|Topics in Calculus|
Limits of functions
Mean value theorem
Calculus (Latin, calculus, a small stone used for counting) is a branch in mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem of calculus. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient.
Historically, calculus was called "the calculus of infinitesimals", or "infinitesimal calculus". More generally, calculus (plural calculi) may refer to any method or system of calculation guided by the symbolic manipulation of expressions. Some examples of other well-known calculi are propositional calculus, variational calculus, lambda calculus, pi calculus, and join calculus.
The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1820 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum. From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus. The method of exhaustion was later used in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Chongzhi used what would later be called Cavalieri's principle to find the volume of a sphere.
Around AD 1000, the Islamic mathematician Ibn al-Haytham (Alhacen) was the first to derive the formula for the sum of the fourth powers of an arithmetic progression, using a method that is readily generalizable to finding the formula for the sum of any higher integral powers, which he used to perform an integration. In the 11th century, the Chinese polymath Shen Kuo developed 'packing' equations that dealt with integration. In the 12th century, the Indian mathematician, Bhāskara II, developed an early derivative representing infinitesimal change, and he described an early form of Rolle's theorem. Also in the 12th century, the Persian mathematician Sharaf al-Dīn al-Tūsī discovered the derivative of cubic polynomials, an important result in differential calculus. In the 14th century, Indian mathematician Madhava of Sangamagrama, along with other mathematician-astronomers of the Kerala school of astronomy and mathematics, described special cases of Taylor series, which are treated in the text Yuktibhasa.
In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimal thin cross-sections. The ideas were similar to Archimedes' in The Method, but this treatise was lost until the early part of the twentieth century. Cavalieri's work was not well respected since his methods can lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.
The formal study of calculus combined Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1675.
The product rule and chain rule, the notion of higher derivatives, Taylor series, and analytical functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics. In his publications, 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 systematized into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for manipulating 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 – he often spent days determining appropriate symbols for concepts.
Leibniz and Newton are usually both credited with the invention of 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, but Leibniz published 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 mathematicians for many years, to the detriment of English mathematics. 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. Today, both Newton and Leibniz are given credit for developing calculus independently. 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. In the 19th century, calculus was put on a much more rigorous footing by mathematicians such as Cauchy, Riemann, and Weierstrass (see (ε, δ)-definition of limit). It was also during this period that the ideas of calculus were generalized to Euclidean space and the complex plane. Lebesgue generalized the notion of the integral so that virtually any function has an integral, while Laurent Schwartz extended differentiation in much the same way.
Calculus is a ubiquitous topic in most modern high schools and universities around the world.
While some of the ideas of calculus were developed earlier in Egypt, Greece, China, India, Iraq, Persia, and Japan, the modern 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 can be used to compute the trajectory of a shuttle docking at a space station or the amount of snow in a driveway.
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 gave several famous examples of such paradoxes. Calculus provides tools, especially the limit and the infinite series, which resolve the paradoxes.
In mathematics, foundations refers to the rigorous development of a subject from precise axioms and definitions. 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.
There is more than one rigorous approach to the foundation of calculus. The usual one today is via the concept of limits defined on the continuum of real numbers. An alternative is nonstandard analysis, in which the real number system is augmented with infinitesimal and infinite numbers, as in the original Newton-Leibniz conception. The foundations of calculus are included in the field of real analysis, which contains full definitions and proofs of the theorems of calculus as well as generalizations such as measure theory and distribution theory.
Calculus is usually developed by manipulating very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like numbers but which are, in some sense, "infinitely small". An infinitesimal number dx could be greater than 0, but less than any number in the sequence 1, ½, ⅓, ... and less than any positive real number. Any integer multiple of an infinitesimal is still infinitely small, i.e., infinitesimals do not satisfy the Archimedean property. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. This 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 limits. Limits describe the value of a function at a certain input in terms of its values at nearby input. They capture small-scale behavior, just like infinitesimals, but use the ordinary 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 are the easiest way to provide rigorous foundations for calculus, and for this reason they are the standard approach.
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 mathematical jargon, the derivative is a linear operator which inputs a function and outputs a second function. 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 it produces turns out to be the doubling function.)
The most common symbol for a derivative is an apostrophe-like mark called prime. Thus, the derivative of the function of f is f′, pronounced "f prime." For instance, if f(x) = x2 is the squaring function, then f′(x) = 2x is the doubling function.
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. (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
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 impossible. 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. Let f(x) = x2 be the squaring function.
The slope of 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 similar computation to the one above shows that the derivative of the squaring function is the doubling function.
A common notation, introduced by Leibniz, for the derivative in the example above is
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:
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 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 is the antiderivative, the inverse operation to the derivative. F is an indefinite integral of f when f is a derivative of F. (This use of upper- and lower-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 area between the graph of the input and the x-axis. The technical definition of the definite integral is the limit of a sum of areas of rectangles, called a Riemann sum.
A motivating example is the distances traveled in a given time.
If the speed is constant, only multiplication is needed, but if the speed changes, then we need a more powerful method of finding the distance. 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.
If f(x) in the diagram on the left 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 , an elongated S (the S stands for "sum"). The definite integral is written as:
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
is to be understood as an operator that takes a function as an input and gives a number, the area, as an output; dx is not a number, and is not being multiplied by f(x).
The indefinite integral, or antiderivative, is written:
Functions differing by only a constant have the same derivative, and therefore 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 is given by:
An undetermined constant like C in the antiderivative is known as a 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 massive 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.
Physics makes particular use of calculus; all concepts in classical mechanics are interrelated 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. In the subfields of electricity and magnetism calculus can be used to find the total flux of electromagnetic fields. A more historical example of the use of calculus in physics is Newton's second law of motion, it expressly uses the term "rate of change" which refers to the derivative: The rate of change of momentum of a body is equal to the resultant force acting on the body and is in the same direction. Even the common expression of Newton's second law as Force = Mass × Acceleration involves differential calculus because acceleration can be expressed as the derivative of velocity. 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.
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.
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.
In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow.
Calculus can be used to find approximate solutions to equations, in 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.
Welcome to the Wikiversity Calculus Learning Center.
The Calculus Learning Center is a Wikiversity content development project where participants create, organize and develop learning resources for Calculus.
The goal of the department of calculus is to familiarize the student with the tools of modern calculus from developing the analytical machinery of limits, derivatives and integration to our use of set theory and basic topology to apply single-variable calculus to more general spaces to the development of the language of differential forms. The student should be prepared to study differential geometry from any modern introductory text at the conclusion of their study of calculus. The inspiration for our style of teaching comes from the texts of Courant, Apostol and Spivak, with whom all mathematicians are undoubtably familiar. A good mix of analysis (this is a course in pure mathematics, so don't hold back proofs) and application with well-chosen exercises and problems is what we should strive for in our lessons.
Wikibooks has a well developed Calculus Book in 10 sections which this Wikiversity webcourse should mirror to maximise resources.
|Chapter||Wikibook Chapter||Wikiversity Topics||Wikiversity for Review & Development|
|1.||Precalculus||*The Real Numbers and Their Development||*Foundations of calculus|
|*Introduction to Calculus Overview Page|
|**1. Introduction to Calculus|
|4.||Integration||**Integration by Substitution||Trigonometric Substitutions|
|**Monte Carlo Integration|
|5.||Parametric equations||Parametric equations|
|6.||Polar equations||Polar equations|
|7.||Sequences and series||Sequences and series|
|9.||Multivariable and differential calculus||Multivariable Calculus|
The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this department, you can list your name here (this can help small departments grow and the participants communicate better; for large departments a list of active participants is not needed).
This wikibook aims to be a quality calculus textbook through which users may master the discipline. Standard topics such as limits, differentiation and integration are covered as well as several others. Please contribute wherever you feel the need.
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Calculus is a branch of mathematics which looks at things that change over time. It tries to say what type of change it is and how big it is using functions at the exact moment at which the change is taking place. There are two different types of calculus. Differential calculus divides things into small (different) pieces and tells us how they change from one moment to the next, while integral calculus joins (integrates) the small pieces together and tells us how much of something is made by a change. It is used in many different fields such as physics, astronomy, biology, engineering, economics, medicine and sociology.
The word Calculus is derived from the Latin for "small stone".
In the 1670s and 1680s, Sir Isaac Newton in England and Gottfried Leibniz in Germany figured out calculus at the same time, working separately from each other. Newton wanted to make new math to predict where to see planets in the sky, because astronomy has always been a popular and useful form of science, and knowing more about the motions of the objects in the night sky was important for navigation of ships. Leibniz wanted to work out the space (area) under a curve (a line which is not straight). Many years later, the two men argued over who discovered it first. Scientists from England supported Newton, but scientists from the rest of Europe supported Leibniz. Most mathematicians today agree that both men share the credit equally. Some parts of modern calculus come from Newton, such as its uses in physics. Other parts come from Leibniz, such as the symbols used to write it.
They were not the first people to use mathematics to describe the physical world—Aristotle and Pythagoras came earlier, as did Galileo, who said that mathematics was the language of science. But they were the first to design a system that describes how things change over time and can predict how they will change in the future.
Differential calculus is the process of finding out the rate of change of a variable compared to another variable. It can be used to find the speed of a moving object or the slope of a curve, figure out the maximum or minimum points of a curve, or find answers to problems in the electricity and magnetism areas of physics, among many other uses.
Many amounts can be variables, which can change their value unlike numbers such as 5 or 200. Some examples of variables are distance and time. The speed of an object is how far it travels in a particular time. So if a town is 80 kilometres (50 miles) away and a person in a car gets there in one hour, they have traveled at an average speed of 80 kilometres (50 miles) per hour. But this is only an average—they may have been traveling faster at some times (on a highway) and slower at others (at a traffic light or on a small street where people live). Imagine a driver trying to figure out a car's speed using only its odometer (distance meter) and clock, without a speedometer!
Until calculus was invented, the only way to work this out was to cut the time into smaller and smaller pieces, so the average speed over the smaller time would get closer and closer to the actual speed at a point in time. This was a very long and hard process and had to be done each time people wanted to work something out.
A very similar problem is to find the slope (how steep it is) at any point on a curve. The slope of a straight line is easy to work out—it is simply how much it goes up (y or vertical) divided by how much it goes across (x or horizontal). On a curve, though, the slope is a variable (has different values at different points) because the line bends. But if the curve was to be cut into very, very small pieces, the curve at the point would look almost like a very short straight line. So to work out its slope, a straight line can be drawn through the point with the same slope as the curve at that point. If it is done exactly right, the straight line will have the same slope as the curve, and is called a tangent. But there is no way to know (without very complicated mathematics) whether the tangent is exactly right, and our eyes are not accurate enough to be certain whether it is exact or simply very close.
What Newton and Leibniz found was a way to work out the slope (or the speed in the distance example) exactly using simple and logical rules. They divided the curve into an infinite number of very small pieces. They then chose points on either side of the range they were interested in and worked out tangents at each. As the points moved closer together towards the point they were interested in, the slope approached a particular value as the tangents approached the real slope of the curve. They said that this particular value it approached was the actual slope.
Let's say we have a function y = f(x). f is short for function, so this equation means "y is a function of x". This tells us that how high y is on the vertical axis depends on what x (the horizontal axis) is at that time. For example with the equation y = x², we know that if x is 1, then y will be 1; if x is 3, then y will be 9; if x is 20, then y will be 400.
If we use y = x², the derivative produced using this method is 2x, or 2 multiplied by x. So we know without having to draw any tangent lines that at any point on the curve f(x) = x², the derivative f'(x) (marked with an apostrophe) will be 2x at any point. This process of working out a slope using limits is called differentiation, or finding the derivative.
Leibniz came to the same result, but called h "dx", which means "a tiny amount of x". He called the resulting change in f(x) "dy", which means "a tiny amount of y". Leibniz's notation is used by more books because it is easy to understand when the equations become more complicated. In Leibniz notation:
Mathematicians have grown this basic theory to make simple algebra rules which can be used to find the derivative of almost any function.
The main idea in calculus is called the "Fundamental Theorem of Calculus". This main idea is that the two calculus processes, differential and integral calculus, are opposites. That is, a person can use differential calculus to undo an integral calculus process. Also, a person can use integral calculus to undo a differential calculus method, just like if you add a number to another number you can 'undo' it by taking away that number.
The method integral calculus uses to find areas of shapes is to break the shape up into many small boxes, and add up the area of each of the boxes. This gives an approximation to the area. If the boxes are made narrower and narrower, then there are more and more of them, and the area of all the boxes becomes very close to the area of the shape. One of the main ideas of calculus is that we can imagine having an infinite number of these boxes, each infinitely narrow, and then we would have the exact area of the shape.