Proof of mean value theorem using infinitesimals

#PROOF OF MEAN VALUE THEOREM USING INFINITESIMALS SERIES#

He argues that since f(a)=f(b), the function is never larger or smaller at any of the particular points. Or when we are using both the maxima and the minima, he states that f(x) occurs at endpoints. In a more detailed example, we can see that since f(x) is differentiable on (a,b) and therefore c is an extremum which concludes that f вЂ(c) = 0. RolleвЂ™s theorem uses f(x) as a continuous function on the closed interval which enables it to achieve the maximum and the minimum points on. The existence of the special case under the stated hypothesis MVT is because it guarantees the existence of point c with f вЂ(c) equals to f(b) вЂ“ f(a) divided by b-a which then equals to f(b) вЂ“ f(a) divided by b-a resulting to zero.

RolleвЂ™s Theorem as discovered by Michel Rolle in 1691 applied a special case of Mean Value Theorem which stated that вЂњif f(x) is a continuous an and differentiable on (a,b) and if f(a)=f(b) then there is some c in the interval (a,b) such that f вЂ(c) = 0вЂќ (Calculus Quest, online). In the 15th century, the mean value theorem came into existence and was described by another Indian geometer Parameshvara between 1370 to 1460 from the Kerala School of astronomy and mathematics (Connor and Robertson, 2000). For instance, in order to solve the equation x3 + a = bx, al Tusi finds the maximum point of the curve by using this equation y= bx- x3.

He devised a formula for finding maxima and minima of curves and derivation function as applied in solving cubic equations which may not have positive solutions. In the the12th century, a Persian mathematician al Tusi discovered the derivative of cubic polynomials through equations. Their contributions collectively discovered scales for measuring length (Indus inch) and inches, an accurate measure for determining the length of the excavated building (Connor & Robertson, 2000).

#PROOF OF MEAN VALUE THEOREM USING INFINITESIMALS SERIES#

HarappaвЂ™s, another Indian mathematician, developed a system of weights and measures and in his analysis discovered that series apart from being decimal in nature, can either be multiplied or divided by 2 and would give main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200 and 500 (Connor & Robertson, 2000). Another contribution came from Madhava who brought new perspectives in mathematics that provided a foundation for mathematicians to build their theories on (Connor & Robertson, 2000 Kartz, 1995).

There would have been no transition between ancient calculus and modern mathematics without the great contributions of the 12th-century Indian geometer Bhaskara II who defined and simplified the derivative and differential coefficient methods along with RolleвЂ™s theorem. Another Indian classic, Manjula, developed an equation in the 10th century that eventually led to the development of the 12th century Bhaskara II derivative concepts and RolleвЂ™s Theorem (Smith, 2007 Kartz, 1995)). By 500 AD, Aryabhata had introduced a notion of infinitesimals, based on basic differential calculus to solve integer equations that arose from his astronomical theories. The mathematics we have today is owed to the great contributors of Indian Mathematicians and astronomers. Their resembling concepts of derivations and Integration can be recognized in the works of Fermat and Democritus on tangents and finding maxima and minima. The outstanding work of the previous centuryвЂ™s geometers such as the great Greek Democritus, the works of the Chinese mathematician Liu Hui and Archimedes, to name just a few brought together the techniques of solving calculus problems.

They confronted the problems of finding areas and volumes of various solids- a term referred to as integration a phrase used throughout the paper. The subsequent development of calculus is largely due to the ancient geometerвЂ™s natural curiosity and their demands of application that provided solutions to mathematical problems. Though the applications interplay between discreetness and continuity, these two geometers have described discreet objects by continuous models and use the same applications for solving continuous problems. To understand their theorem, they clearly understood infinity. Although calculus received numerous criticisms at the time of invention, mathematicians succeeded in putting calculus on a firm foundation. The Cartesian coordinates and modern symbolism are what gave Isaac Newton and Leibniz Gottfried insights into creating calculus and our focus is how they used the Fundamental Theorem of Calculus into transforming peopleвЂ™s theories into a powerful tool. Greek mathematicians separated algebra and geometry. We discuss why itвЂ™s important to study calculus and identify the main areas of interest.

Proof of mean value theorem using infinitesimals