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English Guide: Practical Tips for Better English Communication

article thumbnailIf you are like most people, you would like to achieve professional growth and success in your career. Typically, career success would mean promotions, greater responsibilities and accompanying financial rewards. If you look around you, you will often find that the people who progress and move up the ladder are those who demonstrate passion at work, are proactive, and of course, perform on the job at hand with the desired or...
article thumbnailIt is never a pleasant task to call up the customer service of one of your service providers and complain about any issue you may have encountered, whether it is a faulty bill or inadequate service you have received. Nevertheless, making unpleasant telephone conversations to lodge a complaint is a part of life that one must be prepared for. How you go about making this conversation can go a long way in determining whether you get the desired...
article thumbnail  Let us say that your current English speaking skills aren’t great. You aren’t very confident of being able to communicate confidently and effectively in English with anyone, be it a small group or a large one. However, you are very determined to improve your spoken English skills rapidly and prove to yourself that you can do it, while making a good impression on those around you. Taking up an English speaking course is obviously one way to...
article thumbnailIf you are planning to take the IELTS test, you would be aware that the test comprises of four different modules: Listening, Reading, Writing and Speaking. For many IELTS test takers, the speaking test is often the most difficult. This is not very surprising because it is akin to speaking in public. It is a known fact that ‘public speaking’ is one of the scariest tasks for people.      There are three sections in the  speaking test in IELTS. The...
article thumbnailThere is a quote attributed to the writer Stephen King that says that “if you want to be a writer, there are two things you must do: Read a lot and write a lot.” You can extend that quote to say the following:  “To be a fluent English speaker, you must listen a lot and speak a lot.” While there is no substitute to this mantra, we will give you some more details of how you could work towards becoming a more fluent speaker.          To start with, let’s...
article thumbnailIt is well known that the quality of communication between two or more individuals, impacts the outcome of business meetings. While the exact content of the meeting and its structure are critical, certain external factors like the energy and enthusiasm of the participants in a meeting, also play a very important role in the success or failure of a meeting. In this article, we will offer some simple tips on how you can improve the chances of a...
article thumbnailA career in sales can be very exciting and lucrative for people who do well in it. Sales brings  in revenues for any organisation, and revenues help companies move forward. As long as one is performing well and continues to bring in sales, one’s job is fairly secure because in times of adversity businesses slash costs, while trying to keep revenues high. Yet, success in a sales career isn’t guaranteed for everyone. It takes certain personality...
article thumbnailWhen it comes to learning spoken English, many people often mistake the purpose of the learning. They tend to think that acquiring English speaking skills is about speaking with a British or American accent. They couldn’t be farther from truth. The true purpose should be to learn to speak English correctly, fluently, confidently and in manner that the listener understands the communication without much difficulty.      Too much emphasis is laid on...
article thumbnail  In a recent article, we wrote about the difference between improve and improvise and how to use these two similar terms appropriately. Continuing this series on using similar-sounding English words correctly, let us consider the use of the words “affect” and “effect”.  Both terms  mean “impact” or “result” of something, but their usage  varies depending on the context in which they are being used.     Affect The word “affect” is generally used in the...
article thumbnailEnglish language has plenty of words that seem very similar but mean very different. It is quite easy to ignore the subtle difference in the words themselves and consider them to mean the same thing. This results in incorrect usage of the words when writing or speaking in English, which may confuse the listener or create a negative impression about your English Communication skills. Let’s focus on two words, improve and improvise; the former is a...
article thumbnailSomeone has rightly said, “In the mundane, nothing is sacred. In sacredness, nothing is mundane.” Mundane is boring. There comes a time when you feel saturated with the routine tasks you are doing and you feel like doing different and challenging things. Your ardent desire to take up additional work and prove your worth overwhelms you. But this needs to be communicated effectively to your boss. Your boss should be aware that you are geared up...
article thumbnailGratitude is of vital importance in human life. We need to thank people who have, directly or indirectly, explicitly or implicitly, made a positive change in our life. ‘Thank You’ is the least that we can say. So never shy away from that. In professional life too, gratefulness is of utmost importance, especially when someone has helped you climb up the career ladder.       It’s your promotion party and you have to give a thank you speech – this...
article thumbnail  We all have encountered situations wherein there were unavoidable or unforeseen circumstances because of which we had to extend our leave. For example, sickness, family issues, marriage etc. Now here comes the uncomfortable question – how to ask the boss for a leave extension. Quite tricky, huh?     When you are already on leave and then you have to request for extension of leave, it should be done in such a way that it is granted without much...
article thumbnailA few months back, a video of a job prospect venting out his frustrations on having to speak in English at job interviews, went viral. The protagonist of the video, a youngster, who could speak English fluently was angry that he was always asked to speak in English, and he let out his frustration with a series of expletives in Hindi. The fact that the video went viral and was shared by a large audience in the country, especially among...
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article thumbnailSomeone has rightly said, “Good conversation starts with good listening”. Our teachers have often told us that we wouldn’t learn anything new from talking but we will definitely learn something new from listening. How true! There’s a lot of buzz about spoken English these days, but people tend to forget that English listening is also an integral and inseparable part of English communication, which cannot be overlooked.No wonder then the...
article thumbnailIndia is undergoing a digital revolution. The country has embarked on the ‘Digital India’ movement that has been officially promoted by the Indian government. It is therefore not surprising to see a jump in career opportunities in the field of digital marketing, especially search engine marketing (SEM). This field attracts a lot of freshers and young job aspirants. Having good English skills will enable a person to take advantage of the growing...
article thumbnailGood manners will open doors that the best education cannot.” Etiquette is the science of living. It embraces everything.People all over the world are judged, liked, disliked, remembered or looked down with contempt based on their manners. Good manners speak a lot about your personality.       It is imperative that you display your best behaviour when you are in a public place especially when eating out.  When you walk into a restaurant you don’t want to...
article thumbnailGood teamwork is essential not only for the growth of any organisation but also for the personal growth of individuals in the organisation. An essential component of achieving good team bonding is when team members express positive thoughts about fellow team members to them directly.  Sincere compliments and praise go a long way in making a person feel good and feel motivated to perform better for the team and the organisation.     There are a...
article thumbnailAt the outset, let us state that we hope you never have to apply for a compassionate leave. However, the reality is that most of us, at some stage of our life, end up having to request for such leaves. This article is written to specifically help you in case of such an eventuality.     Before we get into the mechanics of applying for compassionate leave, let us first understand what is compassionate leave. It is leave granted to an employee upon...
article thumbnailThe use of idioms in regular communication serves to add more ‘colour’ to the communication and makes it more interesting and engaging. Therefore, it is quite common to see the usage of lots of English idioms, especially sports-related idioms, in any office communication. As a young professional, adapting yourself to an office environment, it will help to not only be familiar with some of the common idioms but also learn to use them in your...
article thumbnailThe Indian IT & ITES industry has been one of the major employers of the young, educated workforce in the last two-three decades. The demand for qualified manpower in this sector with diverse technical skills continues to be high. Many IT professionals therefore believe that their technical skills such as programming will be good enough to give them a good and long, successful career. This is far from the truth. Excellent English communication skills...
article thumbnailThis is another article written specifically to help visitors to our blog avoid some common grammar mistakes and improve their spoken English skills. In this piece, we will focus on a common mistake that we keep hearing and reading from so many English speakers these days: the use of “one of the….” in sentences.   To start with, let us take some examples of the type of sentences that you may come across in everyday conversation. Pay specific...
blackboard Two important aspects of any language are the grammar and the vocabulary. While you focus on improving your English vocabulary, it is also important to ensure that you work on grammar as well.The truth is that grammar is the foundation and the building block for becoming proficient in English, while vocabulary lends beauty to the language.It is also true that fixing grammar issues is more challenging than building on your vocabulary. So, the...
english-can-may An important aspect of communication between any two individuals is seeking and obtaining permission to carry out any specific activity or task. Right from our childhood and school days, we have been taught to ask permission from our parents or teachers, whether this is to take or do something. Whether you will be successful in getting the permission, depends on the way you ask. Here are some key considerations to bear in mind. Be polite when you...

INTRODUCTION TO INTEGRATION

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Integration

introductiontointegration1Calculus is centered on the concept of the derivative.  If a function ‘f’ is differentiable in an interval I, then its derivative f’ exists at each point of I.  We can think about the question, if f’ exists at each point of I then what will be the function? Such functions are called anti-derivatives.  The formula that gives all these anti-derivatives is called the indefinite integral of the function and the process of finding anti-derivatives is called integration.  The development of integral calculus arises out of the efforts of solving the problems of the following types:
i)    The problem of finding a function whenever its derivative is given.
ii)    The problem of finding the area bounded by the graph of a function under certain conditions.
These two types of problems lead to indefinite and definite integrals, which together constitute the Integral Calculus.
    

Integration as an Inverse Process of Differentiation

Integration is the inverse process of differentiation.  If we are given the derivative of a function and asked to find its primitive (original function), the process is called integration or anti-differentiation.
Let us consider few examples,
i)    d (tanx)  = sec2x                  …………….(1)
     ___    
     dx

The function sec2x is the derived function of tanx, so we say that tanx is the anti-derivative of sec2x
ii)    d (logx) =  1
      __           ___
      dx            x                        ……………..(2)  

The function 1/x is the derived function of logx, so we that logx is the           anti-derivative of 1/x.
We know that the derivative of any real number C, treated as a Constant Function is zero. And hence we can write (1) and (2) as
         d (tanx+ C)= Sec2x        d(logx +C) = 1
        __                               ___             __
        dx                                dx               x

Hence the anti-derivatives (integrals) given above are not unique.  There exists infinitely many integrals of each these functions which can be obtained by choosing C arbitrarily from the set of real numbers.  Hence C is referred to as arbitrary constant.
In general, if there is a function F such that d F(x) = f(x) for all xЄI then for any arbitrary real number C,                                __           
                                                               dx

                               d [F(x)+C] = f(x), x Є I
                               __                             
                               dx

We introduce the new symbol, ∫f(x) dx which will represent the entire class of anti derivatives read as the indefinite integral of ‘f’ w. r. t x.
Symbolically, we write ∫f(x) dx = F(x) + C
f(x) in ∫f(x)dx is the integrand, ‘x’ in f(x) is the variable of integration, ‘C’ is the constant of integration.

Geometrical interpretation of indefinite integral

The statement ∫f(x) dx=F(x) + C=y, represents a family of curves.  The different values of C will correspond to different members of this family and these members can be obtained by shifting any one of the curves parallel to itself.  This is the geometrical interpretation of indefinite integral.

Some properties of indefinite integral

1)    The process of differentiation and integration are inverses of each other which results in
                 ∫f’(x)dx=f(x)+C, where C is any arbitrary constant.
2)    Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent.
Let ‘f’ and ‘g’ be two functions such that
                    d ∫f(x) dx= d ∫g(x) dx
                   __             __                  
                   dx             dx

3)    ∫[f(x)+g(x)]dx = ∫f(x)dx + ∫g(x)dx
4)    For any real number ‘k’, ∫k f(x) dx= k ∫f(x) dx
5)    Properties 3 and 4 can be generalized to a finite number of functions f1, f2 ……….. fn and the real numbers k1, k2…………kn giving
∫ [k1f1(x) + k2f2(x) +…..+ knfn(x)] dx = k1∫f1(x)dx + k2∫f2(x)dx +       …….+kn∫fn(x)dx.


Comparison between differentiation and integration

1.    Both are operations on functions.
2.    Both satisfy the property of linearity.  For any two constants k1 and k2,
a)    d[k1f1(x)+k2f2(x)] = k1d[f1(x)] + k2 d[f2(x)]
      __                            __               __   
      dx                            dx                dx

b)    ∫[k1f1(x)+k2f2(x)] dx= k1∫f1(x) dx + k2∫f2(x)dx
3.    All the functions are not differentiable, in the same way all the functions are not integrable.
4.    The derivative of a function if it exists, is unique, but the integral of a function is not unique.
5.    The derivative of a polynomial function is a polynomial function with power less than one, but the integral of a polynomial function is a polynomial function with power more than one.
6.     We can find derivative at a point, but the integral of a function exist only for an interval not at a point.
7.    The geometrical meaning of a derivative is that it is the slope of tangent to a corresponding curve at a point but the geometrical meaning of an indefinite integral is a family of curves placed parallel to each other having parallel tangents at the points of intersection of the curves of the family with the lines orthogonal to the axis representing the variable of integration.

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