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
Ingeometry, thetangent line (or simplytangent) to a planecurve at a givenpoint is, intuitively, thestraight line that "just touches" the curve at that point.Leibniz defined it as the line through a pair ofinfinitely close points on the curve.[1][2] More precisely, a straight line is tangent to the curvey =f(x) at a pointx =c if the line passes through the point(c,f(c)) on the curve and hasslopef'(c), wheref' is thederivative off. A similar definition applies tospace curves and curves inn-dimensionalEuclidean space.
The point where the tangent line and the curve meet orintersect is called thepoint 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 atangent line approximation, the graph of theaffine function that best approximates the original function at the given point.[3]
Similarly, thetangent plane to asurface at a given point is theplane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions indifferential geometry and has been extensively generalized;seeTangent space.
The word "tangent" comes from theLatintangere, "to touch".
Euclid makes several references to the tangent (ἐφαπτομένηephaptoménē) to a circle in book III of theElements (c. 300 BC).[4] InApollonius' workConics (c. 225 BC) he defines a tangent as beinga line such that no other straight line could fall between it and the curve.[5]
Archimedes (c. 287 – c. 212 BC) found the tangent to anArchimedean spiral by considering the path of a point moving along the curve.[5]
In the 1630sFermat developed the technique ofadequality to calculate tangents and other problems in analysis and used this to calculate tangents to the parabola. The technique of adequality is similar to taking the difference between and and dividing by a power of. IndependentlyDescartes used hismethod of normals based on the observation that the radius of a circle is always normal to the circle itself.[6]
These methods led to the development ofdifferential calculus in the 17th century. Many people contributed.Roberval discovered a general method of drawing tangents, by considering a curve as described by a moving point whose motion is the resultant of several simpler motions.[7]René-François de Sluse andJohannes Hudde found algebraic algorithms for finding tangents.[8] Further developments included those ofJohn Wallis andIsaac Barrow, leading to the theory ofIsaac Newton andGottfried Leibniz.
An 1828 definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it".[9] This old definition preventsinflection points from having any tangent. It has been dismissed and the modern definitions are equivalent to those ofLeibniz, who defined the tangent line as the line through a pair ofinfinitely close points on the curve; in modern terminology, this is expressed as: the tangent to a curve at a pointP on the curve is thelimit of the line passing through two points of the curve when these two points tends toP.
The intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines (secant lines) passing through two points,A andB, those that lie on the function curve. The tangent atA is the limit when pointB approximates or tends toA. The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability." For example, if two circular arcs meet at a sharp point (a vertex) then there is no uniquely defined tangent at the vertex because the limit of the progression of secant lines depends on the direction in which "pointB" approaches the vertex.
At most points, the tangent touches the curve without crossing it (though it may, when continued, cross the curve at other places away from the point of tangent). A point where the tangent (at this point) crosses the curve is called aninflection point.Circles,parabolas,hyperbolas andellipses do not have any inflection point, but more complicated curves do have, like the graph of acubic function, which has exactly one inflection point, or a sinusoid, which has two inflection points per eachperiod of thesine.
Conversely, it may happen that the curve lies entirely on one side of a straight line passing through a point on it, and yet this straight line is not a tangent line. This is the case, for example, for a line passing through the vertex of atriangle and not intersecting it otherwise—where the tangent line does not exist for the reasons explained above. Inconvex geometry, such lines are calledsupporting lines.
At each point, the moving line is always tangent to thecurve. Its slope is thederivative; green marks positive derivative, red marks negative derivative and black marks zero derivative. The point (x,y) = (0,1) where the tangent intersects the curve, is not amax, or a min, but is apoint of inflection. (Note: the figure contains the incorrect labeling of 0,0 which should be 0,1)
The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or thetangent line problem, was one of the central questions leading to the development ofcalculus in the 17th century. In the second book of hisGeometry,René Descartes[10]said of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know".[11]
Suppose that a curve is given as the graph of afunction,y =f(x). To find the tangent line at the pointp = (a,f(a)), consider another nearby pointq = (a +h,f(a +h)) on the curve. Theslope of thesecant line passing throughp andq is equal to thedifference quotient
As the pointq approachesp, which corresponds to makingh smaller and smaller, the difference quotient should approach a certain limiting valuek, which is the slope of the tangent line at the pointp. Ifk 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 valuek. The precise mathematical formulation was given byCauchy in the 19th century and is based on the notion oflimit. Suppose that the graph does not have a break or a sharp edge atp and it is neither plumb nor too wiggly nearp. Then there is a unique value ofk such that, ash approaches 0, the difference quotient gets closer and closer tok, and the distance between them becomes negligible compared with the size ofh, ifh 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 functionf. This limit is thederivative of the functionf atx =a, denotedf ′(a). Using derivatives, the equation of the tangent line can be stated as follows:
Calculus provides rules for computing the derivatives of functions that are given by formulas, such as thepower function,trigonometric functions,exponential function,logarithm, and their various combinations. 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. For these points the functionf isnon-differentiable. There are two possible reasons for the method of finding the tangents based on the limits and derivatives to fail: either the geometric tangent exists, but it is a vertical line, which cannot be given in the point-slope form since it does not have a slope, or the graph exhibits one of three behaviors that precludes a geometric tangent.
The graphy =x1/3 illustrates the first possibility: here the difference quotient ata = 0 is equal toh1/3/h =h−2/3, which becomes very large ash approaches 0. This curve has a tangent line at the origin that is vertical.
The graphy =x2/3 illustrates another possibility: this graph has acusp at the origin. This means that, whenh approaches 0, the difference quotient ata = 0 approaches plus or minus infinity depending on the sign ofx. Thus both branches of the curve are near to the half vertical line for whichy=0, but none is near to the negative part of this line. Basically, there is no tangent at the origin in this case, but in some context one may consider this line as a tangent, and even, inalgebraic geometry, as adouble tangent.
The graphy = |x| of theabsolute value function consists of two straight lines with different slopes joined at the origin. As a pointq approaches the origin from the right, the secant line always has slope 1. As a pointq approaches the origin from the left, the secant line always has slope −1. Therefore, there is no unique tangent to the graph at the origin. Having two different (but finite) slopes is called acorner.
Finally, since differentiability implies continuity, thecontrapositive statesdiscontinuity implies non-differentiability. Any such jump or point discontinuity will have no tangent line. 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 byy =f(x) then the slope of the tangent isso by thepoint–slope formula the equation of the tangent line at (X, Y) is
where (x, y) are the coordinates of any point on the tangent line, and where the derivative is evaluated at.[12]
When the curve is given byy =f(x), the tangent line's equation can also be found[13] by usingpolynomial division to divide by; if the remainder is denoted by, then the equation of the tangent line is given by
When the equation of the curve is given in the formf(x, y) = 0 then the value of the slope can be found byimplicit differentiation, giving
The equation of the tangent line at a point (X,Y) such thatf(X,Y) = 0 is then[12]
This equation remains true if
in which case the slope of the tangent is infinite. If, however,
the tangent line is not defined and the point (X,Y) is said to besingular.
Foralgebraic curves, computations may be simplified somewhat by converting tohomogeneous coordinates. Specifically, let the homogeneous equation of the curve beg(x, y, z) = 0 whereg is a homogeneous function of degreen. Then, if (X, Y, Z) lies on the curve,Euler's theorem impliesIt follows that the homogeneous equation of the tangent line is
The equation of the tangent line in Cartesian coordinates can be found by settingz=1 in this equation.[14]
To apply this to algebraic curves, writef(x, y) as
where eachur is the sum of all terms of degreer. The homogeneous equation of the curve is then
Applying the equation above and settingz=1 produces
as the equation of the tangent line.[15] The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.[14]
The line perpendicular to the tangent line to a curve at the point of tangency is called thenormal line to the curve at that point. The slopes of perpendicular lines have product −1, so if the equation of the curve isy =f(x) then slope of the normal line is
and it follows that the equation of the normal line at (X, Y) is
Similarly, if the equation of the curve has the formf(x, y) = 0 then the equation of the normal line is given by[17]
The angle between two curves at a point where they intersect is defined as the angle between their tangent lines at that point. More specifically, two curves are said to be tangent at a point if they have the same tangent at a point, and orthogonal if their tangent lines are orthogonal.[18]
The limaçon trisectrix: a curve with two tangents at the origin.
The formulas above fail when the point is asingular point. In this case there may be two or more branches of the curve that pass through the point, each branch having its own tangent line. When the point is the origin, the equations of these lines can be found for algebraic curves by factoring the equation formed by eliminating all but the lowest degree terms from the original equation. Since any point can be made the origin by a change of variables (or bytranslating the curve) this gives a method for finding the tangent lines at any singular point.
For example, the equation of thelimaçon trisectrix shown to the right is
Expanding this and eliminating all but terms of degree 2 gives
which, when factored, becomes
So these are the equations of the two tangent lines through the origin.[19]
When the curve is not self-crossing, the tangent at a reference point may still not be uniquely defined because the curve is not differentiable at that point although it is differentiable elsewhere. In this case theleft 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). For example, the curvey = |x | is not differentiable atx = 0: its left and right derivatives have respective slopes −1 and 1; the tangents at that point with those slopes are called the left and right tangents.[20]
Sometimes the slopes of the left and right tangent lines are equal, so the tangent lines coincide. This is true, for example, for the curvey =x2/3, for which both the left and right derivatives atx = 0 are infinite; both the left and right tangent lines have equationx = 0.
Thetangent plane to asurface at a given pointp is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane atp, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close top as these points converge top. Mathematically, if the surface is given by a function, the equation of the tangent plane at point can be expressed as:
.
Here, and are the partial derivatives of the function with respect to and respectively, evaluated at the point. In essence, the tangent plane captures the local behavior of the surface at the specific pointp. It's a fundamental concept used in calculus and differential geometry, crucial for understanding how functions change locally on surfaces.
^In "Nova Methodus pro Maximis et Minimis" (Acta Eruditorum, Oct. 1684), Leibniz appears to have a notion of tangent lines readily from the start, but later states: "modo teneatur in genere, tangentem invenire esse rectam ducere, quae duo curvae puncta distantiam infinite parvam habentia jungat, seu latus productum polygoni infinitanguli, quod nobis curvae aequivalet", ie. defines the method for drawing tangents through points infinitely close to each other.
^Thomas L. Hankins (1985).Science and the Enlightenment. Cambridge University Press. p. 23.ISBN9780521286190.
^Katz, Victor J. (2008).A History of Mathematics (3rd ed.). Addison Wesley. p. 510.ISBN978-0321387004.
^Wolfson, Paul R. (2001). "The Crooked Made Straight: Roberval and Newton on Tangents".The American Mathematical Monthly.108 (3):206–216.doi:10.2307/2695381.JSTOR2695381.
^Katz, Victor J. (2008).A History of Mathematics (3rd ed.). Addison Wesley. pp. 512–514.ISBN978-0321387004.
^Noah Webster,American Dictionary of the English Language (New York: S. Converse, 1828), vol. 2, p. 733,[1]
