Related rates
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 with respect to time; because science and engineering relate quantities to each other, the methods of related rates have broad applications in these fields. Differentiation with respect to time or one of the other variables requires application of the chain rule, since most problems involve several variables. Fundamentally, if a function F is defined such that F = f the derivative of the function F can be taken with respect to another variable. We assume x is a function of t, i.e. x = g. F = f, so F ′ = f ′ ⋅ g ′ Written in Leibniz notation, this is: d F d t = d f d x ⋅ d x d t; the value of this is: if it is known how x changes with respect to t we can determine how F changes with respect to t and vice versa. We can extend this application of the chain rule with the sum, difference and quotient rules of calculus, etc. e.g.
If F = G + H 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 and the rate of change, to be found. 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; 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 caused by plugging in the known values for the variables before finding the derivative with respect to time. Doing so will yield an incorrect result, since if those values are substituted for the variables before differentiation, those variables will become constants. A 10-meter ladder is leaning against the wall of a building, the base of the ladder is sliding away from the building at a rate of 3 meters per second.
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, the height of the ladder on the wall, y, represent the sides of a right triangle with the ladder as the hypotenuse, h; 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,describes the relationship between x, y and h, for a right triangle. Differentiating both sides of this equation with respect to time, t, yields d d t = d d t Step 3: When solved for the wanted rate of change, dy/dt, gives us d d t
Shell integration
Shell integration is a means of calculating the volume of a solid of revolution, when integrating along an axis perpendicular to the axis of revolution. This is in contrast to disc integration which integrates along the axis parallel to the axis of revolution; the shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the xy-plane around the y-axis. Suppose the cross-section is defined by the graph of the positive function f on the interval; the formula for the volume will be: 2 π ∫ a b x f d x If the function is of the y coordinate and the axis of rotation is the x-axis the formula becomes: 2 π ∫ a b y f d y If the function is rotating around the line x = h or y = k, the formulas become: { 2 π ∫ a b f d x, if h ≤ a < b 2 π ∫ a b f d x, if a < b ≤ h and { 2 π ∫ a b f d y, if k ≤ a < b 2 π ∫ a b f d y, if a < b ≤ k The formula is derived by computing the double integral in polar coordinates. Consider the volume, depicted below, whose cross section on the interval is defined by: y = 2 2 In the case of disk integration we would need to solve for x given y.
Because the volume is hollow in the middle we will find two functions, one that defines the inner solid and one that defines the outer solid. After integrating these two functions with the disk method we subtract them to yield the desired volume. With the shell method all we need is the following formula: 2 π ∫ 1 2 x 2 2 d x By expanding the polynomial the integral becomes simple. In the end we find. Solid of revolution Disc integration Weisstein, Eric W. "Method of Shells". MathWorld. Frank Ayres, Elliott Mendelson. Schaum's Outlines: Calculus. McGraw-Hill Professional 2008, ISBN 978-0-07-150861-2. Pp. 244–248
Contour integration
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not found by using only real variable methods. Contour integration methods include direct integration of a complex-valued function along a curve in the complex plane application of the Cauchy integral formula application of the residue theoremOne method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums. In complex analysis a contour is a type of curve in the complex plane. In contour integration, contours provide a precise definition of the curves on which an integral may be suitably defined. A curve in the complex plane is defined as a continuous function from a closed interval of the real line to the complex plane: z: → C.
This definition of a curve coincides with the intuitive notion of a curve, but includes a parametrization by a continuous function from a closed interval. This more precise definition allows us to consider what properties a curve must have for it to be useful for integration. In the following subsections we narrow down the set of curves that we can integrate to only include ones that can be built up out of a finite number of continuous curves that can be given a direction. Moreover, we will restrict the "pieces" from crossing over themselves, we require that each piece have a finite continuous derivative; these requirements correspond to requiring that we consider only curves that can be traced, such as by a pen, in a sequence of steady strokes, which only stop to start a new piece of the curve, all without picking up the pen. Contours are defined in terms of directed smooth curves; these provide a precise definition of a "piece" of a smooth curve. A smooth curve is a curve z: → C with a non-vanishing, continuous derivative such that each point is traversed only once, with the possible exception of a curve such that the endpoints match.
In the case where the endpoints match the curve is called closed, the function is required to be one-to-one everywhere else and the derivative must be continuous at the identified point. A smooth curve, not closed is referred to as a smooth arc; the parametrization of a curve provides a natural ordering of points on the curve: z comes before z if x < y. This leads to the notion of a directed smooth curve, it is most useful to consider curves independent of the specific parametrization. This can be done by considering equivalence classes of smooth curves with the same direction. A directed smooth curve can be defined as an ordered set of points in the complex plane, the image of some smooth curve in their natural order. Note that not all orderings of the points are the natural ordering of a smooth curve. In fact, a given smooth curve has only two such orderings. A single closed curve can have any point as its endpoint, while a smooth arc has only two choices for its endpoints. Contours are the class of curves.
A contour is a directed curve, made up of a finite sequence of directed smooth curves whose endpoints are matched to give a single direction. This requires that the sequence of curves γ1,…,γn be such that the terminal point of γi coincides with the initial point of γi+1, ∀ i, 1 ≤ i < n. This includes. A single point in the complex plane is considered a contour; the symbol + is used to denote the piecing of curves together to form a new curve. Thus we could write a contour Γ, made up of n curves as Γ = γ 1 + γ 2 + ⋯ + γ n; the contour integral of a complex function f: C → C is a generalization of the integral for real-valued functions. For continuous functions in the complex plane, the contour integral can be defined in analogy to the line integral by first defining the integral along a directed smooth curve in terms of an integral over a real valued parameter. A more general definition can be given in terms of partitions of the contour in analogy with the partition of an interval and the Riemann integral.
In both cases the integral over a contour is defined as the sum of the integrals over the directed smooth curves that make up the contour. To define the contour integral in this way one must first consider the integral, over a real variable, of a complex-valued function. Let f: R → C be a complex-valued function of a real variable, t; the real and imaginary parts of f are denoted as u and v so that f = u + i v. The integral of the complex-valued function f over the interval is given by ∫ a b f d t = ∫ a b d t
Limit of a function
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: this means f gets closer and closer to L as x moves closer and closer to p. More 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 close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist; the notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the limit: a function is continuous if all of its limits agree with the values of the function, it appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.
Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique to define continuous functions. However, his work was not known during his lifetime. Cauchy discussed variable quantities and limits and defined continuity of y = f by saying that an infinitesimal change in x produces an infinitesimal change in y in his 1821 book Cours d'analyse, while claims that he only gave a verbal definition. Weierstrass first introduced the epsilon-delta definition of limit in the form it is written today, he 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, 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. She is walking towards the horizontal position given by x; 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 answer L. What does it mean to say that her altitude approaches L? It means that her altitude gets nearer and nearer to L except for a possible small error in accuracy. For example, suppose we set a particular accuracy goal for our traveler: she must get within ten meters of L, 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, her altitude is always ten meters or less from L. The accuracy goal is changed: can she get within one vertical meter? Yes. If she is anywhere within seven horizontal meters of p her altitude always remains within one meter from the target L. In summary, to say that the traveler's altitude approaches L as her horizontal position approaches p means that for every target accuracy goal, however small it may be, there is some neighborhood of p whose altitude fulfills that accuracy goal.
The initial informal statement can now be explicated: The limit of a function f as x approaches p is a number L with the following property: given any target distance from L, there is a distance from p within which the values of f remain within the target distance. This explicit statement is quite close to the formal 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, but not equal, to p; the following definitions are the accepted ones for the limit of a function in various contexts. Suppose f: R → R is defined on the real line and p,L ∈ R, it is said the limit of f, as x approaches p, is L and written lim x → p f = L, if the following property holds: For every real ε > 0, there exists a real δ > 0 such that for all real x, 0 < | x − p | < δ implies | f − L | < ε. The value of the limit does not depend on the value of f, nor that p be in the domain of f. A more general definition applies for functions defined on subsets of the real line.
Let be an open interval in R, p a point of. Let f be a real-valued function defined on all of except at p itself, it is said that the limit of f, as x approaches p, is L if, for every real ε > 0, there exists a real δ > 0 such that 0 < | x − p | < δ and x ∈ implies | f − L | < ε. Here again the limit does not depend on f being well-defined; the letters ε and δ can be understood as "error" and "distance", in fact Cauchy used ε as an abbreviation for "error" in some of his work, though in his definition of continuity he used an infinitesimal α rather than either ε or δ. In these terms, the error in the measurement of the value at the limit can be made as small as desired by reducing the distance to the limit point; as discussed below this definition works for functions in a more general context. The idea that δ and ε represent distances helps suggest these generalizations. Alternatively x may approach p from
Mean value theorem
In mathematics, the mean value theorem states that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. 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 if f is a continuous function on the closed interval, differentiable on the open interval there exists a point c in such that: f ′ = f − f b − a, 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 Michel Rolle in 1691; the mean value theorem in its modern form was stated and proved by Augustin Louis Cauchy in 1823. Let f: → R be a continuous function on the closed interval, differentiable on the open interval, where a < b.
There exists some c in such that f ′ = f − f b − a. The mean value theorem is a generalization of Rolle's theorem, which assumes f = f, so that the right-hand side above is zero; the mean value theorem is still valid in a more general setting. One only needs to assume that f: → R is continuous on, that for every x in the limit lim h → 0 f − f h exists as a finite number or equals ∞ or − ∞. If finite, that limit equals f ′. An example where this version of the theorem applies is given by the real-valued cube root function mapping x → x 1 3, whose derivative tends to infinity at the origin. Note that the theorem, as stated, is false if a differentiable function is complex-valued instead of real-valued. For example, define f = e x i for all real x. F − f = 0 = 0 while f ′ ≠ 0 for any real x; these formal statements are known as Lagrange's Mean Value Theorem. The expression f − f b − a gives the slope of the line joining the points and, a chord of the graph of f, while f ′ gives the slope of the tangent to the curve at the point.
Thus the Mean value theorem says that given any chord of a smooth curve, we can find a point lying between the end-points of the chord suc
Improper integral
In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number, ∞, − ∞, or in some instances as both endpoints approach limits. Such an integral is written symbolically just like a standard definite integral, in some cases with infinity as a limit of integration. An improper integral is a limit of the form: lim b → ∞ ∫ a b f d x, lim a → − ∞ ∫ a b f d x, or lim c → b − ∫ a c f d x, lim c → a + ∫ c b f d x, in which one takes a limit in one or the other endpoints. By abuse of notation, improper integrals are written symbolically just like standard definite integrals with infinity among the limits of integration; when the definite integral exists, this ambiguity is resolved as both the proper and improper integral will coincide in value. One is able to compute values for improper integrals when the function is not integrable in the conventional sense because of a singularity in the function or because one of the bounds of integration is infinite.
The original definition of the Riemann integral does not apply to a function such as 1 / x 2 on the interval [1, ∞), because in this case the domain of integration is unbounded. However, the Riemann integral can be extended by continuity, by defining the improper integral instead as a limit ∫ 1 ∞ 1 x 2 d x = lim b → ∞ ∫ 1 b 1 x 2 d x = lim b → ∞ = 1; the narrow definition of the Riemann integral does not cover the function 1 / x on the interval. The problem here is. However, the improper integral does exist if understood as the limit ∫ 0 1 1 x d x = lim a → 0 + ∫ a 1 1 x d x = lim a → 0 + = 2. Sometimes integrals may have two singularities. Consider, for example, the function 1/ integrated from 0 to ∞. At the lower bound, as x goes to 0 the function goes to ∞, the upper bound is itself ∞, though the function goes to 0, thus this is a doubly improper integral. Integrated, from 1 to 3, an ordinary Riemann sum suffices to produce a result of π/6. To integrate from 1 to ∞, a Riemann sum is not possible.
However, any finite upper bound, say t, gives a well-defined result, 2 arctan − π/2. This has a finite limit as t goes to infinity, namely π/2; the integral from 1/3 to 1 allows a Riemann sum as well, coincidentally again producing π/6. Replacing 1/3 by an arbitrary positive value s is safe, giving π/2 − 2 arctan. This, has a finite limit as s goes to zero, namely π/2. Combining the limits of the two fragments, the result of this improper integral is ∫ 0 ∞ d x x = lim s → 0 + ∫ s 1 d x x + lim t
Calculus
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 major branches, differential calculus, integral calculus; these two branches are related to each other by the fundamental theorem of calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. Modern calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in science 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, broadly called mathematical analysis. Calculus has been called "the calculus of infinitesimals", or "infinitesimal calculus"; the term calculus is used for naming specific methods of calculation or notation as well as some theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, process calculus.
Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz but elements of it appeared in ancient Greece in China and the Middle East, still again in medieval Europe and in India. The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus, but the formulas are simple instructions, with no indication as to method, some of them lack major components. From the age of Greek mathematics, Eudoxus used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes developed this idea further, inventing heuristics which resemble the methods of integral calculus; the method of exhaustion was 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, established a method that would be called Cavalieri's principle to find the volume of a sphere.
In the Middle East, Hasan Ibn al-Haytham, Latinized as Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid. In the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. Madhava of Sangamagrama and the Kerala School of Astronomy and Mathematics thereby stated components of calculus. A complete theory encompassing these components is now well known in the Western world as the Taylor series or infinite series approximations. However, they were not able to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, turn calculus into the great problem-solving tool we have today". In Europe, the foundational work was a treatise written by Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections.
The ideas were similar to Archimedes' in The Method, but this treatise is believed to have been lost in the 13th century, was only rediscovered in the early 20th century, so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, the infinitesimal quantities he introduced were disreputable at first; the formal study of calculus brought together Cavalieri's 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, James Gregory, the latter two proving the second fundamental theorem of calculus around 1670. The product rule and chain rule, the notions of higher derivatives and Taylor series, of analytic functions were introduced by Isaac Newton in an idiosyncratic notation which he used to solve problems of mathematical physics.
In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a cycloid, many other problems discussed in his Principia Mathematica. In other work, he developed series expansions for functions, including fractional and irrational powers, it was clear that he understood the principles of the Taylor series, he did not publish all these discoveries, at this time infinitesimal methods were still considered disreputable. These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, accused of plagiarism by Newton, he is now regarded as an independ