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In mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry.1 Geometry
In plane geometry, a straight line is tangent to a curve, at some point, if both line and curve pass through the point with the same direction; such a line is the best straight-line approximation to the curve at that point. The curve, at point P, has the same slope as a tangent passing through P. The slope of a tangent line can be approximated by a secant line. It is a mistake to think of tangents as lines which intersect a curve at only one single point. There are tangents which intersect curves at several points (as in the following example), and there are non-tangential lines which intersect curves at only one single point. (Note that in the important case of a circle, however, the tangent line will intersect the curve at only one point.)
In the following diagram, a red line intersects the black curve at two points. It is tangent to the curve at one point.
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In higher-dimensional geometry, one can define the tangent plane for a surface in an analogous way to the tangent line for a curve. In general, one can have an (n - 1)-dimensional tangent hyperplane to an (n - 1)-dimensional manifold.
1.1 Quote
- "And I dare say that this is not only the most useful and general [concept] in geometry, that I know, but even that I ever desire to know." Descartes ( 1637)
2 Calculus
A "formal" definition of the tangent requires calculus. Specifically, suppose a curve is the graph of some functionIn mathematics, a function is a relation such that each element of a set (the domain is associated with a unique element of another (possibly the same) set (the codomain not to be confused with the range . The concept of a function is fundamental to virtu, y = f(x), and we are interested in the point (x0, y0) where y0 = f(x0). The curve has a non-vertical tangent at the point (x0, y0) if and only ifIn mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if . It is often, not always, written italicized: iff''. Although "P iff Q" is most standard, common alternative phrases include "P the function is differentiable at x0. In this case, the slopeIn mathematics, the slope (or gradient especially where three or more dimensions are discussed) of a straight line (within a Cartesian coordinate system) is a measure for the "steepness" of said line. This pages focuses on such slopes. With an understandi of the tangent is given by f '(x0). The curve has a vertical tangent at (x0, y0) if and only if the slope approaches plus or minus infinityInfinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. In theology, for instance in the work of Duns Scotus, the infinity of God carries the sense not so much of quantity (leading to the question as one approaches the point from either side.
Above, it was noted that a secant can be used to approximate a tangent; it could be said that the slope of a secant approaches the slope (or direction) of the tangent, as the secants' points of intersection approach each other. Should one also understand the notion of a limitIn mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets "close" to either some point, or infinity; or the behavior of a sequence's elements, as their index approaches infinity. Limits are used in calculu; one might understand how that concept is applicable to those discussed here, via calculus. In essence, calculus was developed (in part) as a means to find the slopes of tangents; this challenge, being known as the tangent line problem, is solvable via NewtonKneller's portrait of 1689. Sir Isaac Newton ( December 25, 1642 March 20, 1727 by the Julian calendar then in use; or January 4, 1643 March 31, 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemis's difference quotient.
Should one know the slope of a tangent, to some function; then, one can determine an equation for the tangent. For example, an understanding of the power rule will help one determine that the slope of x3, at x = 2, is 12. Using the point-slope equation, one can write an equation for this tangent: y - 8 = 12(x - 2) = 12x - 24; or: y = 12x - 16
