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Mathematics Tutor USA

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Basics of Algebra
Equations and Inequalities
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Trigonometry
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Basics of Geometry
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Congruent Triangles
Properties of Triangles
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Similarity
Trigonometry
Right Triangles
Circles
Surface Area and Volume
Constructions
Transformations
Basics of Calculus
Functions and Binary Relations
Graphs of Function and Relations
Straight Lines
The General Equations of Straight Lines
Sequence and Limit of Sequences
Squeeze Theorem
prove of important limits
Questions on Limit
Limit of a Function
Differentiability
first principle
derivative first formula y= x^n
derivative 2nd formula y=(a + bx)^n
product rule of differentiation y=UV
quotient rule of differentiation y=u/v

Mathematics Worksheets

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 Integration Made Easy

By Masood Amir

Mathematics Worksheets Mathematics Worksheets, ASVA provide calculus and Algebra worksheets.

Calculus:

To measure change or variation of a function with respect to the independent variable we use Calculus.

Differential Calculus: (Mathematics Worksheets)

Differrential Calculus used to measure change or variation of a function in a very small invterval of time.

Integral Calculus:

The branch of calculus used to measure changes or variation over an interval of independent variable, called Integral calculus, e.g to find length of curve, the area of region and the volume of a solid in a specified period of time.

The technique used to measure such changes or variation called “Integration” or “Antiderivatives”. It a reverse process of differentiation.

Mathematically, Integration  defined as “ If f'(x) represents the differential coefficient of f(x), we need to find f(x), if we have f'(x) or dy/dx.

Integration

Notation: ''∫'' used to show the integration, it is a symbol of ``S'' derived from the word ``Sum''. i.e. Integration is a process in which we have to sum up the derivatives over a specified interval and to find the function.

Techniques of Integration:

As we know that integration is the reverse process of differentiation, our problem is to find the function f(x) or Y, when f'(X) or dy/dx is given.

dy/dx &= f'(X)

∫dy &= ∫f '(X)dx

Y &= f(x) is our solution

Ist Formula of Integration (1st Rule of Integration)

Indefinite Integration:

Ist Formula of Integration (Ist Rule of Integration):

Let

    \[ \int dy = \int x^n dx \]

    \[ y = \frac{x^{n+1}}{n+1} + C \]

Why ``C'':

In the process of differentiation, we eliminate constant, as the derivative of a constant is “zero“.

So, In functions like x^n, x^n + 6, x^n - 3, x^n + k the derivatives of all of them is X^{n-1} , in finding the anti derivative of X^{n-1} , we put a constant “C”, as we don’t know which constant was present in the original function, and  can be found If we have initial boundary values (Definite Integral).

Example: Solve \int x^n \, dx

Solution:

    \[ \int dy = \int x^3 dx \]

    \[ y = \frac{x^{3+1}}{3+1} + C \]

    \[ y = \frac{x^{4}}{4} + C \]

Example: Solve

    \[ \int (x^3 + x^2 + 5x + 6) \, dx \]

Solution:

    \[ \int (x^3 + x^2 + 5x + 6) \, dx \]

    \[ \int x^3 \, dx + \int x^2 \, dx + \int 5x \, dx + \int 6 \, dx \]

    \[ \frac{x^4}{4} + \frac{x^3}{3} + \frac{5x^2}{2} + 6x + C \]

Worksheet # 1

Find the Integral of the following:

SET 1 

(1)   \begin{align*} 1. & \quad \int (x^3 - 4x^2 + 5x - 6) \, dx \\ 2. & \quad \int (3x^5 - 4x^3 + 3x^2) \, dx \\ 3. & \quad \int (ax^5 - bx^4) \, dx \\ 4. & \quad \int \left( \frac{x^3}{2} - \frac{5x^4}{3} + 3x^2 \right) \, dx \\ 5. & \quad \int (4\sqrt{3}x^2 - 2x) \, dx \\ 6. & \quad \int \left( 3x + 5x^2 - \frac{x^3}{2} - 0.4x^4 \right) \, dx \\ 7. & \quad \int \left( x(8x - \frac{1}{2}) \right) \, dx \\ 8. & \quad \int (2 - x)(4 + 3x) \, dx \\ 9. & \quad \int \left( x^{-3} + x^{-4} \right) \, dx \\ 10. & \quad \int \left( (2x^3 - 3)(3x^4) \right) \, dx \\ 11. & \quad \int \left( 4x^7 + 3x^{12} - 5x^8 + 2x - 1 \right) \, dx \\ 12. & \quad \int \left( ax^3 - bx^2 + cx - d \right) \, dx \\ 13. & \quad \int \left( \frac{1}{x^3} + \frac{2}{x^2} - 6 \right) \, dx \\ 14. & \quad \int \left( -3x - 8 + 2\sqrt{x} \right) \, dx \\ 15. & \quad \int \left( (x^3 - 5)(2x + 5) \right) \, dx \\ 16. & \quad \int \left( 7x - 6 + 5\sqrt{x} \right) \, dx \end{align*}

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MA Economics papers

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MA Economics Past Papers

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MA Economics Papers

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We offer comprehensive support for MA economics papers. Our experienced tutors cover various subjects, including microeconomics, public finance, advanced economic statistics, macroeconomics, economics of Islam, mathematical economics, economics of planning, agricultural economics, national income analysis and accounting, and comparative economics. With in-depth knowledge and expertise in these areas, our tutors ensure students have a solid foundation in MA economics.

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MA Economics Papers list:

Micro Economics Paper (I)
1999 Public Finance
2000 Public Finance
2001 Public Finance
2002 Public Finance
2003 Public Finance
2004 Public Finance
2005
2006
2007
2008
2009
2010
2011
2012

Advance Economics Statistics (III)

Economics of Planning

Paper (IV)

1999 National Income Analysis & Accounting
2000 National Income Analysis & Accounting
2001 National Income Analysis & Accounting
2002 National Income Analysis & Accounting
2003 National Income Analysis & Accounting
2004 National Income Analysis & Accounting
2005
2006
2007
2008
2009
2010
2011
2012

Agricultural Economics Paper (V-A)

1999 Agricultural Economics
2000 Agricultural Economics
2001 Agricultural Economics
2002 Agricultural Economics
2003 Agricultural Economics
2004 Agricultural Economics
2005
2006
2007
2008
2009
2010
2011
2012

Mathematical Economics

Paper (V-C)

History Tuition

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History Tuition

History Tuition
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History

History is indeed a field that involves research and inquiry, as historians seek to uncover and understand facts about the past. It is through this investigative process that historians can piece together narratives and gain insights into different historical events, societies, cultures, and individuals.

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We offer a range of specialized online professors dedicated to providing top-quality tuition in various subjects, including history, sociology, psychology, genetics, English literature, and linguistics.
Our history experts are highly knowledgeable and experienced, catering to students studying O level, A level, and even preparing for the GRE history exam.
History, as an essential discipline, involves thorough research and inquiry to uncover facts and gain a comprehensive understanding of the past.
Our dedicated history tutors are here to guide you through this exciting investigative journey.

History and Science:

Similarities in Research and Inquiry

As such, history and science belong to the same kind of knowledge. Both are a kind of research or inquiry in their own way.

The Nature of Science :

Science consists of forms of thought by which it asks questions and tries to answer them.

Facts and Discovery in Science:

There are some important facts about science which we must know and understand. It is a common belief that science collects known facts and arranges them in suitable groups. But this is only one of its many aspects. Basically, science seizes upon unknown things and tries to discover them. If playing patience with known facts does not give answers to our questions, it is not scientifically valuable.

Purpose of Science:

It can be scientifically valuable only when it serves as a means towards the inquiry and discovery of unknown things that we wish to inquire and discover. This is the essential end and purpose of science. In this way, science begins from the knowledge of our ignorance. It starts from an understanding of our own lack of knowledge. Only when we know fully well that we do not know a particular thing, we fasten upon it and try to know it. This particular unknown thing may be the origin of parliament, the causes of cancer, the chemical composition of the sun, or the way to make a pump work without the help of human or animal energy. Thus, unknown things turn into known facts, and science grows.

History as a Science:

Science is about finding things out. In much the same way, history is also about finding things out. Therefore, history is a science.

What is the Object of Hisory?

All sciences differ from one another. One science finds out thing of one kind another science finds out things of another kinds.
Likewise, history finds out register or things that have been done. In other words history finds out past actions of human beings who lived in the past.
This statement gives rise to a large variety of questions. Conflicting views are expressed about many of them.
As such they may be answered in a large variety of ways. Never less, history all along remains the science of register. It is always an effort to find out answers to questions about past activities of human beings who lived in the past.
No one can refute this statement. No one can prove it wrong.

How Does History Proceed?

In other words it means how is the procedure or method which help history grow from age to age.
Obviously, the growth of history depends entirely upon the interpretation of evidence.
To be more explicit, interpretation of evidence is the explanation of the full meaning and significance of the evidence in question.
In the case of history evidence is made of a number of things which we meet here and there, now and then, taken up together.
These we may call documents. These documents, or rather historical documents to be more exact are of such nature that if history studies them, it finds answers to its questions about past human actions.
A large number of questions can be asked about the characteristics of evidence and the ways to interpret it. Needless to say, these questions are much too early and out of the place to ask at this junction.
However, ther are different ways to answer them.
But the procedure or method of history remains basically composed of interpretation of evidence, all the same.
By no stretch of imagination can there be any two opinions about this statement among historians.

Lastly, What is History For?

This is perhaps the toughest question of all the four. It will be difficult to answer them.
One will have to cover a wider field to find an answer for the question. Various aspects of history will have  taken into account.
At the same time, various other aspects of a rather general nature will have taken into account as well.
When we say that something is ‘for’ something, we mean that there is a marked difference between the two which distinguished one from the other as separate.
Simple logic can explain the truth of this statement. If we distinctly separate things. We clearly see that A is the thing which is good for B, while B is the thing for which A is good.
Obviously the two are not the same. Instead, they are rather different. Here I give the answer to the question and hoping that the historians answer will accept answers.
However, I believe that it will lead to a host of hard questions. The answer is “History is for human self-knowledge, In other words this means that history helps man to know and understand himself, most of us think that the more we know ourselves, the better it is.
Self-knowledge always stands us in good stead.

The Character Of Happy Life

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The Character of Happy Life

By Sir Henry WontonSir Henry Wonton: 

Sir Henry Wonton was a distinguished poet, scholar and diplomat. He had an eventful of rewards and hazards of public life, which, however, found a tranquil, close in his last days. Writing from his personal experience, he contributed a rich share to the English poetic literature of his time.

The Character of A happy Life

 

The Poem: ‘The Character of Happy Life’ is a lovely little poem written by Sir Henry Wonton. It is concerned with the life and virtues of a good man. Also It is deeply religious in spirit and simple but dignified in expression. Finally It lays emphasis not only on the promise of a heavenly reward but also on the enjoyment of a sever life of peaceful virtue on this search. As such, Walton’s virtues man gets the best of both the words. A sharp sense of contrast, between the uneasy life of an ambitious man and the contended life of a man satisfied with his obscurity, runs right through his poem. Writing from his personal experience, he wishes to tell us that rewards of ambitious life were very dazzling indeed, but its anxieties and dangers were also all too real.

Context: These lines are from the poem, ‘The Character of a Happy Life’ written by Sir Henry Wonton. It tells us about the qualities essential for a truly happy life.

Explanation:

Indepencdence, Honesty of thought and Truthfulness

(1): The poet says that the characteristics of a happy man are independence, honesty of thought and truthfulness. He has not learnt to be the slave of others. He is upright in all his thoughts actions. His honesty shields him from all dangers. He believes only in pure, simple truth. It guides him in the day-to-day affairs of his life. Naturally therefore, he enjoys the blessings of real happiness.

Self-control and freedom from worldly ambitions

(2): The poet says that the characteristics of a happy man are self-control and freedom from worldly ambitions. He is not the slave of his derives and passions. He does not fear death; the world is not too much with him. So he does not have any anxieties, either. He does not care for fame or fortune. As such he does not worry in the least, for what people say about him in private. Naturally, therefore, he enjoys the blessings of real happiness.

Freedom from Envy, Dislike for Praise:

(3): The poet says that the characteristics of a happy man are freedom from envy of worldly success, dislike for praise and diplomacy and love for virtue and goodness. He does not envy anybody who has risen to high position by the stroke of good fortune or through his own wickedness. He does not seek praise for it corrupts character and provokes the enmity of the envious. However, insincere praise may be ironic or mocking. So he is totally unaware of the wounds inflicted by praise. His strong ignorant of the principles of statecraft. All that he knows and understands is absolute goodness. He hears no evil, he sees not evil, he speaks no evil and he does no evil. Naturally, therefore, he enjoys the blessing of real happiness.

Freedom from rumors, care for his conscience

(4): The poet says that the characteristics of a happy man are freedom from rumors, care for his conscience and lack of wealth and position. He keeps his life free from all kinds of rumors. He listens only to the inner voice of his conscience. As his sincere guide, it is a strong hold, which shelters him from all kinds of evils. The world is not too much with him. So his state is not so great that it can feel flatters or tempt people to accuse him and help them to plunder him when he in ruined. Naturally, therefore, he enjoys the blessings of real happiness.

Happy man are piety and simple life

(5): The poet says that the characteristics of a happy man are piety and simple life. He worshiped God with sincere devotion. He does not pray for worldly things. Instead, he prays for His kindness and mercy only. He spends his innocent life in plain living and high thinking which is his motto. He either studies good books or sits in the company of noble friends. Naturally, therefore, he enjoys the blessings of real happiness.

Independence , Povert and Contentment

(6): The poet says that the characteristics of a happy man are independence, poverty of contentment. He is the slave neither of another’s will nor of his own passions. Indeed he is his own master. The world is not too much with him. He has no desire for wealth or worldly cares; he leads an obscure life of peaceful virtue. As much, he has no fear of a downfall either. True he has no riches, but he has happiness, is the greatest wealth a man can have. Naturally, therefore, he enjoys the blessings of a truly real happiness.

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Free online englihs notes XI Karachi Board, “The Character of Happy Life” written by Sir Henry Wonton, Important questions, Reference to context.

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