Fundamental theorem of calculus

Summary

The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). The two operations are inverses of each other apart from a constant value which depends on where one starts to compute area. The first part of the theorem, the first fundamental theorem of calculus, states that for a function f , an antiderivative or indefinite integral F may be obtained as the integral of f over an interval with a variable upper bound. This implies the existence of antiderivatives for continuous functions. Conversely, the second part of the theorem, the second fundamental theorem of calculus, states that the integral of a function f over a fixed interval is equal to the change of any antiderivative F between the en

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EPFL2022

Strictly Real Fundamental Theorem Of Algebra Using Polynomial Interlacing

Soham Basu

Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss's first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

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Massive Open Online Labs (MOOLs): An Innovative Solution to Achieving SDGs in the Global South

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In engineering education, laboratories represent an important academic resource as they provide practical training in addition to the fundamental theories. However, the acquisition of new machinery and the maintenance of the equipment imply a large investment that only a limited number of universities can afford. This paper represents innovative online education activities through a collaborative widespread network with the global south countries, deploying remote laboratories in electrical, mechanical and control engineering at a large scale within MOOC infrastructures.

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