1.
Calculus
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Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two branches, differential calculus, and integral calculus, these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the notions of convergence of infinite sequences. Generally, modern calculus is considered to have developed in the 17th century by Isaac Newton. Today, calculus has widespread uses in science, engineering and economics, Calculus is a part of modern mathematics education. A course in calculus is a gateway to other, more advanced courses in mathematics devoted to the study of functions and limits, Calculus has historically been called the calculus of infinitesimals, or infinitesimal calculus. Calculus is also used for naming some methods of calculation or theories of computation, such as calculus, calculus of variations, lambda calculus. The ancient period introduced some of the ideas that led to integral calculus, the method of exhaustion was later discovered independently in China by Liu Hui in the 3rd century AD in order to find the area of a circle. In the 5th century AD, Zu Gengzhi, son of Zu Chongzhi, indian mathematicians gave a non-rigorous method of a sort of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, Cavalieris 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 Cavalieris infinitesimals with the calculus of finite differences developed in Europe at around the same time, pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, 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 considered disreputable. These ideas were arranged into a calculus of infinitesimals by Gottfried Wilhelm Leibniz. He is now regarded as an independent inventor of and contributor to calculus, unlike Newton, Leibniz paid a lot of attention to the formalism, often spending days determining appropriate symbols for concepts. Leibniz and Newton are usually credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the used in calculus today
2.
Fundamental theorem of calculus
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The fundamental theorem of calculus is a theorem that links the concept of the derivative of a function with the concept of the functions integral. This part of the guarantees the existence of antiderivatives for continuous functions. This part of the theorem has key practical applications because it simplifies the computation of definite integrals. The fundamental theorem of calculus relates differentiation and integration, showing that two operations are essentially inverses of one another. Before the discovery of this theorem, it was not recognized that two operations were related. Ancient Greek mathematicians knew how to compute area via infinitesimals, an operation that we would now call integration, the first published statement and proof of a rudimentary form of the fundamental theorem, strongly geometric in character, was by James Gregory. Isaac Barrow proved a more generalized version of the theorem, while his student Isaac Newton completed the development of the mathematical theory. Gottfried Leibniz systematized the knowledge into a calculus for infinitesimal quantities, for a continuous function y = f whose graph is plotted as a curve, each value of x has a corresponding area function A, representing the area beneath the curve between 0 and x. The function A may not be known, but it is given that it represents the area under the curve. The area under the curve between x and x + h could be computed by finding the area between 0 and x + h, then subtracting the area between 0 and x, in other words, the area of this “sliver” would be A − A. There is another way to estimate the area of this same sliver, as shown in the accompanying figure, h is multiplied by f to find the area of a rectangle that is approximately the same size as this sliver. So, A − A ≈ f h In fact, this becomes a perfect equality if we add the red portion of the excess area shown in the diagram. So, A − A = f h + Rearranging terms, as h approaches 0 in the limit, the last fraction can be shown to go to zero. This is true because the area of the red portion of region is less than or equal to the area of the tiny black-bordered rectangle. More precisely, | f − A − A h | = | Red Excess | h ≤ h | f − f | h = | f − f |, by the continuity of f, the latter expression tends to zero as h does. Therefore, the left-hand side tends to zero as h does and that is, the derivative of the area function A exists and is the original function f, so, the area function is simply an antiderivative of the original function. Computing the derivative of a function and “finding the area” under its curve are opposite operations and this is the crux of the Fundamental Theorem of Calculus. Intuitively, the theorem states that the sum of infinitesimal changes in a quantity over time adds up to the net change in the quantity
3.
Limit of a function
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In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below, informally, a function f assigns an output f to every input x. We say the function has a limit L at an input p, more specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L. On the other hand, if some inputs very close to p are taken to outputs that stay a distance apart. The notion of a limit has many applications in modern calculus, in particular, the many definitions of continuity employ the limit, roughly, a function is continuous if all of its limits agree with the values of the function. It also appears in the definition of the derivative, in the calculus of one variable, however, his work was not known during his lifetime. Weierstrass first introduced the definition of limit in the form it is usually written today. He also introduced the notations lim and limx→x0, the modern notation of placing the arrow below the limit symbol is due to Hardy in his book A Course of Pure Mathematics in 1908. Imagine a person walking over a landscape represented by the graph of y = f and her horizontal position is measured by the value of x, much like the position given by a map of the land or by a global positioning system. Her altitude is given by the coordinate y and she is walking towards the horizontal position given by x = p. As she gets closer and closer to it, she notices that her altitude approaches L, if asked about the altitude of x = p, she would then answer L. What, then, does it mean to say that her altitude approaches L. It means that her altitude gets nearer and nearer to L except for a small error in accuracy. For example, suppose we set a particular goal for our traveler. She reports back that indeed she can get within ten meters of L, since she notes that when she is within fifty horizontal meters of p, the accuracy goal is then changed, can she get within one vertical meter. If she is anywhere within seven meters of p, then her altitude always remains within one meter from the target L. This explicit statement is quite close to the definition of the limit of a function with values in a topological space. To say that lim x → p f = L, means that ƒ can be made as close as desired to L by making x close enough, the following definitions are the generally accepted ones for the limit of a function in various contexts. Suppose f, R → R is defined on the real line, the value of the limit does not depend on the value of f, nor even that p be in the domain of f
4.
Continuous function
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In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function, a continuous function with a continuous inverse function is called a homeomorphism. Continuity of functions is one of the concepts of topology. The introductory portion of this focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the general case of functions between two metric spaces. In order theory, especially in theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article, as an example, consider the function h, which describes the height of a growing flower at time t. By contrast, if M denotes the amount of money in an account at time t, then the function jumps at each point in time when money is deposited or withdrawn. A form of the definition of continuity was first given by Bernard Bolzano in 1817. Cauchy defined infinitely small quantities in terms of quantities. The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s but the work wasnt published until the 1930s, all three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of continuity in 1872. This is not a definition of continuity since the function f =1 x is continuous on its whole domain of R ∖ A function is continuous at a point if it does not have a hole or jump. A “hole” or “jump” in the graph of a function if the value of the function at a point c differs from its limiting value along points that are nearby. Such a point is called a discontinuity, a function is then continuous if it has no holes or jumps, that is, if it is continuous at every point of its domain. Otherwise, a function is discontinuous, at the points where the value of the function differs from its limiting value, there are several ways to make this definition mathematically rigorous. These definitions are equivalent to one another, so the most convenient definition can be used to determine whether a function is continuous or not. In the definitions below, f, I → R. is a function defined on a subset I of the set R of real numbers and this subset I is referred to as the domain of f
5.
Mean value theorem
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This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval. More precisely, if a function f is continuous on the closed interval and it is one of the most important results in real analysis. A special case of this theorem was first described by Parameshvara, from the Kerala school of astronomy and mathematics in India, in his commentaries on Govindasvāmi and Bhāskara II. A restricted form of the theorem was proved by Rolle in 1691, the result was what is now known as Rolles theorem, the mean value theorem in its modern form was stated and proved by Cauchy in 1823. Let f, → R be a function on the closed interval, and differentiable on the open interval. Then there exists c in such that f ′ = f − f b − a. The mean value theorem is a generalization of Rolles theorem, which assumes f = f, the mean value theorem is still valid in a slightly more general setting. One only needs to assume that f, → R is continuous on, If finite, that limit equals f ′. An example where this version of the theorem applies is given by the cube root function mapping x → x 13. Note that the theorem, as stated, is if a differentiable function is complex-valued instead of real-valued. For example, define f = e x i for all real x, then f − f =0 =0 while f ′ ≠0 for any real x. Thus the Mean value theorem says that given any chord of a smooth curve, the following proof illustrates this idea. Define g = f − r x, where r is a constant, since f is continuous on and differentiable on, the same is true for g. We now want to choose r so that g satisfies the conditions of Rolles theorem, Assume that f is a continuous, real-valued function, defined on an arbitrary interval I of the real line. If the derivative of f at every point of the interval I exists and is zero. Proof, Assume the derivative of f at every point of the interval I exists and is zero. Let be an open interval in I. By the mean value theorem, there exists a point c in such that 0 = f ′ = f − f b − a, thus, f is constant on the interior of I and thus is constant on I by continuity
6.
Rolle's theorem
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If a real-valued function f is continuous on a proper closed interval, differentiable on the open interval, and f = f, then there exists at least one c in the open interval such that f ′ =0. This version of Rolles theorem is used to prove the mean value theorem and it is also the basis for the proof of Taylors theorem. Indian mathematician Bhāskara II is credited with knowledge of Rolles theorem, although the theorem is named after Michel Rolle, Rolles 1691 proof covered only the case of polynomial functions. His proof did not use the methods of calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem, the name Rolles theorem was first used by Moritz Wilhelm Drobisch of Germany in 1834 and by Giusto Bellavitis of Italy in 1846. For a radius r >0, consider the function f = r 2 − x 2, x ∈ and its graph is the upper semicircle centered at the origin. This function is continuous on the interval and differentiable in the open interval. Since f = f, Rolles theorem applies, and indeed, note that the theorem applies even when the function cannot be differentiated at the endpoints because it only requires the function to be differentiable in the open interval. If differentiability fails at a point of the interval, the conclusion of Rolles theorem may not hold. Consider the absolute value function f = | x |, x ∈, then f = f, but there is no c between −1 and 1 for which the derivative is zero. This is because that function, although continuous, is not differentiable at x =0, note that the derivative of f changes its sign at x =0, but without attaining the value 0. The theorem cannot be applied to this function, clearly, because it does not satisfy the condition that the function must be differentiable for x in the open interval. However, when the differentiability requirement is dropped from Rolles theorem, f will still have a number in the open interval. The second example illustrates the generalization of Rolles theorem, Consider a real-valued. If the right- and left-hand limits agree for every x, then they agree in particular for c, if f is convex or concave, then the right- and left-hand derivatives exist at every inner point, hence the above limits exist and are real numbers. This generalized version of the theorem is sufficient to prove convexity when the derivatives are monotonically increasing. Since the proof for the version of Rolles theorem and the generalization are very similar. In particular, if the derivative exists, it must be zero at c, by assumption, f is continuous on, and by the extreme value theorem attains both its maximum and its minimum in
7.
Differential calculus
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In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two divisions of calculus, the other being integral calculus. The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, the derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation, geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point. Differential calculus and integral calculus are connected by the theorem of calculus. Differentiation has applications to nearly all quantitative disciplines, for example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration. The derivative of the momentum of a body equals the applied to the body. The reaction rate of a reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials, derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena, derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra. Suppose that x and y are real numbers and that y is a function of x and this relationship can be written as y = f. If f is the equation for a line, then there are two real numbers m and b such that y = mx + b. In this slope-intercept form, the m is called the slope and can be determined from the formula, m = change in y change in x = Δ y Δ x. It follows that Δy = m Δx, a general function is not a line, so it does not have a slope. Geometrically, the derivative of f at the point x = a is the slope of the tangent line to the function f at the point a and this is often denoted f ′ in Lagranges notation or dy/dx|x = a in Leibnizs notation. Since the derivative is the slope of the approximation to f at the point a. If every point a in the domain of f has a derivative, for example, if f = x2, then the derivative function f ′ = dy/dx = 2x
8.
Derivative
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The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a tool of calculus. For example, the derivative of the position of an object with respect to time is the objects velocity. The derivative of a function of a variable at a chosen input value. The tangent line is the best linear approximation of the function near that input value, for this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of real variables. In this generalization, the derivative is reinterpreted as a transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables and it can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation, the reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration, differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative, the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x, If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. The simplest case, apart from the case of a constant function, is when y is a linear function of x. This formula is true because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x. Thus, since y + Δ y = y + m Δ x and this gives an exact value for the slope of a line. If the function f is not linear, however, then the change in y divided by the change in x varies, differentiation is a method to find an exact value for this rate of change at any given value of x. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the value of the ratio of the differences Δy / Δx as Δx becomes infinitely small
9.
Differential of a function
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In calculus, the differential represents the principal part of the change in a function y = f with respect to changes in the independent variable. The differential dy is defined by d y = f ′ d x, where f ′ is the derivative of f with respect to x, one also writes d f = f ′ d x. The precise meaning of the variables dy and dx depends on the context of the application, traditionally, the variables dx and dy are considered to be very small, and this interpretation is made rigorous in non-standard analysis. The quotient dy/dx is not infinitely small, rather it is a real number, the use of infinitesimals in this form was widely criticized, for instance by the famous pamphlet The Analyst by Bishop Berkeley. Augustin-Louis Cauchy defined the differential without appeal to the atomism of Leibnizs infinitesimals, in physical treatments, such as those applied to the theory of thermodynamics, the infinitesimal view still prevails. Courant & John reconcile the use of infinitesimal differentials with the mathematical impossibility of them as follows. The differentials represent finite non-zero values that are smaller than the degree of accuracy required for the purpose for which they are intended. Thus physical infinitesimals need not appeal to a corresponding mathematical infinitesimal in order to have a precise sense, following twentieth-century developments in mathematical analysis and differential geometry, it became clear that the notion of the differential of a function could be extended in a variety of ways. In real analysis, it is desirable to deal directly with the differential as the principal part of the increment of a function. This leads directly to the notion that the differential of a function at a point is a functional of an increment Δx. This approach allows the differential to be developed for a variety of more sophisticated spaces, in non-standard calculus, differentials are regarded as infinitesimals, which can themselves be put on a rigorous footing. The differential is defined in modern treatments of calculus as follows. The differential of a function f of a real variable x is the function df of two independent real variables x and Δx given by d f = d e f f ′ Δ x. One or both of the arguments may be suppressed, i. e. one may see df or simply df, if y = f, the differential may also be written as dy. The partial differential is therefore ∂ y ∂ x 1 d x 1 involving the partial derivative of y with respect to x1. The total differential is then defined as d y = ∂ y ∂ x 1 Δ x 1 + ⋯ + ∂ y ∂ x n Δ x n. Since, with this definition, d x i = Δ x i, in measurement, the total differential is used in estimating the error Δf of a function f based on the errors Δx, Δy. of the parameters x, y. As they are assumed to be independent, the analysis describes the worst-case scenario, the absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign
10.
Total differential
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In calculus, the differential represents the principal part of the change in a function y = f with respect to changes in the independent variable. The differential dy is defined by d y = f ′ d x, where f ′ is the derivative of f with respect to x, one also writes d f = f ′ d x. The precise meaning of the variables dy and dx depends on the context of the application, traditionally, the variables dx and dy are considered to be very small, and this interpretation is made rigorous in non-standard analysis. The quotient dy/dx is not infinitely small, rather it is a real number, the use of infinitesimals in this form was widely criticized, for instance by the famous pamphlet The Analyst by Bishop Berkeley. Augustin-Louis Cauchy defined the differential without appeal to the atomism of Leibnizs infinitesimals, in physical treatments, such as those applied to the theory of thermodynamics, the infinitesimal view still prevails. Courant & John reconcile the use of infinitesimal differentials with the mathematical impossibility of them as follows. The differentials represent finite non-zero values that are smaller than the degree of accuracy required for the purpose for which they are intended. Thus physical infinitesimals need not appeal to a corresponding mathematical infinitesimal in order to have a precise sense, following twentieth-century developments in mathematical analysis and differential geometry, it became clear that the notion of the differential of a function could be extended in a variety of ways. In real analysis, it is desirable to deal directly with the differential as the principal part of the increment of a function. This leads directly to the notion that the differential of a function at a point is a functional of an increment Δx. This approach allows the differential to be developed for a variety of more sophisticated spaces, in non-standard calculus, differentials are regarded as infinitesimals, which can themselves be put on a rigorous footing. The differential is defined in modern treatments of calculus as follows. The differential of a function f of a real variable x is the function df of two independent real variables x and Δx given by d f = d e f f ′ Δ x. One or both of the arguments may be suppressed, i. e. one may see df or simply df, if y = f, the differential may also be written as dy. The partial differential is therefore ∂ y ∂ x 1 d x 1 involving the partial derivative of y with respect to x1. The total differential is then defined as d y = ∂ y ∂ x 1 Δ x 1 + ⋯ + ∂ y ∂ x n Δ x n. Since, with this definition, d x i = Δ x i, in measurement, the total differential is used in estimating the error Δf of a function f based on the errors Δx, Δy. of the parameters x, y. As they are assumed to be independent, the analysis describes the worst-case scenario, the absolute values of the component errors are used, because after simple computation, the derivative may have a negative sign
11.
Second derivative
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In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. In Leibniz notation, a = d v d t = d 2 x d t 2, on the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a second derivative bows downward. The second derivative of a function f is denoted f ″. That is, f ″ = ′ When using Leibnizs notation for derivatives and this notation is derived from the following formula, d 2 y d x 2 = d d x. Given the function f = x 3, the derivative of f is the function f ′ =3 x 2, the second derivative of f is the derivative of f′, namely f ″ =6 x. The second derivative of a function f measures the concavity of the graph of f, a function whose second derivative is positive will be concave up, meaning that the tangent line will lie below the graph of the function. Similarly, a function whose derivative is negative will be concave down. If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, a point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, the relation between the second derivative and the graph can be used to test whether a stationary point for a function is a local maximum or a local minimum. Specifically, If f ′ ′ <0 then f has a maximum at x. If f ′ ′ >0 then f has a minimum at x. If f ′ ′ =0, the second derivative test says nothing about the point x, the reason the second derivative produces these results can be seen by way of a real-world analogy. Consider a vehicle that at first is moving forward at a great velocity, the same is true for the minimum, with a vehicle that at first has a very negative velocity but positive acceleration. It is possible to write a single limit for the second derivative, the limit is called the second symmetric derivative. Note that the symmetric derivative may exist even when the second derivative does not. The expression on the right can be written as a quotient of difference quotients. This limit can be viewed as a version of the second difference for sequences
12.
Implicit function
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In mathematics, an implicit equation is a relation of the form R =0, where R is a function of several variables. For example, the equation of the unit circle is x 2 + y 2 −1 =0. An implicit function is a function that is defined implicitly by an implicit equation, thus, an implicit function for y in the context of the unit circle is defined implicitly by x 2 + f 2 −1 =0. This implicit equation defines f as a function of x only if −1 ≤ x ≤1, the implicit function theorem provides conditions under which a relation defines an implicit function. A common type of function is an inverse function. If f is a function of x, then the function of f. This solution is x = f −1, intuitively, an inverse function is obtained from f by interchanging the roles of the dependent and independent variables. Stated another way, the function gives the solution for x of the equation R = y − f =0. Example The product log is a function giving the solution for x of the equation y − x ex =0. An algebraic function is a function satisfies a polynomial equation whose coefficients are themselves polynomials. Algebraic functions play an important role in analysis and algebraic geometry. A simple example of a function is given by the left side of the unit circle equation. Solving for y gives a solution, y = ±1 − x 2. But even without specifying this explicit solution, it is possible to refer to the solution of the unit circle equation. Nevertheless, one can refer to the implicit solution y = g involving the multi-valued implicit function g. Not every equation R =0 implies a graph of a single-valued function, another example is an implicit function given by x − C =0 where C is a cubic polynomial having a hump in its graph. Thus, for a function to be a true function it might be necessary to use just part of the graph. An implicit function can sometimes be successfully defined as a function only after zooming in on some part of the x-axis
13.
Related rates
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In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is usually with respect to time, because science and engineering often relate quantities to each other, the methods of related rates have broad applications in these fields. Because problems involve several variables, differentiation with respect to time or one of the other variables requires application of the chain rule. Fundamentally, if a function F is defined such that F = f and we assume x is a function of t, i. e. x = g. Then F = f, so F ′ = f ′ ⋅ g ′ Written in Leibniz notation, the value of this is, if it is known how x changes with respect to t, then we can determine how F changes with respect to t and vice versa. We can extend this application of the rule with the sum, difference, product and quotient rules of calculus. If F = G + H then d F d x ⋅ d x d t = d G d y ⋅ d y d t + d H d z ⋅ d z d t. The most common way to approach related rates problems is the following, Identify the known variables, including rates of change, construct an equation relating the quantities whose rates of change are known to the quantity whose rate of change is to be found. Differentiate both sides of the equation with respect to time, often, the chain rule is employed at this step. Substitute the known rates of change and the known quantities into the equation, solve for the wanted rate of change. Errors in this procedure are often caused by plugging in the values for the variables before finding the derivative with respect to time. A 10-meter ladder is leaning against the wall of a building, how fast is the top of the ladder sliding down the wall when the base of the ladder is 6 meters from the wall. The distance between the base of the ladder and the wall, x, and the height of the ladder on the wall, y, represent the sides of a triangle with the ladder as the hypotenuse. The objective is to find dy/dt, the rate of change of y with respect to time, t, when h, x and dx/dt, the rate of change of x, are known. Step 1, x =6 h =10 d x d t =3 d h d t =0 d y d t =. Step 2, From the Pythagorean theorem, the equation x 2 + y 2 = h 2, step 4 &5, Using the variables from step 1 gives us, d y d t = h d h d t − x d x d t y. D y d t =10 ×0 −6 ×3 y = −18 y, in doing such, the top of the ladder is sliding down the wall at a rate of 9⁄4 meters per second. Because one physical quantity often depends on another, which, in turn depends on others, such as time and this section presents an example of related rates kinematics and electromagnetic induction
14.
Taylor's theorem
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In calculus, Taylors theorem gives an approximation of a k-times differentiable function around a given point by a k-th order Taylor polynomial. For analytic functions the Taylor polynomials at a point are finite order truncations of its Taylor series. The exact content of Taylors theorem is not universally agreed upon, indeed, there are several versions of it applicable in different situations, and some of them contain explicit estimates on the approximation error of the function by its Taylor polynomial. Taylors theorem is named after the mathematician Brook Taylor, who stated a version of it in 1712, yet an explicit expression of the error was not provided until much later on by Joseph-Louis Lagrange. An earlier version of the result was already mentioned in 1671 by James Gregory, Taylors theorem is taught in introductory level calculus courses and it is one of the central elementary tools in mathematical analysis. Within pure mathematics it is the point of more advanced asymptotic analysis. Taylors theorem also generalizes to multivariate and vector valued functions f, R n → R m on any dimensions n and m and this generalization of Taylors theorem is the basis for the definition of so-called jets which appear in differential geometry and partial differential equations. If a real-valued function f is differentiable at the point a then it has an approximation at the point a. This means that there exists a function h1 such that f = f + f ′ + h 1, here P1 = f + f ′ is the linear approximation of f at the point a. The graph of y = P1 is the tangent line to the graph of f at x = a, the error in the approximation is R1 = f − P1 = h 1. Note that this goes to zero a little bit faster than x − a as x tends to a, if we wanted a better approximation to f, we might instead try a quadratic polynomial instead of a linear function. Instead of just matching one derivative of f at a, we can match two derivatives, thus producing a polynomial that has the slope and concavity as f at a. The quadratic polynomial in question is P2 = f + f ′ + f ″22, Taylors theorem ensures that the quadratic approximation is, in a sufficiently small neighborhood of the point a, a better approximation than the linear approximation. Specifically, f = P2 + h 22, lim x → a h 2 =0. Here the error in the approximation is R2 = f − P2 = h 22 which, given the behavior of h 2. Similarly, we might get better approximations to f if we use polynomials of higher degree. In general, the error in approximating a function by a polynomial of degree k will go to zero a little bit faster than k as x tends to a. Find the smallest degree k for which the polynomial Pk approximates f to within an error on a given interval
15.
Product rule
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In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated as ′ = f ′ ⋅ g + f ⋅ g ′ or in the Leibniz notation d d x = u ⋅ d v d x + v ⋅ d u d x. In differentials notation, this can be written as d = u d v + v d u, discovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials. Here is Leibnizs argument, Let u and v be two functions of x. Then the differential of uv is d = ⋅ − u ⋅ v = u ⋅ d v + v ⋅ d u + d u ⋅ d v. Since the term du·dv is negligible, Leibniz concluded that d = v ⋅ d u + u ⋅ d v, suppose we want to differentiate ƒ = x2 sin. By using the rule, one gets the derivative ƒ = 2x sin + x2cos. This follows from the rule since the derivative of any constant is zero. This, combined with the sum rule for derivatives, shows that differentiation is linear, the rule for integration by parts is derived from the product rule, as is the quotient rule. Let h = f g, and suppose that f and g are each differentiable at x and we want to prove that h is differentiable at x and that its derivative h is given by f g + f g. To do this f g − f g is added to the numerator to permit its factoring, a rigorous proof of the product rule can be given using the definition of the derivative as a limit, and the basic properties of limits. Let h = f g, and suppose that f and g are each differentiable at x0 and we want to prove that h is differentiable at x0 and that its derivative h′ is given by f′ g + f g′. Let Δh = h − h, note that although x0 is fixed, Δh depends on the value of Δx, which is thought of as being small. The function h is differentiable at x0 if the limit lim Δ x →0 Δ h Δ x exists, as with Δh, let Δf = f − f and Δg = g − g which, like Δh, also depends on Δx. Then f = f + Δf and g = g + Δg, using the basic properties of limits and the definition of the derivative, we can tackle this term-by term. First, lim Δ x →0 = f ′ g, similarly, lim Δ x →0 = f g ′. The third term, corresponding to the small rectangle, winds up being negligible because Δf Δg vanishes to second order. Then, f g − f g = − f g = f ′ g h + f g ′ h + O Taking the limit for small h gives the result, Let f = uv and suppose u and v are positive functions of x
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Chain rule
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In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. This can be more explicitly in terms of the variable. Let F = f ∘ g, or equivalently, F = f for all x, then one can also write F ′ = f ′ g ′. The chain rule may be written, in Leibnizs notation, in the following way. If a variable z depends on the y, which itself depends on the variable x, so that y and z are therefore dependent variables, then z, via the intermediate variable of y. The chain rule states, d z d x = d z d y ⋅ d y d x. In integration, the counterpart to the rule is the substitution rule. The chain rule seems to have first been used by Leibniz and he used it to calculate the derivative of a + b z + c z 2 as the composite of the square root function and the function a + b z + c z 2. He first mentioned it in a 1676 memoir, the common notation of chain rule is due to Leibniz. LHôpital uses the chain rule implicitly in his Analyse des infiniment petits, the chain rule does not appear in any of Leonhard Eulers analysis books, even though they were written over a hundred years after Leibnizs discovery. Suppose that a skydiver jumps from an aircraft, assume that t seconds after his jump, his height above sea level in meters is given by g =4000 −4. 9t2. One model for the pressure at a height h is f =101325 e−0. 0001h. These two equations can be differentiated and combined in ways to produce the following data, g′ = −9. 8t is the velocity of the skydiver at time t. F′ = −10. 1325e−0. 0001h is the rate of change in pressure with respect to height at the height h and is proportional to the buoyant force on the skydiver at h meters above sea level. Is the atmospheric pressure the skydiver experiences t seconds after his jump, ′ is the rate of change in atmospheric pressure with respect to time at t seconds after the skydivers jump and is proportional to the buoyant force on the skydiver at t seconds after his jump. The chain rule gives a method for computing ′ in terms of f′, while it is always possible to directly apply the definition of the derivative to compute the derivative of a composite function, this is usually very difficult. The utility of the rule is that it turns a complicated derivative into several easy derivatives. The chain rule states that, under conditions, ′ = f ′ ⋅ g ′