The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point. The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. The process of finding a derivative is called differentiation. Differentiation and the derivative The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y divided by x is a line. y + Δy = f(x + Δx) = m (x + Δx) + b = m x + m Δx + b = y + m Δx. It follows that Δy = m Δx. This gives an exact value for the slope of a line. Rate of change as a limit value Figure 1. Figure 2. Figure 3. Figure 4. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences Δy / Δx as Δx becomes infinitely small. Notation Rigorous definition
CalculusHistory Modern calculus was developed in 17th century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (see the Leibniz–Newton calculus controversy), but elements of it have appeared in ancient Greece, China, medieval Europe, India, and the Middle East. Ancient 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. Medieval Modern 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 infinitesimally thin cross-sections. These ideas were arranged 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. Leibniz and Newton are usually both credited with the invention of calculus. Foundations
e (mathematical constant)Functions f(x) = ax are shown for several values of a. e is the unique value of a, such that the derivative of f(x) = ax at the point x = 0 is equal to 1. The blue curve illustrates this case, ex. For comparison, functions 2x (dotted curve) and 4x (dashed curve) are shown; they are not tangent to the line of slope 1 and y-intercept 1 (red). 2.71828182845904523536028747135266249775724709369995... (sequence A001113 in OEIS). The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies An account starts with $1.00 and pays 100 percent interest per year. 1.
IntegralA definite integral of a function can be represented as the signed area of the region bounded by its graph. The term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written: However, the integrals discussed in this article are termed definite integrals. The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century. Integrals and derivatives became the basic tools of calculus, with numerous applications in science and engineering. History Pre-calculus integration The next significant advances in integral calculus did not begin to appear until the 16th century. Newton and Leibniz The major advance in integration came in the 17th century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz. Formalizing integrals Historical notation or
Euclidean algorithmEuclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because remainder EA is less than CD. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. In mathematics, the Euclidean algorithm[a], or Euclid's algorithm, is a method for computing the greatest common divisor (GCD) of two (usually positive) integers, also known as the greatest common factor (GCF) or highest common factor (HCF). The GCD of two positive integers is the largest integer that divides both of them without leaving a remainder (the GCD of two integers in general is defined in a more subtle way). The main principle is that the GCD does not change if the smaller number is subtracted from the larger number.
Vector calculusVector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration. Vector calculus plays an important role in differential geometry and in the study of partial differential equations. Vector calculus was developed from quaternion analysis by J. Basic objects The basic objects in vector calculus are scalar fields (scalar-valued functions) and vector fields (vector-valued functions). Vector operations Algebraic operations The basic algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field, and consist of: scalar multiplication vector addition dot product or
ParabolaThe parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas. Strictly, the adjective parabolic should be applied only to things that are shaped as a parabola, which is a two-dimensional shape. However, as shown in the last paragraph, the same adjective is commonly used for three-dimensional objects, such as parabolic reflectors, which are really paraboloids. Sometimes, the noun parabola is also used to refer to these objects. Part of a parabola (blue), with various features (other colours). Introductory images Click on any image to enlarge it. Description of final image History The earliest known work on conic sections was by Menaechmus in the fourth century BC. Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. Thus: Substituting: is
Multivariable calculusTypical operations Limits and continuity A study of limits and continuity in multivariable calculus yields many counter-intuitive results not demonstrated by single-variable functions. For example, there are scalar functions of two variables with points in their domain which give a particular limit when approached along any arbitrary line, yet give a different limit when approached along a parabola. approaches zero along any line through the origin. , it has a limit of 0.5. Continuity in each argument is not sufficient for multivariate continuity: For instance, in the case of a real-valued function with two real-valued parameters, , continuity of in for fixed and continuity of does not imply continuity of . It is easy to check that all real-valued functions (with one real-valued argument) that are given by are continuous in (for any fixed ). are continuous as is symmetric with regards to and . itself is not continuous as can be seen by considering the sequence (for natural if See also
CardioidA cardioid generated by a rolling circle around another circle and tracing one point on the edge of it. A cardioid given as the envelope of circles whose centers lie on a given circle and which pass through a fixed point on the given circle. The name was coined by de Castillon in 1741 but had been the subject of study decades beforehand. Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk. A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid, (any 2d plane containing the 3d straight line of the microphone body.) In three dimensions, the cardioid is shaped like an apple centred on the microphone which is the "stalk" of the apple. Equations Based on the rolling circle description, with the fixed circle having the origin as its center, and both circles having radius a, the cardioid is given by the following parametric equations: or Inverse curve