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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.
The slope between these two points is. This expression is called a difference quotient. It is not possible to discover the behavior at a by setting h to zero because this would require dividing by zero , which is undefined.
The derivative is defined by taking the limit as h tends to zero, meaning that it considers the behavior of f for all small values of h and extracts a consistent value for the case when h equals zero:. Geometrically, the derivative is the slope of the tangent line to the graph of f at a.
The tangent line is a limit of secant lines just as the derivative is a limit of difference quotients. For this reason, the derivative is sometimes called the slope of the function f. Here is a particular example, the derivative of the squaring function at the input 3. The slope of the tangent line to the squaring function at the point 3, 9 is 6, that is to say, it is going up six times as fast as it is going to the right.
The limit process just described can be performed for any point in the domain of the squaring function. This defines the derivative function of the squaring function, or just the derivative of the squaring function for short. A computation similar to the one above shows that the derivative of the squaring function is the doubling function.
Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers, dy being the infinitesimally small change in y caused by an infinitesimally small change dx applied to x.
For example:. In this usage, the dx in the denominator is read as "with respect to x ". Another example of correct notation could be:. Even when calculus is developed using limits rather than infinitesimals, it is common to manipulate symbols like dx and dy as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as the total derivative.
Integral calculus is the study of the definitions, properties, and applications of two related concepts, the indefinite integral and the definite integral. The process of finding the value of an integral is called integration. In technical language, integral calculus studies two related linear operators. The indefinite integral , also known as the antiderivative , is the inverse operation to the derivative.
F is an indefinite integral of f when f is a derivative of F. This use of lower- and upper-case letters for a function and its indefinite integral is common in calculus.
The definite integral inputs a function and outputs a number, which gives the algebraic sum of areas between the graph of the input and the x-axis. The technical definition of the definite integral involves the limit of a sum of areas of rectangles, called a Riemann sum.
If the speed is constant, only multiplication is needed, but if the speed changes, a more powerful method of finding the distance is necessary.
One such method is to approximate the distance traveled by breaking up the time into many short intervals of time, then multiplying the time elapsed in each interval by one of the speeds in that interval, and then taking the sum a Riemann sum of the approximate distance traveled in each interval.
The basic idea is that if only a short time elapses, then the speed will stay more or less the same. However, a Riemann sum only gives an approximation of the distance traveled. We must take the limit of all such Riemann sums to find the exact distance traveled. When velocity is constant, the total distance traveled over the given time interval can be computed by multiplying velocity and time. In the diagram on the left, when constant velocity and time are graphed, these two values form a rectangle with height equal to the velocity and width equal to the time elapsed.
Therefore, the product of velocity and time also calculates the rectangular area under the constant velocity curve. This connection between the area under a curve and distance traveled can be extended to any irregularly shaped region exhibiting a fluctuating velocity over a given time period. For each small segment, we can choose one value of the function f x.
Call that value h. The sum of all such rectangles gives an approximation of the area between the axis and the curve, which is an approximation of the total distance traveled. The definite integral is written as:. In a formulation of the calculus based on limits, the notation. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator.
Functions differing by only a constant have the same derivative, and it can be shown that the antiderivative of a given function is actually a family of functions differing only by a constant.
The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration. The fundamental theorem of calculus states that differentiation and integration are inverse operations. More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the fundamental theorem of calculus provides a practical way of computing definite integrals.
It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration. The fundamental theorem of calculus states: If a function f is continuous on the interval [ a , b ] and if F is a function whose derivative is f on the interval a , b , then.
Furthermore, for every x in the interval a , b ,. This realization, made by both Newton and Leibniz , who based their results on earlier work by Isaac Barrow , was key to the proliferation of analytic results after their work became known. The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulas for antiderivatives.
It is also a prototype solution of a differential equation. Differential equations relate an unknown function to its derivatives, and are ubiquitous in the sciences. Calculus is used in every branch of the physical sciences, actuarial science , computer science , statistics , engineering , economics , business , medicine , demography , and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired.
It allows one to go from non-constant rates of change to the total change or vice versa, and many times in studying a problem we know one and are trying to find the other. Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are related through calculus.
The mass of an object of known density , the moment of inertia of objects, as well as the total energy of an object within a conservative field can be found by the use of calculus.
An example of the use of calculus in mechanics is Newton's second law of motion: Starting from knowing how an object is accelerating, we use calculus to derive its path.
Maxwell's theory of electromagnetism and Einstein 's theory of general relativity are also expressed in the language of differential calculus. Chemistry also uses calculus in determining reaction rates and radioactive decay.
In biology, population dynamics starts with reproduction and death rates to model population changes. Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used with linear algebra to find the "best fit" linear approximation for a set of points in a domain.
Or it can be used in probability theory to determine the probability of a continuous random variable from an assumed density function. In analytic geometry , the study of graphs of functions, calculus is used to find high points and low points maxima and minima , slope, concavity and inflection points. Green's Theorem , which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a planimeter , which is used to calculate the area of a flat surface on a drawing.
For example, it can be used to calculate the amount of area taken up by an irregularly shaped flower bed or swimming pool when designing the layout of a piece of property.
Discrete Green's Theorem , which gives the relationship between a double integral of a function around a simple closed rectangular curve C and a linear combination of the antiderivative's values at corner points along the edge of the curve, allows fast calculation of sums of values in rectangular domains.
For example, it can be used to efficiently calculate sums of rectangular domains in images, in order to rapidly extract features and detect object; another algorithm that could be used is the summed area table. In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow.
From the decay laws for a particular drug's elimination from the body, it is used to derive dosing laws. In nuclear medicine, it is used to build models of radiation transport in targeted tumor therapies. In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate both marginal cost and marginal revenue.
Calculus is also used to find approximate solutions to equations; in practice it is the standard way to solve differential equations and do root finding in most applications. Examples are methods such as Newton's method , fixed point iteration , and linear approximation. For instance, spacecraft use a variation of the Euler method to approximate curved courses within zero gravity environments.
Meanwhile, calculations with infinitesimals persisted and often led to correct results. This led Abraham Robinson to investigate if it were possible to develop a number system with infinitesimal quantities over which the theorems of calculus were still valid.
The theory of non-standard analysis is rich enough to be applied in many branches of mathematics. As such, books and articles dedicated solely to the traditional theorems of calculus often go by the title non-standard calculus. This is another reformulation of the calculus in terms of infinitesimals. Based on the ideas of F.
Lawvere and employing the methods of category theory , it views all functions as being continuous and incapable of being expressed in terms of discrete entities. One aspect of this formulation is that the law of excluded middle does not hold in this formulation.
Constructive mathematics is a branch of mathematics that insists that proofs of the existence of a number, function, or other mathematical object should give a construction of the object. As such constructive mathematics also rejects the law of excluded middle. Reformulations of calculus in a constructive framework are generally part of the subject of constructive analysis. From Wikipedia, the free encyclopedia.
This article is about the branch of mathematics. For other uses, see Calculus disambiguation. Branch of mathematics.
Limits of functions Continuity.
Mean value theorem Rolle's theorem. Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem.
Fractional Malliavin Stochastic Variations. Glossary of calculus. Main article: History of calculus. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking. Main articles: Limit of a function and Infinitesimal.
Differential calculus. Leibniz's notation. Fundamental theorem of calculus. Non-standard calculus. Smooth infinitesimal analysis. Constructive analysis. Outline of calculus. Retrieved 15 September An Introduction to the History of Mathematics 4th ed.
New York: Holt, Rinehart and Winston. A brief introduction to the infinitesimal calculus. The Macmillan Company. A comparison of Archimdes' and Liu Hui's studies of circles. Chinese studies in the history and philosophy of science and technology.
A history of mathematics 3rd ed. Boston, MA: Early Transcendentals 3 ed. Number theory. An approach through history. From Hammurapi to Legendre. Birkhauser Boston, Inc.
Single Variable, Volume 1 illustrated ed. Extract of p. The Early Mathematical Manuscripts of Leibniz. Cosimo, Inc. By Cupillari, Antonella illustrated ed.
Edwin Mellen Press. Agnes Scott College. History of Western Philosophy. The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibniz believed in actual infinitesimals, but although this belief suited his metaphysics it had no sound basis in mathematics.
Weierstrass, soon after the middle of the nineteenth century, showed how to establish the calculus without infinitesimals, and thus at last made it logically secure. Next came Georg Cantor, who developed the theory of continuity and infinite number. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicists. Jan 05, Anita rated it it was amazing.
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