Calculus Paper

8/12/2019 Calculus Paper

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2 Calculus[Math Project] By Shikha Sebastian

Calculus is the mathematical study of change.

Two main topics that come under calculus are

1) Differential calculus(concerning with slopes and rates of change)2) Integral calculus(concerning accumulation of quantities and area under curves)

In this paper I have focus on differential calculus and only mentioned integral calculus.

Differential CalculusThis is the study of the definition, properties and application of the derivatives of a function.

HistoryThe concept of a derivative in the sense of a tangent line is a very old one, familiar to Greek geometerssuch as Euclid (c. 300 BC), Archimedes (c. 287 – 212 BC) and Apollonius of Perga (c. 262 – 190

BC).Archimedes also introduced the use of infinitesimals, although these were primarily used to studyareas and volumes rather than derivatives and tangents; see Archimedes' use of infinitesimals.

The use of infinitesimals to study rates of change can be found in Indian mathematics, perhaps as earlyas 500 AD, when the astronomer and mathematician Aryabhata (476 – 550) used infinitesimals to studythe motion of the moon. The use of infinitesimals to compute rates of change was developed significantlyby Bhāskara II (1114– 1185); indeed, it has been argued that many of the key notions of differentialcalculus can be found in his work, such as "Rolle's theorem". The Persian mathematician, Sharaf al- Dīnal- Tūsī (1135– 1213), was the first to discover the derivative of cubic polynomials, an important result indifferential calculus; his Treatise on Equations developed concepts related to differential calculus, such asthe derivative function and the maxima and minima of curves, in order to solve cubic equations whichmay not have positive solutions.

The modern development of calculus is usually credited to Isaac Newton (1643 – 1727) and GottfriedLeibniz (1646 – 1716), who provided independentand unified approaches to differentiation and derivatives.The key insight, however, that earned them this credit, was the fundamental theorem of calculus relatingdifferentiation and integration: this rendered obsolete most previous methods for computing areas andvolumes, which had not been significantly extended since the time of Ibn al-Haytham (Alhazen). For theirideas on derivatives, both Newton and Leibniz built on significant earlier work by mathematicians such asIsaac Barrow (1630 –1677), René Descartes (1596 – 1650), Christiaan Huygens (1629 – 1695), BlaisePascal (1623 – 1662) and John Wallis (1616 – 1703). Isaac Barrow is generally given credit for the earlydevelopment of the derivative. Nevertheless, Newton and Leibniz remain key figures in the history ofdifferentiation, not least because Newton was the first to apply differentiation to theoretical physics, whileLeibniz systematically developed much of the notation still used today.

Since the 17th century many mathematicians have contributed to the theory of differentiation. In the 19thcentury, calculus was put on a much more rigorous footing by mathematicians such as Augustin LouisCauchy (1789 – 1857), Bernhard Riemann (1826 – 1866), and Karl Weierstrass (1815 –1897). It was alsoduring this period that the differentiation was generalized to Euclidean space and the complex plane.




Differentiation and SlopesThe slope(m) of a graph is generally found by the formula

But this is not very accurate in curves where the slope is continuously changing.

When we find the slope of a tagent of a point we are finding this points derivative.

The process of finding the derivative of a function is called differentiation.

Differentiating the equation of a graph gives you the equation of the slope of the graph. For curves itgives you the slope of a particular point by finding the slope of its tangent.

In the following graph of the function, we see the geometric tangent at the point (a, f(a)).

There are two ways of showing showing a derivative. For y, we use dy/dx. For f(x) we use a prime andshow it as f’(x).




Formulas A few formulas commonly used to find derivatives are shown the picture below.

And for slightly more complicated derivatives.




Chain RuleThe chain rule is a formula for finding the derivative of the composition of two or more functions.

Practical usageONE example is: If you know the instantanious velocity of an object at any instant then the questionarises to determine the position of the object at that instant. Differentianton cn be used for this.

Integeral calculusIntigrals are also called antiderivatives. This is because when you intigrate a derivative, you get originalequation of the curve(without the constant) and n unknown constant c/k.

References

All of the project is hand typed(except history which was taken directly from

http://en.wikipedia.org/wiki/Differential_calculus#History_of_differentiation ). I do not own theimages in this paper, they have all been takes from google images. My information was takenfrom a few books, http://en.wikipedia.org and http://thecalculuspage.com

http://en.wikipedia.org/wiki/Differential_calculus#History_of_differentiation


http://en.wikipedia.org/



http://thecalculuspage.com/






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