Calculus is themathematical study of continuous change, in the same way thatgeometry is the study of shape, andalgebra is the study of generalizations ofarithmetic operations.

Originally calledinfinitesimal calculus or "the calculus ofinfinitesimals", it has two major branches,differential calculus andintegral calculus. The former concerns instantaneousrates of change, and theslopes ofcurves, while the latter concerns accumulation of quantities, andareas under or between curves. These two branches are related to each other by thefundamental theorem of calculus. They make use of the fundamental notions ofconvergence ofinfinite sequences andinfinite series to a well-definedlimit.^[1] It is the "mathematical backbone" for dealing with problems where variables change with time or another reference variable.^[2]

Infinitesimal calculus was formulated separately in the late 17th century byIsaac Newton andGottfried Wilhelm Leibniz.^[3][4] Later work, includingcodifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus is widely used inscience,engineering,biology, and even has applications insocial science and other branches of math.^[5][6]

Inmathematics education,calculus is an abbreviation of bothinfinitesimal calculus andintegral calculus, which denotes courses of elementarymathematical analysis.

InLatin, the wordcalculus means “small pebble”, (thediminutive ofcalx, meaning "stone"), a meaning which stillpersists in medicine. Because such pebbles were used for counting out distances,^[7] tallying votes, and doingabacus arithmetic, the word came to be the Latin word forcalculation. In this sense, it was used in English at least as early as 1672, several years before the publications of Leibniz and Newton, who wrote their mathematical texts in Latin.^[8]

In addition to differential calculus and integral calculus, the term is also used for naming specific methods of computation or theories that imply some sort of computation. Examples of this usage includepropositional calculus,Ricci calculus,calculus of variations,lambda calculus,sequent calculus, andprocess calculus. Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems asBentham'sfelicific calculus, and theethical calculus.

Modern calculus was developed in 17th-century Europe byIsaac Newton andGottfried Wilhelm Leibniz (independently of each other, first publishing around the same time) but elements of it first appeared in ancient Egypt and later Greece, then in China and the Middle East, and still later again in medieval Europe and India.

Calculations ofvolume andarea, one goal of integral calculus, can be found in theEgyptianMoscow papyrus (c. 1820 BC), but the formulae are simple instructions, with no indication as to how they were obtained.[9]^[10]

Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematicianEudoxus of Cnidus (c. 390–337 BC) developed themethod of exhaustion to prove the formulas for cone and pyramid volumes.

During theHellenistic period, this method was further developed byArchimedes (c. 287 – c. 212 BC), who combined it with a concept of theindivisibles—a precursor toinfinitesimals—allowing him to solve several problems now treated by integral calculus. InThe Method of Mechanical Theorems he describes, for example, calculating thecenter of gravity of a solidhemisphere, the center of gravity of afrustum of a circularparaboloid, and the area of a region bounded by aparabola and one of itssecant lines.^[11]

The method of exhaustion was later discovered independently inChina byLiu Hui in the 3rd century AD to find the area of a circle.^[12][13] In the 5th century AD,Zu Gengzhi, son ofZu Chongzhi, established a method^[14][15] that would later be calledCavalieri's principle to find the volume of asphere.

In the Middle East,Hasan Ibn al-Haytham, Latinized as Alhazen (c. 965 – c. 1040 AD) derived a formula for the sum offourth powers. He determined the equations to calculate the area enclosed by the curve represented by${\displaystyle y=x^k}$  (which translates to the integral${\displaystyle \int x^{k}dx}$  in contemporary notation), for any given non-negative integer value of${\displaystyle k}$ .^[16]He used the results to carry out what would now be called anintegration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of aparaboloid.^[17]

Bhāskara II (c. 1114–1185) was acquainted with some ideas of differential calculus and suggested that the "differential coefficient" vanishes at an extremum value of the function.^[18] In his astronomical work, he gave a procedure that looked like a precursor to infinitesimal methods. Namely, if${\displaystyle x\approx y}$  then${\displaystyle \sin (y)-\sin (x)\approx (y-x)\cos (y).}$  This can be interpreted as the discovery thatcosine is the derivative ofsine.^[19] In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions.Madhava of Sangamagrama and theKerala School of Astronomy and Mathematics stated components of calculus. They studied series equivalent to the Maclaurin expansions of⁠${\displaystyle \sin (x)}$ ⁠,⁠${\displaystyle \cos (x)}$ ⁠, and⁠${\displaystyle \arctan (x)}$ ⁠ more than two hundred years before their introduction in Europe.^[20] According toVictor J. Katz they were not able to "combine many differing ideas under the two unifying themes of thederivative and theintegral, show the connection between the two, and turn calculus into the great problem-solving tool we have today".^[17]

Johannes Kepler's workStereometria Doliorum (1615) formed the basis of integral calculus.^[21] Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse.^[22]

Significant work was a treatise, the origin being Kepler's methods,^[22] written byBonaventura 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' inThe 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 thecalculus of finite differences developed in Europe at around the same time.Pierre de Fermat, claiming that he borrowed fromDiophantus, introduced the concept ofadequality, which represented equality up to an infinitesimal error term.^[23] The combination was achieved byJohn Wallis,Isaac Barrow, andJames Gregory, the latter two proving predecessors to thesecond fundamental theorem of calculus around 1670.^[24][25]

Theproduct rule andchain rule,^[26] the notions ofhigher derivatives andTaylor series,^[27] and ofanalytic functions^[28] were used byIsaac Newton in an idiosyncratic notation which he applied to solve problems ofmathematical 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 acycloid, and many other problems discussed in hisPrincipia Mathematica (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of theTaylor series. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.^[29]

These ideas were arranged into a true calculus of infinitesimals byGottfried Wilhelm Leibniz, who was originally accused ofplagiarism by Newton.^[30] He is now regarded as anindependent inventor of and contributor to calculus. 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 theproduct rule andchain rule, in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation.^[31]

Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to generalphysics. Leibniz developed much of the notation used in calculus today.^{[32]: 51–52} The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series.

When Newton and Leibniz first published their results, there wasgreat controversy over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be published in hisMethod 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 theRoyal Society. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics.^[33] A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of fluxions", a term that endured in English schools into the 19th century.^[34]: 100 The first complete treatise on calculus to be written in English and use the Leibniz notation was not published until 1815.^[35]

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 andintegral calculus was written in 1748 byMaria Gaetana Agnesi.^[36][37]

In calculus,foundations refers to therigorous development of the subject fromaxioms and definitions. In early calculus, the use ofinfinitesimal quantities was thought unrigorous and was fiercely criticized by several authors, most notablyMichel Rolle andBishop Berkeley. Berkeley famously described infinitesimals as theghosts of departed quantities in his bookThe 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.^[38]

Several mathematicians, includingMaclaurin, tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work ofCauchy andWeierstrass, a way was finally found to avoid mere "notions" of infinitely small quantities.^[39] The foundations of differential and integral calculus had been laid. In Cauchy'sCours d'Analyse, we find a broad range of foundational approaches, including a definition ofcontinuity in terms of infinitesimals, and a (somewhat imprecise) prototype of an(ε, δ)-definition of limit in the definition of differentiation.^[40] In his work, Weierstrass formalized the concept oflimit and eliminated infinitesimals (although his definition can validatenilsquare 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".Bernhard Riemann used these ideas to give a precise definition of the integral.^[41] It was also during this period that the ideas of calculus were generalized to thecomplex plane with the development ofcomplex analysis.^[42]

In modern mathematics, the foundations of calculus are included in the field ofreal analysis, which contains full definitions andproofs of the theorems of calculus. The reach of calculus has also been greatly extended.Henri Lebesgue inventedmeasure theory, based on earlier developments byÉmile Borel, and used it to define integrals of all but the mostpathological functions.^[43]Laurent Schwartz introduceddistributions, which can be used to take the derivative of any function whatsoever.^[44]

Limits are not the only rigorous approach to the foundation of calculus. Another way is to useAbraham Robinson'snon-standard analysis. Robinson's approach, developed in the 1960s, uses technical machinery frommathematical logic to augment the real number system withinfinitesimal andinfinite numbers, as in the original Newton-Leibniz conception. The resulting numbers are calledhyperreal numbers, and they can be used to give a Leibniz-like development of the usual rules of calculus.^[45] There is alsosmooth infinitesimal analysis, which differs from non-standard analysis in that it mandates neglecting higher-power infinitesimals during derivations.^[38] Based on the ideas ofF. W. Lawvere and employing the methods ofcategory theory, smooth infinitesimal analysis views all functions as beingcontinuous and incapable of being expressed in terms ofdiscrete entities. One aspect of this formulation is that thelaw of excluded middle does not hold.^[38] The law of excluded middle is also rejected inconstructive mathematics, 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. Reformulations of calculus in a constructive framework are generally part of the subject ofconstructive analysis.^[38]

While many of the ideas of calculus had been developed earlier inGreece,China,India,Iraq, Persia, andJapan, the use of calculus began in Europe, during the 17th century, when Newton and Leibniz built on the work of earlier mathematicians to introduce its basic principles.^[13][29][46] The Hungarian polymathJohn von Neumann wrote of this work,

The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. 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.^[47]

Applications of differential calculus include computations involvingvelocity andacceleration, theslope of a curve, andoptimization.^{[48]: 341–453} Applications of integral calculus include computations involving area,volume,arc length,center of mass,work, andpressure.^{[48]: 685–700} More advanced applications includepower series andFourier 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 involvingdivision by zero or sums of infinitely many numbers. These questions arise in the study ofmotion and area. Theancient Greek philosopherZeno of Elea gave several famous examples of suchparadoxes. Calculus provides tools, especially thelimit and theinfinite series, that resolve the paradoxes.^[49]

Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was byinfinitesimals. These are objects which can be treated like real numbers but which are, in some sense, "infinitely small". For example, an infinitesimal number could be greater than 0, but less than any number in the sequence 1, 1/2, 1/3, ... and thus less than any positivereal number. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols${\displaystyle dx}$  and${\displaystyle dy}$  were taken to be infinitesimal, and the derivative${\displaystyle dy/dx}$  was their ratio.^[38]

The infinitesimal approach fell out of favor in the 19th century because it was difficult to make the notion of an infinitesimal precise. In the late 19th century, infinitesimals were replaced within academia by theepsilon, delta approach tolimits. Limits describe the behavior of afunction at a certain input in terms of its values at nearby inputs. They capture small-scale behavior using the intrinsic structure of thereal number system (as ametric space with theleast-upper-bound property). In this treatment, calculus is a collection of techniques for manipulating certain limits. Infinitesimals get replaced by sequences of smaller and smaller numbers, and the infinitely small behavior of a function is found by taking the limiting behavior for these sequences. Limits were thought to provide a more rigorous foundation for calculus, and for this reason, they became the standard approach during the 20th century. However, the infinitesimal concept was revived in the 20th century with the introduction ofnon-standard analysis andsmooth infinitesimal analysis, which provided solid foundations for the manipulation of infinitesimals.^[38]

Differential calculus is the study of the definition, properties, and applications of thederivative of a function. The process of finding the derivative is calleddifferentiation. 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 thederivative function or just thederivative of the original function. In formal terms, the derivative is alinear 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 differentiating the squaring function turns out to be the doubling function.^[32]: 32

In more explicit terms the "doubling function" may be denoted byg(x) = 2x and the "squaring function" byf(x) =x². The "derivative" now takes the functionf(x), defined by the expression "x²", as an input, that is all the information—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 output another function, the functiong(x) = 2x, as will turn out.

InLagrange's notation, the symbol for a derivative is anapostrophe-like mark called aprime. Thus, the derivative of a function calledf is denoted byf′, pronounced "f prime" or "f dash". For instance, iff(x) =x² is the squaring function, thenf′(x) = 2x is its derivative (the doubling functiong from above).

If the input of the function represents time, then the derivative represents change concerning time. For example, iff is a function that takes time as input and gives the position of a ball at that time as output, then the derivative off is how the position is changing in time, that is, it is thevelocity of the ball.^{[32]: 18–20}

If a function islinear (that is if thegraph of the function is a straight line), then the function can be written asy =mx +b, wherex is the independent variable,y is the dependent variable,b is they-intercept, and:

${\displaystyle m=\frac{\text{rise}}{\text{run}}=\frac{\text{change in }y}{\text{change in }x}=\frac{\mathrm{\Delta }y}{\mathrm{\Delta }x}.}$ 

This gives an exact value for the slope of a straight line.^[50]: 6 If the graph of the function is not a straight line, however, then the change iny divided by the change inx varies. Derivatives give an exact meaning to the notion of change in output concerning change in input. To be concrete, letf be a function, and fix a pointa in the domain off.(a,f(a)) is a point on the graph of the function. Ifh is a number close to zero, thena +h is a number close toa. Therefore,(a +h,f(a +h)) is close to(a,f(a)). The slope between these two points is

${\displaystyle m=\frac{f(a+h)-f(a)}{(a+h)-a}=\frac{f(a+h)-f(a)}{h}.}$ 

This expression is called adifference quotient. A line through two points on a curve is called asecant line, som is the slope of the secant line between(a,f(a)) and(a +h,f(a +h)). The second line is only an approximation to the behavior of the function at the point a because it does not account for what happens between a and a +h. It is not possible to discover the behavior at a by settingh to zero because this would requiredividing by zero, which is undefined. The derivative is defined by taking thelimit ash tends to zero, meaning that it considers the behavior off for all small values ofh and extracts a consistent value for the case whenh equals zero:

${\displaystyle \underset{h\to 0}{lim}\frac{f(a+h)-f(a)}{h}.}$ 

Geometrically, the derivative is the slope of thetangent line to the graph off ata. 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 functionf.^{[50]: 61–63}

Here is a particular example, the derivative of the squaring function at the input 3. Letf(x) =x² be the squaring function.

 

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 thederivative function of the squaring function or just thederivative 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.^[50]: 63

A common notation, introduced by Leibniz, for the derivative in the example above is

 

In an approach based on limits, the symbol⁠dy/ dx⁠ is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above.^[50]: 74 Leibniz, however, did intend it to represent the quotient of two infinitesimally small numbers,dy being the infinitesimally small change iny caused by an infinitesimally small change dx applied tox. We can also think of⁠d/ dx⁠ as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example:

${\displaystyle \frac{d}{dx}(x^2)=2x.}$ 

In this usage, thedx in the denominator is read as "with respect tox".^[50]: 79 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 anddy as if they were real numbers; although it is possible to avoid such manipulations, they are sometimes notationally convenient in expressing operations such as thetotal derivative.

Integral calculus is the study of the definitions, properties, and applications of two related concepts, theindefinite integral and thedefinite integral. The process of finding the value of an integral is calledintegration.^[48]: 508 The indefinite integral, also known as theantiderivative, is the inverse operation to the derivative.^{[50]: 163–165}F is an indefinite integral off whenf is a derivative ofF. (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 thex-axis. The technical definition of the definite integral involves thelimit of a sum of areas of rectangles, called aRiemann sum.^[51]: 282

A motivating example is the distance traveled in a given time.^[50]: 153 If the speed is constant, only multiplication is needed:

${\displaystyle \mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{e}\mathrm{d}\cdot \mathrm{T}\mathrm{i}\mathrm{m}\mathrm{e}}$ 

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 (aRiemann 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. For example, traveling a steady 50 mph for 3 hours results in a total distance of 150 miles. Plotting the velocity as a function of time yields a rectangle with a height equal to the velocity and a width equal to the time elapsed. Therefore, the product of velocity and time also calculates the rectangular area under the (constant) velocity curve.^[48]: 535 This connection between the area under a curve and the distance traveled can be extended toany irregularly shaped region exhibiting a fluctuating velocity over a given period. Iff(x) represents speed as it varies over time, the distance traveled between the times represented by a andb is the area of the region betweenf(x) and thex-axis, betweenx =a andx =b.

To approximate that area, an intuitive method would be to divide up the distance between a andb into several equal segments, the length of each segment represented by the symbolΔx. For each small segment, we can choose one value of the functionf(x). Call that valueh. Then the area of the rectangle with baseΔx and heighth gives the distance (timeΔx multiplied by speedh) traveled in that segment. Associated with each segment is the average value of the function above it,f(x) =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. A smaller value forΔx will give more rectangles and in most cases a better approximation, but for an exact answer, we need to take a limit asΔx approaches zero.^{[48]: 512–522}

The symbol of integration is${\displaystyle \int }$ , anelongatedS chosen to suggest summation.^[48]: 529 The definite integral is written as:

${\displaystyle {\int }_a^{b}f(x)dx}$ 

and is read "the integral froma tob off-of-x with respect tox." The Leibniz notationdx is intended to suggest dividing the area under the curve into an infinite number of rectangles so that their widthΔx becomes the infinitesimally smalldx.^[32]: 44

The indefinite integral, or antiderivative, is written:

${\displaystyle \int f(x)dx.}$ 

Functions differing by only a constant have the same derivative, and it can be shown that the antiderivative of a given function is a family of functions differing only by a constant.^[51]: 326 Since the derivative of the functiony =x² +C, whereC is any constant, isy′ = 2x, the antiderivative of the latter is given by:

${\displaystyle \int 2xdx=x^2+C.}$ 

The unspecified constantC present in the indefinite integral or antiderivative is known as theconstant of integration.^[52]: 135

Thefundamental theorem of calculus states that differentiation and integration are inverse operations.^[51]: 290 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 functionf iscontinuous on the interval[a,b] and ifF is a function whose derivative isf on the interval(a,b), then

${\displaystyle {\int }_a^{b}f(x)dx=F(b)-F(a).}$ 

Furthermore, for everyx in the interval(a,b),

${\displaystyle \frac{d}{dx}{\int }_a^{x}f(t)dt=f(x).}$ 

This realization, made by bothNewton andLeibniz, was key to the proliferation of analytic results after their work became known. (The extent to which Newton and Leibniz were influenced by immediate predecessors, and particularly what Leibniz may have learned from the work ofIsaac Barrow, is difficult to determine because of the priority dispute between them.^[53]) The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulae forantiderivatives. It is also a prototype solution of adifferential equation. Differential equations relate an unknown function to its derivatives and are ubiquitous in the sciences.^{[54]: 351–352}

Calculus is used in every branch of the physical sciences,^[55]: 1actuarial science,computer science,statistics,engineering,economics,business,medicine,demography, and in other fields wherever a problem can bemathematically modeled and anoptimal solution is desired.^[56] 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.^[57] Calculus can be used in conjunction with other mathematical disciplines. For example, it can be used withlinear algebra to find the "best fit" linear approximation for a set of points in a domain. Or, it can be used inprobability theory to determine theexpectation value of a continuous random variable given aprobability density function.^[58]: 37 Inanalytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope,concavity andinflection points. 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 asNewton's method,fixed point iteration, andlinear approximation. For instance, spacecraft use a variation of theEuler method to approximate curved courses within zero-gravity environments.

Physics makes particular use of calculus; all concepts inclassical mechanics andelectromagnetism are related through calculus. Themass of an object of knowndensity, themoment of inertia of objects, and thepotential energies due to gravitational and electromagnetic forces can all be found by the use of calculus. An example of the use of calculus in mechanics isNewton's second law of motion, which states that the derivative of an object'smomentum concerning time equals the netforce upon it. Alternatively, Newton's second law can be expressed by saying that the net force equals the object's mass times itsacceleration, which is the time derivative of velocity and thus the second time derivative of spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path.^[59]

Maxwell's theory ofelectromagnetism andEinstein's theory ofgeneral relativity are also expressed in the language of differential calculus.^{[60][61]: 52–55} Chemistry also uses calculus in determining reaction rates^[62]: 599 and in studying radioactive decay.^[62]: 814 In biology, population dynamics starts with reproduction and death rates to model population changes.^{[63][64]: 631}

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 aplanimeter, which is used to calculate the area of a flat surface on a drawing.^[65] 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.

In the realm of medicine, calculus can be used to find the optimal branching angle of ablood vessel to maximize flow.^[66] Calculus can be applied to understand how quickly a drug is eliminated from a body or how quickly acancerous tumor grows.^[67]

In economics, calculus allows for the determination of maximal profit by providing a way to easily calculate bothmarginal cost andmarginal revenue.^[68]: 387

• Glossary of calculus
• List of calculus topics
• List of derivatives and integrals in alternative calculi
• List of differentiation identities
• Publications in calculus
• Table of integrals

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