Tangent Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot. Tangent plane to a sphere As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The word tangent comes from the Latin tangere, to touch. History[edit] The first definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it".[1] This old definition prevents inflection points from having any tangent. Pierre de Fermat developed a general technique for determining the tangents of a curve using his method of adequality in the 1630s. Leibniz defined the tangent line as the line through a pair of infinitely close points on the curve. Tangent line to a curve[edit] At each point, the line is always tangent to the curve. Analytical approach[edit] Intuitive description[edit] by but If
Sine For the angle α, the sine function gives the ratio of the length of the opposite side to the length of the hypotenuse. The sine function graphed on the Cartesian plane. In this graph, the angle x is given in radians (π = 180°). The sine and cosine functions are related in multiple ways. The derivative of is . . In mathematics, the sine function is a trigonometric function of an angle. The sine function is commonly used to model periodic phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. The function sine can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.[1] The word "sine" comes from a Latin mistranslation of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.[2] Identities[edit] and
Cosine The cosine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). Let be an angle measured counterclockwise from the x-axis along the arc of the unit circle. is the horizontal coordinate of the arc endpoint. The common schoolbook definition of the cosine of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle adjacent to the angle and the hypotenuse, i.e A convenient mnemonic for remembering the definition of the sine, cosine, and tangent is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent). As a result of its definition, the cosine function is periodic with period . also obeys the identity The definition of the cosine function can be extended to complex arguments using the definition The cosine function has a fixed point at 0.739085... for is where via to
Cardioid A 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[2] but had been the subject of study decades beforehand.[3] 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.) Equations[edit] 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: In the complex plane this becomes or, in rectangular coordinates, or, in the complex plane, With the substitution u=tan t/2, or . is a cardioid.
Parabola The 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. Though not perfectly correct, this usage is generally understood. Part of a parabola (blue), with various features (other colours). Introductory images[edit] Click on any image to enlarge it. Description of final image[edit] History[edit] The earliest known work on conic sections was by Menaechmus in the fourth century BC. Equation in Cartesian coordinates[edit] as the equation of the parabola.
Euclidean algorithm Euclid'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. On the right Nicomachus' example with numbers 49 and 21 resulting in their GCD of 7 (derived from Heath 1908:300). 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).
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... The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier.[6] 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 Jacob Bernoulli discovered this constant by studying a question about compound interest:[6] 1. 2.
Derivative 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. In fact, the derivative at a point of a function of a single variable is the slope of the tangent line to the graph of the function at that point. The process of finding a derivative is called differentiation. Differentiation and the derivative[edit] 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. Notation[edit] Rigorous definition[edit] Example[edit]
Kepler conjecture The Kepler conjecture, named after the 17th-century German mathematician and astronomer Johannes Kepler, is a mathematical conjecture about sphere packing in three-dimensional Euclidean space. It says that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is slightly greater than 74%. In 1998 Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture. Background[edit] Diagrams of cubic close packing (left) and hexagonal close packing (right). Imagine filling a large container with small equal-sized spheres. Experiment shows that dropping the spheres in randomly will achieve a density of around 65%. The Kepler conjecture says that this is the best that can be done—no other arrangement of spheres has a higher average density. Origins[edit] Nineteenth century[edit]