Taylor series. As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin(x) and its Taylor approximations, polynomials of degree 1, 3, 5, 7, 9, 11 and 13. The exponential functionex (in blue), and the sum of the first n+1 terms of its Taylor series at 0 (in red). It is common practice to approximate a function by using a finite number of terms of its Taylor series. Taylor's theorem gives quantitative estimates on the error in this approximation. Definition[edit] The Taylor series of a real or complex-valued function ƒ(x) that is infinitely differentiable at a real or complex number a is the power series which can be written in the more compact sigma notation as where n!
Examples[edit] The Maclaurin series for any polynomial is the polynomial itself. The Maclaurin series for (1 − x)−1 at a = 0 is the geometric series so the Taylor series for x−1 at a = 1 is and the corresponding Taylor series for log(x) at a = 1 is History[edit] Analytic functions[edit] Hyperbolic function. A ray through the origin intercepts the unit hyperbola in the point , where is twice the area between the ray, the hyperbola, and the -axis. For points on the hyperbola below the -axis, the area is considered negative (see animated version with comparison with the trigonometric (circular) functions). In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the equilateral hyperbola. The hyperbolic functions take real values for a real argument called a hyperbolic angle. In complex analysis, the hyperbolic functions arise as the imaginary parts of sine and cosine.
Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.[3] Riccati used Sc. and Cc. Standard algebraic expressions[edit] sinh, cosh and tanh csch, sech and coth (a) cosh(x) is the average of exand e−x and Hence: where. Asymptote. The graph of a function with a horizontal, vertical, and oblique asymptote. A curve intersecting an asymptote infinitely many times and in many places. In analytic geometry, an asymptote (/ˈæsɪmptoʊt/) of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors.[1] In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity.[2][3] The word asymptote is derived from the Greek ἀσύμπτωτος (asumptotos) which means "not falling together," from ἀ priv. + σύν "together" + πτωτ-ός "fallen.
"[4] The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.[5] A "simple" example[edit] Asymptotes of functions[edit] or and In the graph of . Polar coordinate system. Points in the polar coordinate system with pole O and polar axis L.
In green, the point with radial coordinate 3 and angular coordinate 60 degrees, or (3,60°). In blue, the point (4,210°). History[edit] Hipparchus From the 8th century AD onward, astronomers developed methods for approximating and calculating the direction to Makkah (qibla)—and its distance—from any location on the Earth.[3] From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its longitude and latitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles, and whose polar axis is the line through the location and its antipodal point.[4] Conventions[edit] A polar grid with several angles labeled in degrees is mapped onto . Circle[edit] Taylor series. Hyperbolic function. Asymptote.
Polar coordinate system. Taylor series. Hyperbolic function. Asymptote. Polar coordinate system. Hyperbolic function. Asymptote. Polar coordinate system. Polar coordinate system.