The origin of the word ‘Calculus’ comes from Latin word ‘calculus’ meaning ‘small stone’. Students dread Calculus and hate calculus homework. They assume it as extremely tough. But most of them don’t know that if they can only remember the fundamentals of calculus, then the subject is not as difficult as it is thought to be. This article attempts to explain what calculus is all about.
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Definition of calculus and its importance
Calculus is the field of study that explains the change while focusing on rate of change and accumulation. It has two categories: differential calculus and integral calculus.
Calculus was developed out of a need to understand continuously changing quantities. Newton while finding the effect of gravity, which is responsible for failing objects to constantly accelerate, discovered that the speed of an object constantly increases every fraction of a second as it falls. No one before the time of Newton and Leibnitz could answer the speed of a falling object at a frozen instant in time. The solution of this issue came to be known as the derivative (the part of differential calculus).
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Foundation of calculus
Differential calculus
This part of calculus is concerned with continuous change and its application. The study mainly covers properties application of the derivative of a function. Differential is defined by the process of finding the derivative.
The rule for differentiation
The mathematical form of differentiation rule is,
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According to Leibniz, the equation would be,
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You need to think of the rule as ‘multiply by the index and then reduce the index by 1,’ suggests Leibniz.
For example,
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The rule also applies when the fraction is negative,
Negative =
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Fractional = ) =
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or both =
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Example 1:
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Example 2:
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Example 3:
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Integral calculus
This side of calculus resembles the reverse of differentiation. It is used to find areas, volumes, central points and many other things. Integral is the inverse of differentiation. So here instead of multiplying by the index and reducing the index by one, we increase it by one and divide it by the new index.
For example,
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The rule for integration
In other form, this can be written,
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In general,
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Here are few examples:
Example 1: Find the equation of the curve for which
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and passes through the point (1,3).
The equation would be,
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Substituting x = 1 and y = 3 gives
3 = 1+2+C
C = 0
Therefore,
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Example 2: Find the equation of the curve for which
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and passes through the point, (2,9)
Integrating
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Substituting x = 2 and y =9
- 9 = 4 + 2 + C
- C = 3
- y = x2 + x + 3
Fundamental theorem
It is a theorem of calculus that links the concept of the derivative of a function with the concept of function’s integral. The first part of the theorem defines integration of a function that relates to its anti-derivative and can be reversed by differentiation. The second part of the theorem explains a definite integral of a function that can be calculated by using any one of its infinitely many anti-derivatives.
The fundamental theorem of calculus states: If ƒ is continuous on the interval [a,b] and if F is a function whose derivatives of ƒ on the interval is (a,b) then
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So, for every x in the interval (a,b)
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Let’s explain you with an example.
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Here and we can use F(x) = as the anti-derivative, therefore:
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Example 2: Find the value of
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Here,
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and
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can be used as the antiderivative. Therefore,
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Or equivalently,
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These are only the basic concepts. To solve the calculus problems, you need to learn the advanced concepts before solving complex problems of calculus homework. And needless to say, mathematical problems come with a lot of complications, especially when it is calculus. But you can make things easy by taking expert help from professionals.
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