must be thorough with the types of problems and the methods to solve the same.
Keep Practising
This is the most important trick that you need to remember in order to ace any topic especially calculus. Once you have gained an insight for every concept, the next step should be to practice as many problems as possible. Take a good reference book, mark some questions (make sure that you practise questions of each type and difficulty) and start solving them. This will not only boost your confidence but also it will help you in remembering the formulae and methods of problem solving.
kindly approve
Rajdeep
231 Points
4 years ago

DIFFERENTIATION:

Differentiation is the essence of Calculus. A derivative is defined as the instantaneous rate of change in function based on one of its variables. It is similar to finding the slope of tangent to the function at a point.

Suppose you need to find the slope of the tangent line to a graph at point P. The slope can be approximated by drawing a line through the point P and finding the slope by a line that is known as secant line.

 

A function f in x is said to be differentiable at the point x = a if the derivative f'(a) exists at every point in its domain. 

For a function to be differentiable at any point x=a in its domain, it must be continuous at that particular point but vice-versa is necessarily not always true. The domain of f’(x) is defined by the existence of its limits.

If y = f(x) is a function in x, then the derivative of f(x) is given as dy/dx . This is known as the derivative of y with respect to x.

Derivative of a function f(x) signifies the rate of change of the function f(x) with respect to x at a point a lying in its domain.

If the derivative of the function, f’, is known which is differentiable in its domain then we can find the function f. In integral calculus, we call f as the anti-derivative or primitive of the function f’. The method of calculating the anti-derivative is known as anti-differentiation or integration.

 

INTEGRATION:

The integration of a function f(x) is given by F(x) and it is represented by:

\int f(x)dx = F(x) + c

where

R.H.S. of the equation indicates integral of f(x) with respect to x

F(x) is called anti-derivative or primitive.

f(x) is called the integrand.

dx is called the integrating agent.

C is the constant of integration or arbitrary constant.

x is the variable of integration.

 

Hope this helps.

Thanks!

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