File Name: calculus differentiation and integration .zip
Integration and Differentiation are two fundamental concepts in calculus, which studies the change.
Applications of Differentiation
Calculus , originally called infinitesimal calculus or "the calculus of infinitesimals ", 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 and integral calculus ; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves.
These two branches are related to each other by the fundamental theorem of calculus , and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. In mathematics education , calculus denotes courses of elementary mathematical analysis , which are mainly devoted to the study of functions and limits.
The word calculus plural calculi is a Latin word, meaning originally "small pebble" this meaning is kept in medicine — see Calculus medicine. Because such pebbles were used for calculation, the meaning of the word has evolved and today usually means a method of computation. It is therefore used for naming specific methods of calculation and related theories, such as propositional calculus , Ricci calculus , calculus of variations , lambda calculus , and process calculus.
Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz independently of each other, first publishing around the same time but elements of it appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India.
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. From the age of Greek mathematics , Eudoxus c.
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.
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, resembling differentiation, applicable to 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. In Europe, the foundational work was a treatise written by 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. The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time.
Pierre de Fermat , claiming that he borrowed from Diophantus , introduced the concept of adequality , which represented equality up to an infinitesimal error term. The product rule and chain rule ,  the notions of higher derivatives and Taylor series ,  and of analytic functions [ citation needed ] were used by Isaac Newton in an idiosyncratic notation which he applied to solve problems of mathematical physics.
In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid , and many other problems discussed in his Principia Mathematica In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the Taylor series.
He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable. These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz , who was originally accused of plagiarism by Newton. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule , in their differential and integral forms.
Unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series.
By Newton's time, the fundamental theorem of calculus was known. When Newton and Leibniz first published their results, there was great controversy over which mathematician and therefore which country deserved credit. Newton derived his results first later to be published in his Method of Fluxions , but Leibniz published his " Nova Methodus pro Maximis et Minimis " first.
Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus " the science of fluxions ".
Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and integral calculus was written in by Maria Gaetana Agnesi. 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 unrigorous, and was fiercely criticized by a number of authors, most notably Michel Rolle and Bishop Berkeley.
Berkeley famously described infinitesimals as the ghosts of departed quantities in his book The Analyst in 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 years later when, due to the work of Cauchy and Weierstrass , a way was finally found to avoid mere "notions" of infinitely small quantities.
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". Bernhard 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.
In modern mathematics, the foundations of calculus are included in the field of real analysis , which contains full definitions and proofs of the theorems of calculus. The reach of calculus has also been greatly extended. Henri Lebesgue invented measure theory and used it to define integrals of all but the most pathological functions.
Laurent Schwartz introduced distributions , which can be used to take the derivative of any function whatsoever. Limits are not the only rigorous approach to the foundation of calculus.
Another way is to use Abraham Robinson 's non-standard analysis. Robinson's approach, developed in the s, uses technical machinery from mathematical logic to augment the real number system with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The resulting numbers are called hyperreal numbers , and they can be used to give a Leibniz-like development of the usual rules of calculus.
There is also smooth infinitesimal analysis , which differs from non-standard analysis in that it mandates neglecting higher power infinitesimals during derivations. 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 , that resolve the paradoxes. Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like real numbers but which are, in some sense, "infinitely small". From this point of view, calculus is a collection of techniques for manipulating infinitesimals.
The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. However, the concept was revived in the 20th century with the introduction of non-standard analysis and smooth infinitesimal analysis , which provided solid foundations for the manipulation of infinitesimals.
In the late 19th century, infinitesimals were replaced within academia by the epsilon, delta approach to limits. Limits describe the value of a function at a certain input in terms of its values at nearby inputs. They capture small-scale behavior in the context of the real number system.
In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by very small numbers, and the infinitely small behavior of the function is found by taking the limiting behavior for smaller and smaller numbers. Limits were thought to provide a more rigorous foundation for calculus, and for this reason they became the standard approach during the twentieth century.
Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation.
Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near that point. By finding the derivative of a function at every point in its domain, it is possible to produce a new function, called the derivative function or just the derivative of the original function.
In formal terms, the derivative is a linear operator which takes a function as its input and produces a second function as its output.
This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine.
The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function.
The function produced by deriving the squaring function turns out to be the doubling function. The most common symbol for a derivative is an apostrophe -like mark called prime. This notation is known as Lagrange's notation. If the input of the function represents time, then the derivative represents change with respect to time. For example, if f is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of f is how the position is changing in time, that is, it is the velocity of the ball.
This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in y divided by the change in x varies. Derivatives give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let f be a function, and fix a point a in the domain of f.
Introduction to Integration
Show this both in symbolic proper mathematical form as well as in an illustration similar to that shown above. Follow-up question: this circuit will not work as shown if both R values are the same, and both C values are the same as well. Differential calculus Is a subfield of calculus concerned with the study of the rates at which quantities change. Do the next step. Both equations contain an I, and both equations also contain a V. The answer to that question can only be found by looking at the schematic diagram: do the resistor and capacitor share the same current, the same voltage, or both? Special Honors.
Examples of volumes of solids of revolution 5. Exercises Chapter 8. Example 9. Some people call it anti-differentiation.. Evaluating Integrals. Wait for the examples that follow. An integral is the reverse of a derivative, and integral calculus is the opposite of differential calculus.
Implicit Differentiation and Related Rates. Inverse Functions and Their Derivatives. Inverses of Trigonometric Functions. Integrals. The Idea of the Integral.
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A printable version of Calculus is available. A PDF version is available. Wikiversity has learning materials about Calculus. This wikibook aims to be a high quality calculus textbook through which users can master the discipline.
So, All of authors and contributors must check their papers before submission to making assurance of following our anti-plagiarism policies. In the first what is the calculus? Before defining the calculus we can divide it into two categories; the first is differentiation, but the second is the integration. The differentiation: that category relates with the rate of change and the slope of the graphs, whereas we use the limits in finding the derivative of any curve: -The integration: that category relates with the area under and between the graphs, whereas appeared many theorems like Riemann integration, Monte Carlo integration and the fundamental theorem of calculus. Why Us?
Integration can be used to find areas, volumes, central points and many useful things. But it is easiest to start with finding the area under the curve of a function like this:. So you should really know about Derivatives before reading more!
Мысли Стратмора судорожно метались в поисках решения. Всегда есть какой-то выход. Наконец он заговорил - спокойно, тихо и даже печально: - Нет, Грег, извини. Я не могу тебя отпустить.
- Я же угада… - Но она замолкла на полуслове.