Contents 1 History 1.1 Ancient 1.2 Medieval 1.3 Modern 1.4 Foundations 1.5 Significance 2 Principles 2.1 Limits and infinitesimals 2.2 Differential calculus 2.3 Leibniz notation 2.4 Integral calculus 2.5 Fundamental theorem 3 Applications 4 Varieties 4.1 Non-standard calculus 4.2 Smooth infinitesimal analysis 4.3 Constructive analysis 5 See also 5.1 Lists 5.2 Other related topics 6 References 7 Further reading 7.1 Books 7.2 Online books 8 External links

Principles Limits and infinitesimals Main articles: Limit of a function and Infinitesimal 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". 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 positive real number. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols dx and dy were taken to be infinitesimal, and the derivative d y / d x {\displaystyle dy/dx} was simply their ratio. 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 19th century, infinitesimals were replaced 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 the first way to provide rigorous foundations for calculus, and for this reason they are the standard approach. Differential calculus Main article: Differential calculus Tangent line at (x, f(x)). The derivative f′(x) of a curve at a point is the slope (rise over run) of the line tangent to that curve at that point. 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. In more explicit terms the "doubling function" may be denoted by g(x) = 2x and the "squaring function" by f(x) = x2. The "derivative" now takes the function f(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 function g(x) = 2x, as will turn out. The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of a function called f is denoted by f′, pronounced "f prime". For instance, if f(x) = x2 is the squaring function, then f′(x) = 2x is its derivative (the doubling function g from above). 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. If a function is linear (that is, if the graph of the function is a straight line), then the function can be written as y = mx + b, where x is the independent variable, y is the dependent variable, b is the y-intercept, and: m = rise run = change in  y change in  x = Δ y Δ x . {\displaystyle m={\frac {\text{rise}}{\text{run}}}={\frac {{\text{change in }}y}{{\text{change in }}x}}={\frac {\Delta y}{\Delta x}}.} 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. (a, f(a)) is a point on the graph of the function. If h is a number close to zero, then a + h is a number close to a. Therefore, (a + h, f(a + h)) is close to (a, f(a)). The slope between these two points is m = f ( a + h ) − f ( a ) ( a + h ) − a = f ( a + h ) − f ( a ) h . {\displaystyle m={\frac {f(a+h)-f(a)}{(a+h)-a}}={\frac {f(a+h)-f(a)}{h}}.} This expression is called a difference quotient. A line through two points on a curve is called a secant line, so m is the slope of the secant line between (a, f(a)) and (a + h, f(a + h)). The secant 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 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: lim h → 0 f ( a + h ) − f ( a ) h . {\displaystyle \lim _{h\to 0}{f(a+h)-f(a) \over {h}}.} 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. Let f(x) = x2 be the squaring function. The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is determined by considering the limiting value of the slopes of secant lines. Here the function involved (drawn in red) is f(x) = x3 − x. The tangent line (in green) which passes through the point (−3/2, −15/8) has a slope of 23/4. Note that the vertical and horizontal scales in this image are different. f ′ ( 3 ) = lim h → 0 ( 3 + h ) 2 − 3 2 h = lim h → 0 9 + 6 h + h 2 − 9 h = lim h → 0 6 h + h 2 h = lim h → 0 ( 6 + h ) = 6. {\displaystyle {\begin{aligned}f'(3)&=\lim _{h\to 0}{(3+h)^{2}-3^{2} \over {h}}\\&=\lim _{h\to 0}{9+6h+h^{2}-9 \over {h}}\\&=\lim _{h\to 0}{6h+h^{2} \over {h}}\\&=\lim _{h\to 0}(6+h)\\&=6.\end{aligned}}} 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 notation Main article: Leibniz's notation A common notation, introduced by Leibniz, for the derivative in the example above is y = x 2 d y d x = 2 x . {\displaystyle {\begin{aligned}y&=x^{2}\\{\frac {dy}{dx}}&=2x.\end{aligned}}} In an approach based on limits, the symbol dy/dx is to be interpreted not as the quotient of two numbers but as a shorthand for the limit computed above. 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. We can also think of d/dx as a differentiation operator, which takes a function as an input and gives another function, the derivative, as the output. For example: d d x ( x 2 ) = 2 x . {\displaystyle {\frac {d}{dx}}(x^{2})=2x.} In this usage, the dx in the denominator is read as "with respect to x". 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 Main article: Integral 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. A motivating example is the distances traveled in a given time. D i s t a n c e = S p e e d ⋅ T i m e {\displaystyle \mathrm {Distance} =\mathrm {Speed} \cdot \mathrm {Time} } 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. Constant velocity Integration can be thought of as measuring the area under a curve, defined by f(x), between two points (here a and b). When velocity is constant, the total distance traveled over the given time interval can be computed by multiplying velocity and time. For example, travelling a steady 50 mph for 3 hours results in a total distance of 150 miles. 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. If f(x) in the diagram on the right represents speed as it varies over time, the distance traveled (between the times represented by a and b) is the area of the shaded region s. To approximate that area, an intuitive method would be to divide up the distance between a and b into a number of equal segments, the length of each segment represented by the symbol Δx. For each small segment, we can choose one value of the function f(x). Call that value h. Then the area of the rectangle with base Δx and height h gives the distance (time Δx multiplied by speed h) 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 for Δx will give more rectangles and in most cases a better approximation, but for an exact answer we need to take a limit as Δx approaches zero. The symbol of integration is ∫ {\displaystyle \int } , an elongated S (the S stands for "sum"). The definite integral is written as: ∫ a b f ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\,dx.} and is read "the integral from a to b of f-of-x with respect to x." The Leibniz notation dx is intended to suggest dividing the area under the curve into an infinite number of rectangles, so that their width Δx becomes the infinitesimally small dx. In a formulation of the calculus based on limits, the notation ∫ a b ⋯ d x {\displaystyle \int _{a}^{b}\cdots \,dx} is to be understood as an operator that takes a function as an input and gives a number, the area, as an output. The terminating differential, dx, is not a number, and is not being multiplied by f(x), although, serving as a reminder of the Δx limit definition, it can be treated as such in symbolic manipulations of the integral. Formally, the differential indicates the variable over which the function is integrated and serves as a closing bracket for the integration operator. The indefinite integral, or antiderivative, is written: ∫ f ( x ) d x . {\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 actually a family of functions differing only by a constant. Since the derivative of the function y = x2 + C, where C is any constant, is y′ = 2x, the antiderivative of the latter given by: ∫ 2 x d x = x 2 + C . {\displaystyle \int 2x\,dx=x^{2}+C.} The unspecified constant C present in the indefinite integral or antiderivative is known as the constant of integration. Fundamental theorem Main article: Fundamental theorem of calculus 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 ∫ a b f ( x ) d x = F ( b ) − F ( a ) . {\displaystyle \int _{a}^{b}f(x)\,dx=F(b)-F(a).} Furthermore, for every x in the interval (a, b), d d x ∫ a x f ( t ) d t = f ( x ) . {\displaystyle {\frac {d}{dx}}\int _{a}^{x}f(t)\,dt=f(x).} 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.

Applications The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus. 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: historically stated it expressly uses the term "change of motion" which implies the derivative saying The change of momentum of a body is equal to the resultant force acting on the body and is in the same direction. Commonly expressed today as Force = Mass × acceleration, it implies differential calculus because acceleration is the time derivative of velocity or second time derivative of trajectory or spatial position. 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.

Varieties Over the years, many reformulations of calculus have been investigated for different purposes. Non-standard calculus Main article: Non-standard calculus Imprecise calculations with infinitesimals were widely replaced with the rigorous (ε, δ)-definition of limit starting in the 1870s. 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. In 1960, building upon the work of Edwin Hewitt and Jerzy Łoś, he succeeded in developing non-standard analysis. 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. Smooth infinitesimal analysis Main article: Smooth infinitesimal analysis This is another reformulation of the calculus in terms of infinitesimals. Based on the ideas of F. W. 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 analysis Main article: Constructive analysis 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.

See also Mathematics portal Analysis portal Main article: Outline of calculus Lists 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 Other related topics Calculus of finite differences Calculus with polynomials Complex analysis Differential equation Differential geometry Elementary Calculus: An Infinitesimal Approach Fourier series Integral equation Mathematical analysis Multiplicative calculus Multivariable calculus Non-classical analysis Non-Newtonian calculus Non-standard analysis Non-standard calculus Precalculus (mathematical education) Product integral Stochastic calculus Taylor series

References ^ "Calculus". OxfordDictionaries. Retrieved 15 September 2017.  ^ "Differential Calculus - Definition of Differential calculus by Merriam-Webster". Retrieved 15 September 2017.  ^ "Integral Calculus - Definition of Integral calculus by Merriam-Webster". Retrieved 15 September 2017.  ^ Fisher, Irving (1897). A brief introduction to the infinitesimal calculus. New York: The Macmillan Company.  ^ Morris Kline, Mathematical thought from ancient to modern times, Vol. I ^ Archimedes, Method, in The Works of Archimedes ISBN 978-0-521-66160-7 ^ Dun, Liu; Fan, Dainian; Cohen, Robert Sonné (1966). "A comparison of Archimdes' and Liu Hui's studies of circles". Chinese studies in the history and philosophy of science and technology. 130. Springer: 279. ISBN 0-7923-3463-9. ,Chapter , p. 279 ^ Katz, Victor J. (2008). A history of mathematics (3rd ed.). Boston, Mass.: Addison-Wesley. p. 203. ISBN 0-321-38700-7.  ^ Zill, Dennis G.; Wright, Scott; Wright, Warren S. (2009). Calculus: Early Transcendentals (3 ed.). Jones & Bartlett Learning. p. xxvii. ISBN 0-7637-5995-3.  Extract of page 27 ^ a b Katz, V. J. 1995. "Ideas of Calculus in Islam and India." Mathematics Magazine (Mathematical Association of America), 68(3):163–174. ^ "Indian mathematics".  ^ von Neumann, J., "The Mathematician", in Heywood, R. B., ed., The Works of the Mind, University of Chicago Press, 1947, pp. 180–196. Reprinted in Bródy, F., Vámos, T., eds., The Neumann Compedium, World Scientific Publishing Co. Pte. Ltd., 1995, ISBN 981-02-2201-7, pp. 618–626. ^ André Weil: Number theory. An approach through history. From Hammurapi to Legendre. Birkhauser Boston, Inc., Boston, MA, 1984, ISBN 0-8176-4565-9, p. 28. ^ Blank, Brian E.; Krantz, Steven George (2006). Calculus: Single Variable, Volume 1 (illustrated ed.). Springer Science & Business Media. p. 248. ISBN 978-1-931914-59-8.  Extract of page 248 ^ Ferraro, Giovanni (2007). The Rise and Development of the Theory of Series up to the Early 1820s (illustrated ed.). Springer Science & Business Media. p. 87. ISBN 978-0-387-73468-2.  Extract of page 87 ^ Leibniz, Gottfried Wilhelm. The Early Mathematical Manuscripts of Leibniz. Cosimo, Inc., 2008. Page 228. Copy ^ Allaire, Patricia R. (2007). Foreword. A Biography of Maria Gaetana Agnesi, an Eighteenth-century Woman Mathematician. By Cupillari, Antonella (illustrated ed.). Edwin Mellen Press. p. iii. ISBN 978-0-7734-5226-8.  ^ Unlu, Elif (April 1995). "Maria Gaetana Agnesi". Agnes Scott College.  ^ Russell, Bertrand (1946). History of Western Philosophy. London: George Allen & Unwin Ltd. p. 857. 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. "Continuity" had been, until he defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicists. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated.  ^ Grabiner, Judith V. (1981). The Origins of Cauchy's Rigorous Calculus. Cambridge: MIT Press. ISBN 0-387-90527-8.

Further reading Books Boyer, Carl Benjamin (1949). The History of the Calculus and its Conceptual Development. Hafner. Dover edition 1959, ISBN 0-486-60509-4 Courant, Richard ISBN 978-3-540-65058-4 Introduction to calculus and analysis 1. Edmund Landau. ISBN 0-8218-2830-4 Differential and Integral Calculus, American Mathematical Society. Robert A. Adams. (1999). ISBN 978-0-201-39607-2 Calculus: A complete course. Albers, Donald J.; Richard D. Anderson and Don O. Loftsgaarden, ed. (1986) Undergraduate Programs in the Mathematics and Computer Sciences: The 1985–1986 Survey, Mathematical Association of America No. 7. John Lane Bell: A Primer of Infinitesimal Analysis, Cambridge University Press, 1998. ISBN 978-0-521-62401-5. Uses synthetic differential geometry and nilpotent infinitesimals. Florian Cajori, "The History of Notations of the Calculus." Annals of Mathematics, 2nd Ser., Vol. 25, No. 1 (Sep. 1923), pp. 1–46. Leonid P. Lebedev and Michael J. Cloud: "Approximating Perfection: a Mathematician's Journey into the World of Mechanics, Ch. 1: The Tools of Calculus", Princeton Univ. Press, 2004. Cliff Pickover. (2003). ISBN 978-0-471-26987-8 Calculus and Pizza: A Math Cookbook for the Hungry Mind. Michael Spivak. (September 1994). ISBN 978-0-914098-89-8 Calculus. Publish or Perish publishing. Tom M. Apostol. (1967). ISBN 978-0-471-00005-1 Calculus, Volume 1, One-Variable Calculus with an Introduction to Linear Algebra. Wiley. Tom M. Apostol. (1969). ISBN 978-0-471-00007-5 Calculus, Volume 2, Multi-Variable Calculus and Linear Algebra with Applications. Wiley. Silvanus P. Thompson and Martin Gardner. (1998). ISBN 978-0-312-18548-0 Calculus Made Easy. Mathematical Association of America. (1988). Calculus for a New Century; A Pump, Not a Filter, The Association, Stony Brook, NY. ED 300 252. Thomas/Finney. (1996). ISBN 978-0-201-53174-9 Calculus and Analytic geometry 9th, Addison Wesley. Weisstein, Eric W. "Second Fundamental Theorem of Calculus." From MathWorld—A Wolfram Web Resource. Howard Anton, Irl Bivens, Stephen Davis:"Calculus", John Willey and Sons Pte. Ltd., 2002. ISBN 978-81-265-1259-1 Larson, Ron, Bruce H. Edwards (2010). Calculus, 9th ed., Brooks Cole Cengage Learning. ISBN 978-0-547-16702-2 McQuarrie, Donald A. (2003). Mathematical Methods for Scientists and Engineers, University Science Books. ISBN 978-1-891389-24-5 Salas, Saturnino L.; Hille, Einar; Etgen, Garret J. (2007). Calculus: One and Several Variables (10th ed.). Wiley. ISBN 978-0-471-69804-3.  Stewart, James (2012). Calculus: Early Transcendentals, 7th ed., Brooks Cole Cengage Learning. ISBN 978-0-538-49790-9 Thomas, George B., Maurice D. Weir, Joel Hass, Frank R. Giordano (2008), Calculus, 11th ed., Addison-Wesley. ISBN 0-321-48987-X Online books Boelkins, M. (2012). Active Calculus: a free, open text (PDF). Archived from the original on 30 May 2013. Retrieved 1 February 2013.  Crowell, B. (2003). "Calculus". Light and Matter, Fullerton. Retrieved 6 May 2007 from http://www.lightandmatter.com/calc/calc.pdf Garrett, P. (2006). "Notes on first year calculus". University of Minnesota. Retrieved 6 May 2007 from http://www.math.umn.edu/~garrett/calculus/first_year/notes.pdf Faraz, H. (2006). "Understanding Calculus". Retrieved 6 May 2007 from UnderstandingCalculus.com, URL http://www.understandingcalculus.com (HTML only) Keisler, H. J. (2000). "Elementary Calculus: An Approach Using Infinitesimals". Retrieved 29 August 2010 from http://www.math.wisc.edu/~keisler/calc.html Mauch, S. (2004). "Sean's Applied Math Book" (pdf). California Institute of Technology. Retrieved 6 May 2007 from https://web.archive.org/web/20070614183657/http://www.cacr.caltech.edu/~sean/applied_math.pdf Sloughter, Dan (2000). "Difference Equations to Differential Equations: An introduction to calculus". Retrieved 17 March 2009 from http://synechism.org/drupal/de2de/ Stroyan, K.D. (2004). "A brief introduction to infinitesimal calculus". University of Iowa. Retrieved 6 May 2007 from https://web.archive.org/web/20050911104158/http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm (HTML only) Strang, G. (1991). "Calculus" Massachusetts Institute of Technology. Retrieved 6 May 2007 from http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm Smith, William V. (2001). "The Calculus". Retrieved 4 July 2008 [1] (HTML only).