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English Guide: Practical Tips for Better English Communication

socializing in English In our day to day life, we use many words to converse in English. Many of these words used are a combination of a verb and a preposition or a combination of verb and adverb. These words are called Phrasal verbs. These words give an entirely different meaning when combined. It is vital to know their proper meaning in order to use them in a sentence. Till now we have listed many blogs related to phrasal verbs and its usage to describe your day....
car All these years we published blogs that were related to interview skills or English grammar. Just a few days back we started creating more blogs that helped the beginners to learn basic English skills. The basic English skills include vocabularies, phrases, idioms that are used in day to day communication. While the blogs provided information on general topics, today’s blog is for those who love cars, bikes etc. i.e. for men and boys. However,...
funny idiom to improve English 1 In our last blog, we described funny idioms that are often used by the native speakers in the day to day communication. Funny idioms not only add a humorous touch but also make you a pro-English speaker. Let us have a look at the remaining idioms that are funnier than the previous one. 6 Funny idioms you must know1. Chew the fatThis idiom means to chat in a friendly and leisurely way or engage in casual gossip sessions. It is generally said to...
funny idiom to improve English Idioms are an everyday figure of speech which is taken in a light and a humorous way. The figurative meaning of the idioms makes the sentence often go against the logical “rules of language and grammar”.  Native speakers use these idioms daily in the conversation which not only improves English but also helps to get the command over the language. In today’s blog, we’ll teach funny idioms which will help improve your English-speaking skills.5...
confused In life, we have faced many instances where we have misunderstood many things and have ended up confused. We were able to clear a few, while there was a lot of confusion in the others. It is important to clarify and clear out the confusion in order to avoid unexpected results. To understand and clarify the confusion, you are required to articulate the points in an impeccable English with good vocabulary words. Today’s blog will help you with...
food “Food is love food is life” for all the food lovers. It’s hard to define the relation that the food lovers share with food. While people explore cuisines from different places to get a sense of taste, they also learn many new things. Learning new things is easy and so is English. As a foodie, you learn new dishes and its ingredients which indirectly help you to learn new words, spellings and pronunciation. Sometimes describing them is difficult as there...
food vocabulary In our last blog, we described vocabulary related to texture of the food. This blog will introduce more words about food that can be used while communicating. Food vocabulary is useful for everyone – right from the housewives to the food lovers. Have you ever been in the phase where you found it difficult to explain a recipe in English? If your answer is yes, then don’t worry. It’s something that everybody has gone through. There are many words that...
conversation starters When you go to a new place, where you don’t know anyone, it becomes a bit difficult to spend the time. In order to know more about the festival, culture, climate etc. of that place you decide to explore the city. You find a group of locals there and decide to make friendship and know about the city. A fluent English, in this case, is very helpful to start the conversation. You need to be aware of the conversation starters that will encourage...
grammar mistakes Are you a beginner in English? It is common to get confused with similar words or phrases. Speaking English is easy as it just takes confidence and a person to talk with. Don’t be afraid of mistakes, mistakes are the proof that you are trying. Only by making mistake you can learn new things. It is not that you are the only one who is making mistakes. There are many native English speakers who get confused while speaking. The mistakes can be...
teen slang Slang words are informal language words that are used by the teens and youngsters. Today, it’s impossible to be in a nightclub, or at a cutting-edge art gallery, without hearing these slang words. These are nothing but idioms and expressions, which make adults scratch their head. Don’t worry, this blog will explain you slang words along with its meaning.10 slang words you must know1. It sucks / that sucksMeaning - It might not be the right...
misused words literally In our last blog, we discussed words that are commonly misused in English. In this article, we will add some more words that are commonly misused. Knowing the correct usage will help you to enhance your vocabulary as well as spoken English Skills which will help you to speak effectively. Let us look at the words.10 words with incorrect usage1. An effectAn effect means either to influence or to fake whereas to effect means to put into effect.Examples:The new...
you can do it In life, there are many situations which make us sad and feel low. At that moment when someone encourages us with their positive words, we feel light and happy.  Don’t we? We encourage people playing sports, while building relationships with friends, family, colleagues, and while working. In the same manner, when your friend or your loved one feels low, you try to encourage them with positive words by complimenting and appreciating them. In...
give it a shot In our last blog, we taught you few phrases that can be used for encouraging someone. When you use these phrases in your communication, it not only helps you to motivate someone but also improves your English-speaking skills. Let’s have a look at some positive phrases that will improve your English skills and help you encourage someone.6 phrases you must know1. You are almost there!This phrase is best to encourage someone who is about to...
interjections In our earlier blog, we provided a fun guide to interjections which help us in our day to day conversation. Interjections are words for the emotions and expressions that we make. These interjections words depict our emotions like surprise, agreement, disagreement, sarcasm, shock etc. In today’s blog, we will explain interjections to be used to express your emotions which will help you in your everyday conversations using “expressive...
couch potato Idioms play an important role in English language. They help to express your feelings in a figurative manner. In today’s lesson, we will describe people and their characteristics with idiomatic expressions. These people might be your friends, cousins, co-workers and to some your family members. All these characters are different from each other and have a unique differentiating factor through which we identify or know them. Let us learn the...
relationships February is a month of love and people all over the world celebrate their relationships. While not everyone might be in a relation, but everyone might surely be aware of relationships. Today’s English lesson will provide you with some phrases which are used in English communication while conversing about relationships.7 vocabulary terms to talk about relationships1. Long distance relationshipMeaning - A relationship where the two people who are in...
compliment man Women get compliments more often for their looks, smile, the way they walk or talk etc. However, it’s not always that women get compliments; men too get compliments. Men also like to get compliments. Who doesn’t like to get praised for their appearance? As a small gesture, you can compliment them on their appearance – it might be their new hairstyle, beard, moustache etc. The only thing to remember here is to use adjectives like cool, handsome, trendy,...
misused words Words have the tendency to make or break the sentence. When a word is used correctly, it makes a wonderful meaningful sentence. In the same way when the word is wrongly used it creates a wrong sentence. Generally, it happens with non-native English speakers. However, practising these words and knowing their appropriate meaning will help with its correct usage. So, in this article, we are bringing words that are confused and misused.9 words...
you are right A conversation is incomplete without a set of questions. It doesn’t matter with whom you are talking, but it will surely have a question. For e.g. while commuting, if a co-passenger asks whether it’s a Chicago-bound train and you confirm it by replying yes to him. There are many instances when you will have to reply to someone that they are correct. In today’s blog, you will learn English phrases that can be used to agree with someone or to say...
act your age Age is something that is never shared and asking a girl or a woman her age, is considered bad manners. People rarely talk about their age and salary. Although, while conversing in English, people refer to idioms related to age to give a hint about their age,no one wants to reveal their actual age and want to be referred to as a sweet sixteen. Sweet sixteen is regarded as the characteristic age of prettiness and innocence in a girl. In today’s...
article thumbnailLife is full of surprises and shocks. You might never know when a surprise can turn into a shock. There are many times when we warn people to avoid taking too many risks or if they are climbing up a ladder you might warn them to be safe. Sometimes we see formal notices, pictures, posters that act as warnings about dangers. For e.g. If someone is investing all his money in stock market, you might warn them to avoid taking too many risks or...
article thumbnailYou know it's the last month of the year, and you are in celebration and vacation mood. And not to forget, the favorite season, winter. Who doesn’t like winter or let’s say who doesn’t like to sit indoors by the fireside in the mornings and evenings with a hot cup of tea. We all love it. With a week or a few days away from Christmas and New Year, we are already in party mood. People visit their family and friends to spend their time and enjoy...
article thumbnailIn our last blog, we listed five holiday idioms that described the holiday. In today’s blog, we will discuss few more idioms and phrases that are related to holidays. Most of the people in foreign countries use holiday phrases and idioms while conversing in English. So, having a fair knowledge about these with their meaning will come handy when you are on a vacation.5 Holiday phrases and idioms1. Let your hair down Meaning - to allow yourself to...
article thumbnailhttp://eagetutor.com/December is that time of the month when many countries are in a festive mood. Christmas and New Year bring holidays and celebration mood along with them. In many English-speaking countries, they send gifts and greetings to wish their friends and family. If you are planning a vacation in any of the countries especially in December, then knowing some holiday greetings might surely help you. Even if you don’t celebrate...
article thumbnailHair is the personality of a person, especially women. It makes them look beautiful. Everyone has a unique hairstyle which makes their appearance different. While some flaunt it, others nag about it. Whatever you do, you express it with words and phrases. These expressions are nothing but idioms and phrases. These idioms have a figurative meaning. Let us learn some expressions about hair that is generally used by a native speaker.Must know...

LOCAL LINEAR APPROXIMATION

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Introduction

locallinearapproximation1In this section we will learn how derivatives can be used to approximate nonlinear functions by simpler linear functions.  We will also define the differentials dy and dx and use them to interpret the derivative dy/dx as a ratio of differentials.
Let a function ‘f’ is differentiable at x0 and recall that the equation of the tangent line to the graph of the function ‘f’ through P(x0, f(x0)) is y=f(x0)+f’(x0)(x-x0).  Since this line closely approximates the graph of ‘f’ for values of x near xo, if follows that
                       f(x) ≈ f(x0) + f’(x0)(x-x0)       …………………………..(1)

Provided x is close to x0.  We call (1) the local linear approximation of ‘f’ at x0.  Furthermore, it can be shown that (1) is actually the best linear approximation of ‘f’ near x0 in the sense that any other linear function will fail to give as good as approximation to f for values of x very close to x0.  An alternative version of this formula can be obtained by letting ∆x= x-x0 in which case (1) can be expressed as
                       f(x0+∆x) ≈ f(x0) + f’(x0)∆x

In practice we use the following method for local linear approximation
1)    We are given some value to calculate say f(x), and the calculation is difficult.
2)    We see that there is a nearby point x0, where the calculation of both f(x0) and f(x0)’ is relatively easy.
3)    We approximate f(x) by f(x0) + f(x0)’(x-x0)
Example: Find the local linear approximation of f(x) = √x at x0=1
Solution: Given f(x) = √x, f’(x) = 1/(2√x)
The local linear approximation of √x at x0=1 is
√x ≈ √1 + ½ (x-1) = ½ (x+1)

Approximating Changes – Differentials


locallinearapproximation2The second use of derivative is to approximate small changes in a function.  Start at point x and move a small distance given by the independent variable ∆x.  Then define ∆y to be the corresponding change in the value of y=f(x).  We can see that ∆y=f(x+∆x)-f(x)

y+∆y = f(x+∆x)
    ∆y = f(x+∆x)-y
         = f(x+∆x)-f(x)
Let dx be an arbitrary variable and define dy to be f’(x)dx.  Then the ratio of dy to dx is the derivative dy/dx.  dx and dy are called the differentials.
Example: Find the differential dy when y = x18 when x=1
Solution: y=x18
dy = 18x17dx
    = 18dx since 117 = 1

Error Propagation in Applications

In applications, small errors invariably occur in measured quantities.  When these quantities are used in computations, those errors are propagated in turn to the computed quantities.  For example, suppose that in an application the variables x and y are related by a function y=f(x).  If xa is the actual value of x, and it is measured to be x0, then we define the difference dx=x0-xa to be the error in measurement of x.  If the error is positive, the measured value is larger than the actual value and if the error is negative the measured value is smaller than the actual value.  Since y is determined from x by the function y=f(x), the true value of y is f(xa) and the value of y computed from the measured value of x is f(x0). The propagated error in the computed value of y is defined to be f(x0) – f(xa).  If the propagated error is positive, the calculated value of y will be too large and if this error is negative, the calculated value of y will be too small.
The local linear approximation (1) becomes
                f(xa) ≈ f(x0) +f’(x0)(xa-x0)
                        ≈ f(x0) – f’(x0) (x0-xa)
                        ≈ f(x0) – f’(x0) dx

Example: Suppose that the side of a square is measured with a ruler to be 10 inches with a measurement error of at most ±1/32 of an inch.  Use a differential to estimate the error in the computed area of the square.
Solution: The side of a square x and the area of the square y are related by the equation y=x2
Since dy=2xdx and if x=10 then dy=20dx
To say that the measurement error is at most ±1/32 of an inch means that the measurement error dx = x0-xa satisfies the inequalities          -1/32≤dx≤1/32.  Multiplying each term by 20 yields the equivalent inequalities
20(-1/32)≤ dy ≤20(1/32)
-5/8 ≤dy ≤5/8
Since we are using the differential dy to approximate the propagated error, we estimate this propagated error to be between -5/8 and 5/8 of a square inch.  Hence we estimate the propagated error to be at most ±5/8 of a square inch.

Relative error and Percentage error

The ratio of the error in some measured or calculated quantity to the true value of the quantity is called the relative error of the measurement or calculation.  When expressed as a percentage, the relative error is called the percentage error.
For example, suppose that the side of a square is measured to be 10 inches, but the actual length of the side is 9.98 inches, the relative error in this measurement is 0.02/9.98≈0.002004008.

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