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Vertical Tangent
In mathematics, particularly calculus, a vertical tangent is a tangent, tangent line that is Vertical direction, vertical. Because a vertical line has Infinity, infinite slope, a Function (mathematics), function whose graph of a function, graph has a vertical tangent is not Differentiable function, differentiable at the point of tangency. Limit definition A function ƒ has a vertical tangent at ''x'' = ''a'' if the difference quotient used to define the derivative has Limit_of_a_function#Infinite_limits, infinite limit: :\lim_\frac = \quad\text\quad\lim_\frac = . The graph of ƒ has a vertical tangent at ''x'' = ''a'' if the derivative of ƒ at ''a'' is either positive or negative infinity. For a continuous function, it is often possible to detect a vertical tangent by taking the limit of the derivative. If :\lim_ f'(x) = \text then ƒ must have an upward-sloping vertical tangent at ''x'' = ''a''. Similarly, if :\lim_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Calculus
Calculus is the mathematics, 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. Originally called infinitesimal calculus or "the calculus of infinitesimals", it has two major branches, differential calculus and integral calculus. The former concerns instantaneous Rate of change (mathematics), rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus. They make use of the fundamental notions of convergence (mathematics), convergence of infinite sequences and Series (mathematics), infinite series to a well-defined limit (mathematics), limit. It is the "mathematical backbone" for dealing with problems where variables change with time or another reference variable. Infinitesimal calculus was formulated separately ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More precisely, a straight line is tangent to the curve at a point if the line passes through the point on the curve and has slope , where ''f'' is the derivative of ''f''. A similar definition applies to space curves and curves in ''n''-dimensional Euclidean space. The point where the tangent line and the curve meet or intersect is called the ''point of tangency''. The tangent line is said to be "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point. The tangent line to a point on a differentiable curve can also be thought of as a '' tangent line approximation'', the graph of the affine function that best approximates the original function at the given point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Vertical Direction
In astronomy, geography, and related sciences and contexts, a ''Direction (geometry, geography), direction'' or ''plane (geometry), plane'' passing by a given point is said to be vertical if it contains the local gravity direction at that point. Conversely, a direction, plane, or surface is said to be horizontal (or ''leveled'') if it is everywhere perpendicular to the vertical direction. In general, something that is vertical can be drawn from up to down (or down to up), such as the y-axis in the Cartesian coordinate system. Historical definition The word ''horizontal'' is derived from the Latin , which derives from the Greek , meaning 'separating' or 'marking a boundary'. The word ''vertical'' is derived from the late Latin ', which is from the same root as ''vertex'', meaning 'highest point' or more literally the 'turning point' such as in a whirlpool. Girard Desargues defined the vertical to be perpendicular to the horizon in his 1636 book ''Perspective''. Geophysical defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including Guillaume de l'Hôpital, l'Hôpital and Johann Bernoulli, Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or Magnitude (mathematics), magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
