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.
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.
Euclid makes several references to the tangent (ἐφαπτομένη ephaptoménē) to a circle in book III of the Elements (c. 300 BC).
[5] Archimedes (c. 287 – c. 212 BC) found the tangent to an Archimedean spiral by considering the path of a point moving along the curve.
[6] These methods led to the development of differential calculus in the 17th century.
[7] René-François de Sluse and Johannes Hudde found algebraic algorithms for finding tangents.
An 1828 definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it".
The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability."
The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th century.
The slope of the secant line passing through p and q is equal to the difference quotient
As the point q approaches p, which corresponds to making h smaller and smaller, the difference quotient should approach a certain limiting value k, which is the slope of the tangent line at the point p. If k is known, the equation of the tangent line can be found in the point-slope form:
To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value k. The precise mathematical formulation was given by Cauchy in the 19th century and is based on the notion of limit.
Suppose that the graph does not have a break or a sharp edge at p and it is neither plumb nor too wiggly near p. Then there is a unique value of k such that, as h approaches 0, the difference quotient gets closer and closer to k, and the distance between them becomes negligible compared with the size of h, if h is small enough.
This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the derivative of the function f at x = a, denoted f ′(a).
Thus, equations of the tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus.
Calculus also demonstrates that there are functions and points on their graphs for which the limit determining the slope of the tangent line does not exist.
The graph y = x1/3 illustrates the first possibility: here the difference quotient at a = 0 is equal to h1/3/h = h−2/3, which becomes very large as h approaches 0.
The graph y = |x| of the absolute value function consists of two straight lines with different slopes joined at the origin.
As a point q approaches the origin from the right, the secant line always has slope 1.
As a point q approaches the origin from the left, the secant line always has slope −1.
This includes cases where one slope approaches positive infinity while the other approaches negative infinity, leading to an infinite jump discontinuity When the curve is given by y = f(x) then the slope of the tangent is
[12] When the curve is given by y = f(x), the tangent line's equation can also be found[13] by using polynomial division to divide
For algebraic curves, computations may be simplified somewhat by converting to homogeneous coordinates.
[15] The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.
However, it may occur that the tangent line exists and may be computed from an implicit equation of the curve.
In this case the left and right derivatives are defined as the limits of the derivative as the point at which it is evaluated approaches the reference point from respectively the left (lower values) or the right (higher values).
Two distinct circles lying in the same plane are said to be tangent to each other if they meet at exactly one point.
If points in the plane are described using Cartesian coordinates, then two circles, with radii
In essence, the tangent plane captures the local behavior of the surface at the specific point p. It's a fundamental concept used in calculus and differential geometry, crucial for understanding how functions change locally on surfaces.
