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CBSE Class 12 Commerce Mathematics Unit III Calculus

CBSE Class 12 Commerce Mathematics Unit III Calculus

CBSE Class 12 Commerce Mathematics Unit III Calculus:- The goal of the Academic, Training, Innovation and Research unit of Central Board of

CBSE Class 12 Commerce Mathematics Unit III Calculus

Secondary Education is to achieve academic excellence by conceptualizing policies and their operational planning to ensure balanced academic activities in the schools affiliated to the Board. The Unit strives to provide Scheme of Studies, curriculum, academic guidelines, textual material, support material, enrichment activities and capacity building programmed. The unit functions according to the broader objectives set in the National Curriculum Framework-2005 and in consonance with various policies and acts passed by the Government of India from time to time.

CBSE Class 12 Commerce Mathematics Unit III Calculus

CBSE Class 11 Commerce Maths Calculus : Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered

CBSE Class 12 Commerce Mathematics Unit III Calculus

Unit-III: Calculus

  1. Limits and Derivatives

Derivative introduced as rate of change both as that of distance function and geometrically.

Intuitive idea of limit. Limits of polynomials and rational functions, trigonometry, exponential and logarithmic functions. Definition of derivative, relate it to slope of tangent of a curve, derivative of sum, difference, product and quotient of functions. The derivative of polynomial and trigonometric functions.

CBSE Class 12 Commerce Mathematics Unit III Calculus

CBSE Class 12 Commerce Mathematics Unit III Calculus: Calculus (from Latin calculus, literally “small pebble used for counting on an abacus”) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus (concerning rates of change and slopes of curves), and integral calculus (concerning accumulation of quantities and the areas under and between curves); these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Generally, modern calculus is considered to have been developed in the 17th century by Isaac Newton and Siegfried Leibniz. Today, calculus has widespread uses in science, engineering and economics.

Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has historically been called “the calculus of infinitesimals”, or “infinitesimal calculus”. Calculus (plural calculi) is also used for naming some methods of calculation or theories of computation, such as propositional calculus, calculus of variations, lambda calculus, and process calculus.

Complete Details Of CBSE Class 11 Commerce Maths Calculus

CBSE Class 12 Commerce Mathematics Unit III Calculus: The word Calculus comes from Latin meaning “small stone”,  Because it is like understanding something by looking at small pieces. Differential Calculus cuts something into small pieces to find how it changes.  Integral Calculus joins (integrates) the small pieces together to find how much there is.

Differential Equations

In our world things change, and describing how they change often ends up as a Differential Equation: an equation with a function and one or more of its derivatives:

Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (independently from each other, first publishing around the same time) but elements of it have appeared in ancient Greece, and (alphabetically, later on) in China, in the Middle East, again in medieval Europe, and in India.

CBSE Class 12 Commerce Mathematics Unit III Calculus

CBSE Class 12 Commerce Mathematics Unit III Calculus: The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (13th dynasty, c. 1820 BC), but the formulas are simple instructions, with no indication as to method, and some of them lack major components. From the age of Greek mathematics, Eudoxus (c. 408–355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287–212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus. The method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, established a method that would later be called Cavalieri’s principle to find the volume of a sphere.

Medieval

In the Middle East, Alhazen (c. 965 – c. 1040 ce) derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. In the 14th century, Indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. Madhava of Sangamagrama and the Kerala school of astronomy and mathematics thereby stated components of calculus. A complete theory encompassing these components is now well-known in the Western world as the Taylor series or infinite series approximations. However, they were not able to “combine many differing

ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today”.

Modern

Complete Details Of CBSE Class 11 Commerce Maths Calculus In Europe, the foundation work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes’ in The Method, but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri’s work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.

Foundations

In calculus, foundations refers to the rigorous development of the subject from axioms and definitions. In early calculus the use of infinitesimal quantities was thought rigorous, and was fiercely criticized by a number of authors, most notably Michel Role and Bishop Berkeley. Berkeley famously described infinitesimals as the ghosts of departed quantities in his book The Analyst in 1734. Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today.

Several mathematicians, including Maclaurin, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of Cauchy and Weierstrass, a way was finally found to avoid mere “notions” of infinitely small quantities. The foundations of differential and integral calculus had been laid. In Cauchy’s Cours d’Analyse, we find a broad range of foundation approaches, including a definition of continuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an (ε, δ)-definition of limit in the definition of differentiation. In his work Weierstrass formalized the concept of limit and eliminated infinitesimals (although his definition can actually validate nil square infinitesimals). Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called “infinitesimal calculus”. Bern hard Riemann used these ideas to give a precise definition of the integral. It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane.

Significance

While many of the ideas of calculus had been developed earlier in Greece, China, India, Iraq, Persia, and Japan, the use of calculus began in Europe, during the 17th century, when Isaac Newton and Gottfried Wilhelm Leibniz built on the work of earlier mathematicians to introduce its basic principles. The development of calculus was built on earlier concepts of instantaneous motion and area underneath curves.

Applications of differential calculus include computations involving velocity and acceleration, the slope of a curve, and optimization. Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. More advanced applications include power series and Fourier series.

Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums of infinitely many numbers. These questions arise in the study of motion and area. The ancient Greek philosopher Zeno of Elea gave several famous examples of such paradoxes. Calculus provides tools, especially the limit and the infinite series, which resolve the paradoxes

CBSE Class 12 Commerce Mathematics Unit III Calculus

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