3701:
52:
44:
4109:
471:
3712:. 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 by
5402:, 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.
510:. 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 "point
577:
1049:. 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.
2367:
2194:
2778:
3691:
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.
5067:
1774:
1616:
4413:
1886:
1159:
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
4729:
4573:
555:
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 a
3510:
1991:
517:
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 an
189:. 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
2062:
2202:
600:
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 the
3135:
2941:
2631:
3960:
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 the
2616:
3965:
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 curve
3675:
4891:
622:
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".
3835:
3228:
2477:
1306:
1144:
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 a
3396:
1113:=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, in
5144:
5107:
1041:
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 function
3314:
1647:
1012:
769:
3907:
3015:
4884:
3952:
3573:
2848:
1528:
298:
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
4272:
1785:
859:
4588:
4432:
5250:
4264:
4218:
588:; 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 a
4172:
1431:
4825:
1501:
1392:
1215:
1033:, 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.
331:
3419:
1897:
1460:
360:
2783:
as the equation of the tangent line. The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.
1343:
401:
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.
5204:
5184:
5164:
4091:
4067:
380:
2362:{\displaystyle {\frac {\partial g}{\partial x}}(X,Y,Z)\cdot x+{\frac {\partial g}{\partial y}}(X,Y,Z)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,Z)\cdot z=0.}
5368:
6787:
3978:= 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.
909:
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
6775:
3023:
2189:{\displaystyle {\frac {\partial g}{\partial x}}\cdot X+{\frac {\partial g}{\partial y}}\cdot Y+{\frac {\partial g}{\partial z}}\cdot Z=ng(X,Y,Z)=0.}
2773:{\displaystyle {\frac {\partial f}{\partial x}}(X,Y)\cdot x+{\frac {\partial f}{\partial y}}(X,Y)\cdot y+{\frac {\partial g}{\partial z}}(X,Y,1)=0}
428:
An 1828 definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it". This old definition prevents
3233:
the tangent line is not defined. However, it may occur that the tangent line exists and may be computed from an implicit equation of the curve.
2859:
619:
5256:. It's a fundamental concept used in calculus and differential geometry, crucial for understanding how functions change locally on surfaces.
3981:
Sometimes the slopes of the left and right tangent lines are equal, so the tangent lines coincide. This is true, for example, for the curve
6897:
6782:
869:
To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value
2501:
1519:
6765:
6760:
6770:
6755:
5869:
5062:{\displaystyle z-z_{0}={\frac {\partial f}{\partial x}}(x_{0},y_{0})(x-x_{0})+{\frac {\partial f}{\partial y}}(x_{0},y_{0})(y-y_{0})}
4774:
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 at
3584:
486:
The intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines (
6057:
4747:
6750:
3729:
3146:
6367:
6121:
5393:
2397:
1235:
5919:
5570:
5510:
4778:, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to
6865:
6724:
5421:
4070:
3325:
1769:{\displaystyle {\frac {\partial f}{\partial x}}(X,Y)\cdot (x-X)+{\frac {\partial f}{\partial y}}(X,Y)\cdot (y-Y)=0.}
6279:
6195:
5353:
5112:
5075:
3251:
to the curve at that point. The slopes of perpendicular lines have product −1, so if the equation of the curve is
214:
that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in
6860:
6792:
6417:
6272:
6240:
5999:
4034:
3269:
17:
2056:
6928:
6923:
6493:
6470:
6185:
712:
6583:
6521:
6316:
6190:
5862:
5809:
5624:
3846:
2949:
1611:{\displaystyle {\frac {dy}{dx}}=-{\frac {\partial f}{\partial x}}{\bigg /}{\frac {\partial f}{\partial y}}.}
560:
and not intersecting it otherwise—where the tangent line does not exist for the reasons explained above. In
6938:
6069:
6047:
5590:
1017:
Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the
914:
6892:
4408:{\displaystyle \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}\pm r_{2}\right)^{2}.}
1881:{\displaystyle {\frac {\partial f}{\partial y}}(X,Y)=0,\quad {\frac {\partial f}{\partial x}}(X,Y)\neq 0,}
943:
461:
6877:
6643:
6257:
6079:
5804:
4830:
3918:
3521:
2796:
402:
6933:
6262:
6032:
4724:{\displaystyle \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}-r_{2}\right)^{2}.}
4568:{\displaystyle \left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}=\left(r_{1}+r_{2}\right)^{2}.}
465:
191:
804:
6681:
6628:
4751:
3962:
3709:
2005:
433:
432:
from having any tangent. It has been dismissed and the modern definitions are equivalent to those of
6089:
5373:
440:
points on the curve; in modern terminology, this is expressed as: the tangent to a curve at a point
6797:
6568:
6116:
5855:
5358:
5209:
4223:
4177:
37:
5799:
1218:
544:, which has exactly one inflection point, or a sinusoid, which has two inflection points per each
6563:
6235:
5784:
5451:
5252:. In essence, the tangent plane captures the local behavior of the surface at the specific point
4757:
4137:
4042:
1319:) are the coordinates of any point on the tangent line, and where the derivative is evaluated at
1022:
180:
6691:
6573:
6394:
6342:
6148:
6126:
5994:
5437:
3713:
3281:
2908:
2016:
1580:
1397:
1136:
function consists of two straight lines with different slopes joined at the origin. As a point
632:
33:
4789:
3505:{\displaystyle {\frac {\partial f}{\partial y}}(x-X)-{\frac {\partial f}{\partial x}}(y-Y)=0.}
1986:{\displaystyle {\frac {\partial f}{\partial y}}(X,Y)={\frac {\partial f}{\partial x}}(X,Y)=0,}
6817:
6676:
6588:
6245:
6180:
6153:
6143:
6064:
6052:
6037:
6009:
4124:
4030:
3247:
The line perpendicular to the tangent line to a curve at the point of tangency is called the
1468:
874:
565:
394:
215:
47:
Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.
5841:
5670:
Strickland-Constable, Charles, "A simple method for finding tangents to polynomial graphs",
5602:
3720:
1367:
1183:
390:
based on the observation that the radius of a circle is always normal to the circle itself.
301:
6633:
6252:
6099:
5672:
1026:
540:
do not have any inflection point, but more complicated curves do have, like the graph of a
261:
1436:
336:
8:
6653:
6578:
6465:
6422:
6173:
6158:
5989:
5977:
5964:
5924:
5904:
4740:
2787:
1361:
1322:
878:
704:
593:
445:
266:
211:
409:
found algebraic algorithms for finding tangents. Further developments included those of
6742:
6717:
6548:
6501:
6442:
6407:
6402:
6382:
6377:
6372:
6337:
6284:
6267:
6168:
6042:
6027:
5972:
5939:
5778:
5641:
5608:
5543:
5526:
Wolfson, Paul R. (2001). "The
Crooked Made Straight: Roberval and Newton on Tangents".
5303:
5189:
5169:
5149:
4767:
4076:
4052:
4046:
1114:
1093:
398:
365:
284:
207:
4582:
if the distance between their centres is equal to the difference between their radii:
786:
smaller and smaller, the difference quotient should approach a certain limiting value
234:
6882:
6706:
6638:
6460:
6437:
6311:
6304:
6207:
6022:
5914:
5818:
5566:
5506:
5417:
5313:
5308:
4022:
3242:
1140:
approaches the origin from the right, the secant line always has slope 1. As a point
589:
545:
422:
387:
3716:
the curve) this gives a method for finding the tangent lines at any singular point.
616:
6840:
6623:
6536:
6516:
6447:
6357:
6299:
6291:
6225:
6138:
5899:
5894:
5759:
5633:
5535:
5348:
5323:
5318:
3686:
576:
520:
475:
448:
of the line passing through two points of the curve when these two points tends to
429:
76:
2372:
The equation of the tangent line in
Cartesian coordinates can be found by setting
6902:
6887:
6671:
6526:
6506:
6475:
6452:
6432:
6326:
5982:
5929:
5821:
5398:
5369:
Multiplicity (mathematics)#Behavior of a polynomial function near a multiple root
5338:
5291:
4103:
4037:
in the context of curves in R. More generally, tangent vectors are elements of a
2012:
937:). Using derivatives, the equation of the tangent line can be stated as follows:
798:
is known, the equation of the tangent line can be found in the point-slope form:
561:
196:
173:
6812:
6711:
6558:
6511:
6412:
6215:
5760:"Circles For Leaving Certificate Honours Mathematics by Thomas O'Sullivan 1997"
5363:
4018:
4007:
1152:
1133:
1109:. Thus both branches of the curve are near to the half vertical line for which
1018:
596:. (Note: the figure contains the incorrect labeling of 0,0 which should be 0,1)
541:
406:
6230:
5476:
3130:{\displaystyle {\frac {dx}{dt}}(T)\cdot (y-Y)={\frac {dy}{dt}}(T)\cdot (x-X).}
6917:
6686:
6541:
6427:
6131:
6106:
5328:
5275:
5265:
4038:
901:, and the distance between them becomes negligible compared with the size of
437:
221:
91:
points on the curve. More precisely, a straight line is tangent to the curve
88:
80:
4423:
1080:
approaches 0. This curve has a tangent line at the origin that is vertical.
6696:
6666:
6531:
6094:
5343:
4112:
Two pairs of tangent circles. Above internally and below externally tangent
611:
418:
414:
275:
a line such that no other straight line could fall between it and the curve
5835:
2936:{\displaystyle {\frac {dy}{dx}}={\frac {dy}{dt}}{\bigg /}{\frac {dx}{dt}}}
5944:
5886:
4014:
3700:
692:
487:
479:
410:
165:
3957:
So these are the equations of the two tangent lines through the origin.
29:
In mathematics, straight line touching a plane curve without crossing it
6661:
6593:
6347:
6220:
6084:
6074:
6017:
5645:
5547:
5333:
585:
295:
280:
157:
3993:= 0 are infinite; both the left and right tangent lines have equation
51:
6855:
6603:
6598:
5909:
5826:
1030:
533:
383:
43:
5637:
5539:
2611:{\displaystyle g=u_{n}+u_{n-1}z+\dots +u_{1}z^{n-1}+u_{0}z^{n}=0.\,}
1060:
illustrates the first possibility: here the difference quotient at
6850:
6352:
5878:
5283:
606:
557:
529:
60:
4739:"Tangent plane" redirects here. For the geographical concept, see
4108:
881:. Suppose that the graph does not have a break or a sharp edge at
470:
6701:
4128:
3319:
and it follows that the equation of the normal line at (X, Y) is
1891:
in which case the slope of the tangent is infinite. If, however,
537:
199:
that best approximates the original function at the given point.
84:
3704:
The limaçon trisectrix: a curve with two tangents at the origin.
2196:
It follows that the homogeneous equation of the tangent line is
897:
approaches 0, the difference quotient gets closer and closer to
6870:
5934:
3840:
Expanding this and eliminating all but terms of degree 2 gives
1105:= 0 approaches plus or minus infinity depending on the sign of
605:
was one of the central questions leading to the development of
525:
291:
247:
3670:{\displaystyle {\frac {dx}{dt}}(x-X)+{\frac {dy}{dt}}(y-Y)=0.}
5949:
4132:
4026:
2019:. Specifically, let the homogeneous equation of the curve be
688:
581:
436:, who defined the tangent line as the line through a pair of
251:
229:
133:
72:
4116:
Two distinct circles lying in the same plane are said to be
5847:
3830:{\displaystyle (x^{2}+y^{2}-2ax)^{2}=a^{2}(x^{2}+y^{2}).\,}
2015:, computations may be simplified somewhat by converting to
549:
287:
by considering the path of a point moving along the curve.
4426:
between their centres is equal to the sum of their radii,
3223:{\displaystyle {\frac {dx}{dt}}(T)={\frac {dy}{dt}}(T)=0,}
1360:), the tangent line's equation can also be found by using
1151:
Finally, since differentiability implies continuity, the
4786:. Mathematically, if the surface is given by a function
580:
At each point, the moving line is always tangent to the
4033:
at a given point. Tangent vectors are described in the
3413:) = 0 then the equation of the normal line is given by
498:, those that lie on the function curve. The tangent at
179:
The point where the tangent line and the curve meet or
5115:
5078:
4748:
Differential geometry of surfaces § Tangent plane
790:, which is the slope of the tangent line at the point
5816:
5212:
5192:
5172:
5152:
4894:
4833:
4792:
4591:
4435:
4275:
4226:
4180:
4140:
4079:
4055:
3921:
3849:
3732:
3587:
3524:
3422:
3401:
Similarly, if the equation of the curve has the form
3328:
3272:
3149:
3026:
2952:
2862:
2799:
2634:
2504:
2400:
2205:
2065:
1900:
1788:
1650:
1531:
1506:
When the equation of the curve is given in the form
1471:
1439:
1400:
1370:
1325:
1238:
1186:
946:
807:
715:
368:
339:
304:
4045:. Tangent vectors can also be described in terms of
2472:{\displaystyle f=u_{n}+u_{n-1}+\dots +u_{1}+u_{0}\,}
1462:, then the equation of the tangent line is given by
873:. The precise mathematical formulation was given by
5745:Thomas, George B. Jr., and Finney, Ross L. (1979),
5632:(8). Mathematical Association of America: 495–512.
3989:, for which both the left and right derivatives at
5565:(3rd ed.). Addison Wesley. pp. 512–514.
5244:
5198:
5178:
5158:
5138:
5101:
5061:
4878:
4819:
4723:
4567:
4407:
4258:
4212:
4166:
4085:
4061:
3946:
3901:
3829:
3669:
3567:
3504:
3390:
3308:
3222:
3129:
3009:
2935:
2842:
2772:
2610:
2471:
2361:
2188:
1985:
1880:
1768:
1610:
1518:) = 0 then the value of the slope can be found by
1495:
1454:
1425:
1386:
1337:
1301:{\displaystyle y-Y={\frac {dy}{dx}}(X)\cdot (x-X)}
1300:
1209:
1091:illustrates another possibility: this graph has a
1006:
877:in the 19th century and is based on the notion of
853:
763:
374:
354:
325:
5411:
4120:to each other if they meet at exactly one point.
6915:
4000:
2495:. The homogeneous equation of the curve is then
631:Suppose that a curve is given as the graph of a
455:
5622:R. E. Langer (October 1937). "Rene Descartes".
5589:(New York: S. Converse, 1828), vol. 2, p. 733,
5259:
3695:
1996:the tangent line is not defined and the point (
609:in the 17th century. In the second book of his
283:(c. 287 – c. 212 BC) found the tangent to an
4073:of the algebra defined by the set of germs at
3391:{\displaystyle (x-X)+{\frac {dy}{dx}}(y-Y)=0.}
397:in the 17th century. Many people contributed.
5863:
5505:(3rd ed.). Addison Wesley. p. 510.
5139:{\textstyle {\frac {\partial f}{\partial y}}}
5102:{\textstyle {\frac {\partial f}{\partial x}}}
4827:, the equation of the tangent plane at point
4734:
1621:The equation of the tangent line at a point (
83:that "just touches" the curve at that point.
5621:
5146:are the partial derivatives of the function
3708:The formulas above fail when the point is a
2946:giving the equation for the tangent line at
885:and it is neither plumb nor too wiggly near
864:
5611:; Latham, Marcia L. Open Court. p. 95.
5587:American Dictionary of the English Language
4123:If points in the plane are described using
3309:{\displaystyle -1{\bigg /}{\frac {dy}{dx}}}
232:
5870:
5856:
5776:
5416:. Cambridge University Press. p. 23.
4049:. Formally, a tangent vector at the point
1036:
462:Differentiable curve § Tangent vector
273:(c. 225 BC) he defines a tangent as being
6898:Regiomontanus' angle maximization problem
5600:
3898:
3826:
3236:
2988:
2966:
2953:
2607:
2468:
2379:To apply this to algebraic curves, write
1374:
1101:approaches 0, the difference quotient at
1003:
850:
764:{\displaystyle {\frac {f(a+h)-f(a)}{h}}.}
647:). To find the tangent line at the point
250:makes several references to the tangent (
87:defined it as the line through a pair of
6741:
4107:
3699:
3578:then the equation of the normal line is
3515:If the curve is given parametrically by
2621:Applying the equation above and setting
626:
575:
469:
393:These methods led to the development of
50:
42:
6246:Differentiating under the integral sign
5705:
5703:
5684:
5682:
5657:
5655:
5525:
5470:
5468:
4752:Parametric surface § Tangent plane
3902:{\displaystyle a^{2}(3x^{2}-y^{2})=0\,}
3680:
3010:{\displaystyle \,t=T,\,X=x(T),\,Y=y(T)}
14:
6916:
571:
218:and has been extensively generalized;
6122:Inverse functions and differentiation
5851:
5817:
5783:. London: MacMillan and Co. pp.
5206:respectively, evaluated at the point
1221:the equation of the tangent line at (
1097:at the origin. This means that, when
116:if the line passes through the point
5730:
5721:
5712:
5700:
5691:
5679:
5652:
5560:
5500:
5465:
5394:Nova Methodus pro Maximis et Minimis
2039:is a homogeneous function of degree
1007:{\displaystyle y=f(a)+f'(a)(x-a).\,}
5749:, Addison Wesley Publ. Co.: p. 140.
4879:{\displaystyle (x_{0},y_{0},z_{0})}
4266:are tangent to each other whenever
3947:{\displaystyle y=\pm {\sqrt {3}}x.}
3568:{\displaystyle x=x(t),\quad y=y(t)}
3263:) then slope of the normal line is
2843:{\displaystyle x=x(t),\quad y=y(t)}
1180:) then the slope of the tangent is
24:
5920:Free variables and bound variables
5449:
5374:Algebraic curve#Tangent at a point
5127:
5119:
5090:
5082:
4999:
4991:
4925:
4917:
4097:
3687:Angle § Angles between curves
3472:
3464:
3434:
3426:
2734:
2726:
2690:
2682:
2646:
2638:
2491:is the sum of all terms of degree
2317:
2309:
2267:
2259:
2217:
2209:
2135:
2127:
2106:
2098:
2077:
2069:
1950:
1942:
1912:
1904:
1845:
1837:
1800:
1792:
1718:
1710:
1662:
1654:
1596:
1588:
1569:
1561:
889:. Then there is a unique value of
663:)), consider another nearby point
228:The word "tangent" comes from the
164:. A similar definition applies to
25:
6950:
6725:The Method of Mechanical Theorems
5792:
5528:The American Mathematical Monthly
5474:
3719:For example, the equation of the
2853:then the slope of the tangent is
1433:; if the remainder is denoted by
259:) to a circle in book III of the
6280:Partial fractions in integration
6196:Stochastic differential equation
4006:This section is an excerpt from
854:{\displaystyle y-f(a)=k(x-a).\,}
6418:Jacobian matrix and determinant
6273:Tangent half-angle substitution
6241:Fundamental theorem of calculus
5752:
5739:
5664:
5615:
5594:
4035:differential geometry of curves
3546:
2821:
1833:
6494:Arithmetico-geometric sequence
6186:Ordinary differential equation
5747:Calculus and Analytic Geometry
5604:The Geometry of René Descartes
5579:
5554:
5519:
5494:
5443:
5430:
5405:
5386:
5239:
5213:
5056:
5037:
5034:
5008:
4982:
4963:
4960:
4934:
4873:
4834:
4814:
4802:
4253:
4227:
4207:
4181:
3912:which, when factored, becomes
3889:
3860:
3820:
3794:
3772:
3733:
3658:
3646:
3620:
3608:
3562:
3556:
3540:
3534:
3493:
3481:
3455:
3443:
3379:
3367:
3341:
3329:
3208:
3202:
3176:
3170:
3121:
3109:
3103:
3097:
3071:
3059:
3053:
3047:
3004:
2998:
2982:
2976:
2837:
2831:
2815:
2809:
2761:
2743:
2711:
2699:
2667:
2655:
2344:
2326:
2294:
2276:
2244:
2226:
2177:
2159:
1971:
1959:
1933:
1921:
1866:
1854:
1821:
1809:
1779:This equation remains true if
1757:
1745:
1739:
1727:
1701:
1689:
1683:
1671:
1487:
1481:
1449:
1443:
1414:
1401:
1381:
1375:
1295:
1283:
1277:
1271:
1076:, which becomes very large as
997:
985:
982:
976:
962:
956:
844:
832:
823:
817:
782:, which corresponds to making
749:
743:
734:
722:
490:) passing through two points,
349:
343:
320:
308:
32:For the tangent function, see
13:
1:
6317:Integro-differential equation
6191:Partial differential equation
5625:American Mathematical Monthly
5414:Science and the Enlightenment
5379:
5245:{\displaystyle (x_{0},y_{0})}
4259:{\displaystyle (x_{2},y_{2})}
4213:{\displaystyle (x_{1},y_{1})}
4001:Tangent line to a space curve
456:Tangent line to a plane curve
5877:
5842:Tangent and first derivative
5260:Higher-dimensional manifolds
4782:as these points converge to
3696:Multiple tangents at a point
1163:
7:
6471:Generalized Stokes' theorem
6258:Integration by substitution
5844:— An interactive simulation
5805:Encyclopedia of Mathematics
5297:
5270:More generally, there is a
4418:The two circles are called
4167:{\displaystyle r_{1},r_{2}}
3974:| is not differentiable at
1348:When the curve is given by
1168:When the curve is given by
417:, leading to the theory of
362:and dividing by a power of
294:developed the technique of
219:
10:
6955:
6000:(ε, δ)-definition of limit
5838:With interactive animation
5770:
5438:Best Affine Approximations
5412:Thomas L. Hankins (1985).
5263:
4755:
4745:
4738:
4735:Tangent plane to a surface
4101:
4005:
3963:left and right derivatives
3684:
3240:
459:
252:
242:
192:tangent line approximation
31:
6893:Proof that 22/7 exceeds π
6830:
6808:
6734:
6682:Gottfried Wilhelm Leibniz
6652:
6629:e (mathematical constant)
6614:
6486:
6393:
6325:
6206:
6008:
5963:
5885:
5676:, November 2005, 466–467.
5601:Descartes, René (1954) .
1426:{\displaystyle (x-X)^{2}}
865:More rigorous description
514:" approaches the vertex.
506:approximates or tends to
55:Tangent plane to a sphere
6644:Stirling's approximation
6117:Implicit differentiation
6065:Rules of differentiation
5563:A History of Mathematics
5561:Katz, Victor J. (2008).
5503:A History of Mathematics
5501:Katz, Victor J. (2008).
5477:"e-CALCULUS Section 2.8"
5436:Dan Sloughter (2000) . "
5359:Tangent lines to circles
4820:{\displaystyle z=f(x,y)}
1520:implicit differentiation
564:, such lines are called
502:is the limit when point
210:at a given point is the
38:Tangent (disambiguation)
6878:Euler–Maclaurin formula
6783:trigonometric functions
6236:Constant of integration
4758:Normal plane (geometry)
4043:differentiable manifold
2017:homogeneous coordinates
1496:{\displaystyle y=g(x).}
1037:How the method can fail
1023:trigonometric functions
6847:Differential geometry
6692:Infinitesimal calculus
6395:Multivariable calculus
6343:Directional derivative
6149:Second derivative test
6127:Logarithmic derivative
6100:General Leibniz's rule
5995:Order of approximation
5246:
5200:
5180:
5160:
5140:
5103:
5063:
4880:
4821:
4725:
4569:
4409:
4260:
4214:
4168:
4113:
4087:
4063:
3948:
3903:
3831:
3723:shown to the right is
3705:
3671:
3569:
3506:
3392:
3310:
3237:Normal line to a curve
3224:
3131:
3011:
2937:
2844:
2786:If the curve is given
2774:
2612:
2473:
2363:
2190:
1987:
1882:
1770:
1612:
1497:
1456:
1427:
1388:
1387:{\displaystyle f\,(x)}
1339:
1302:
1211:
1210:{\displaystyle dy/dx,}
1008:
855:
765:
597:
483:
466:Frenet–Serret formulas
403:René-François de Sluse
376:
356:
327:
326:{\displaystyle f(x+h)}
233:
56:
48:
36:. For other uses, see
34:Tangent (trigonometry)
6929:Differential topology
6924:Differential geometry
6766:logarithmic functions
6761:exponential functions
6677:Generality of algebra
6555:Tests of convergence
6181:Differential equation
6165:Further applications
6154:Extreme value theorem
6144:First derivative test
6038:Differential operator
6010:Differential calculus
5780:Differential Calculus
5247:
5201:
5181:
5161:
5141:
5104:
5064:
4886:can be expressed as:
4881:
4822:
4746:Further information:
4726:
4570:
4410:
4261:
4215:
4169:
4125:Cartesian coordinates
4111:
4088:
4064:
4025:that is tangent to a
3949:
3904:
3832:
3703:
3672:
3570:
3507:
3393:
3311:
3241:Further information:
3225:
3132:
3012:
2938:
2845:
2775:
2613:
2474:
2376:=1 in this equation.
2364:
2191:
2055:) lies on the curve,
1988:
1883:
1771:
1613:
1498:
1457:
1428:
1389:
1340:
1303:
1212:
1009:
856:
766:
687:)) on the curve. The
627:Intuitive description
603:tangent line problem,
592:, or a min, but is a
579:
473:
460:Further information:
395:differential calculus
377:
357:
328:
216:differential geometry
132:on the curve and has
79:is, intuitively, the
54:
46:
6831:Miscellaneous topics
6771:hyperbolic functions
6756:irrational functions
6634:Exponential function
6487:Sequences and series
6253:Integration by parts
5673:Mathematical Gazette
5354:Tangential component
5210:
5190:
5170:
5150:
5113:
5076:
4892:
4831:
4790:
4589:
4433:
4273:
4224:
4178:
4138:
4077:
4053:
3919:
3847:
3730:
3681:Angle between curves
3585:
3522:
3420:
3326:
3270:
3147:
3024:
2950:
2860:
2797:
2632:
2502:
2398:
2203:
2063:
1898:
1786:
1648:
1529:
1469:
1455:{\displaystyle g(x)}
1437:
1398:
1368:
1323:
1236:
1184:
1027:exponential function
944:
913:. This limit is the
805:
713:
444:on the curve is the
366:
355:{\displaystyle f(x)}
337:
302:
6939:Elementary geometry
6818:List of derivatives
6654:History of calculus
6569:Cauchy condensation
6466:Exterior derivative
6423:Lagrange multiplier
6159:Maximum and minimum
5990:Limit of a sequence
5978:Limit of a function
5925:Graph of a function
5905:Continuous function
5836:Tangent to a circle
5777:J. Edwards (1892).
5609:Smith, David Eugene
5452:"Euclid's Elements"
5278:at each point of a
4741:Local tangent plane
1362:polynomial division
1338:{\displaystyle x=X}
1219:point–slope formula
705:difference quotient
594:point of inflection
584:. Its slope is the
572:Analytical approach
195:, the graph of the
6751:rational functions
6718:Method of Fluxions
6564:Alternating series
6461:Differential forms
6443:Partial derivative
6403:Divergence theorem
6285:Quadratic integral
6053:Leibniz's notation
6043:Mean value theorem
6028:Partial derivative
5973:Indeterminate form
5819:Weisstein, Eric W.
5242:
5196:
5176:
5156:
5136:
5099:
5059:
4876:
4817:
4721:
4580:internally tangent
4565:
4420:externally tangent
4405:
4256:
4210:
4164:
4114:
4083:
4059:
3944:
3899:
3827:
3721:limaçon trisectrix
3706:
3667:
3565:
3502:
3388:
3306:
3220:
3127:
3007:
2933:
2840:
2770:
2608:
2469:
2359:
2186:
1983:
1878:
1766:
1608:
1493:
1452:
1423:
1384:
1335:
1298:
1207:
1115:algebraic geometry
1047:non-differentiable
1004:
851:
761:
598:
484:
372:
352:
323:
285:Archimedean spiral
57:
49:
6934:Analytic geometry
6911:
6910:
6837:Complex calculus
6826:
6825:
6707:Law of Continuity
6639:Natural logarithm
6624:Bernoulli numbers
6615:Special functions
6574:Direct comparison
6438:Multiple integral
6312:Integral equation
6208:Integral calculus
6139:Stationary points
6113:Other techniques
6058:Newton's notation
6023:Second derivative
5915:Finite difference
5314:Osculating circle
5309:Normal (geometry)
5199:{\displaystyle y}
5179:{\displaystyle x}
5159:{\displaystyle f}
5134:
5097:
5006:
4932:
4770:at a given point
4086:{\displaystyle x}
4062:{\displaystyle x}
3936:
3644:
3606:
3479:
3441:
3365:
3304:
3243:Normal (geometry)
3200:
3168:
3095:
3045:
2931:
2904:
2881:
2741:
2697:
2653:
2324:
2274:
2224:
2142:
2113:
2084:
1957:
1919:
1852:
1807:
1725:
1669:
1603:
1576:
1550:
1269:
756:
430:inflection points
423:Gottfried Leibniz
388:method of normals
375:{\displaystyle h}
186:point of tangency
16:(Redirected from
6946:
6841:Contour integral
6739:
6738:
6589:Limit comparison
6498:Types of series
6457:Advanced topics
6448:Surface integral
6292:Trapezoidal rule
6231:Basic properties
6226:Riemann integral
6174:Taylor's theorem
5900:Concave function
5895:Binomial theorem
5872:
5865:
5858:
5849:
5848:
5832:
5831:
5813:
5788:
5764:
5763:
5756:
5750:
5743:
5737:
5736:Edwards Art. 197
5734:
5728:
5727:Edwards Art. 195
5725:
5719:
5718:Edwards Art. 194
5716:
5710:
5709:Edwards Art. 196
5707:
5698:
5697:Edwards Art. 193
5695:
5689:
5688:Edwards Art. 192
5686:
5677:
5668:
5662:
5661:Edwards Art. 191
5659:
5650:
5649:
5619:
5613:
5612:
5607:. Translated by
5598:
5592:
5583:
5577:
5576:
5558:
5552:
5551:
5523:
5517:
5516:
5498:
5492:
5491:
5489:
5487:
5481:
5472:
5463:
5462:
5460:
5458:
5447:
5441:
5434:
5428:
5427:
5409:
5403:
5390:
5349:Tangential angle
5324:Osculating plane
5319:Osculating curve
5251:
5249:
5248:
5243:
5238:
5237:
5225:
5224:
5205:
5203:
5202:
5197:
5185:
5183:
5182:
5177:
5166:with respect to
5165:
5163:
5162:
5157:
5145:
5143:
5142:
5137:
5135:
5133:
5125:
5117:
5108:
5106:
5105:
5100:
5098:
5096:
5088:
5080:
5068:
5066:
5065:
5060:
5055:
5054:
5033:
5032:
5020:
5019:
5007:
5005:
4997:
4989:
4981:
4980:
4959:
4958:
4946:
4945:
4933:
4931:
4923:
4915:
4910:
4909:
4885:
4883:
4882:
4877:
4872:
4871:
4859:
4858:
4846:
4845:
4826:
4824:
4823:
4818:
4730:
4728:
4727:
4722:
4717:
4716:
4711:
4707:
4706:
4705:
4693:
4692:
4674:
4673:
4668:
4664:
4663:
4662:
4650:
4649:
4631:
4630:
4625:
4621:
4620:
4619:
4607:
4606:
4574:
4572:
4571:
4566:
4561:
4560:
4555:
4551:
4550:
4549:
4537:
4536:
4518:
4517:
4512:
4508:
4507:
4506:
4494:
4493:
4475:
4474:
4469:
4465:
4464:
4463:
4451:
4450:
4414:
4412:
4411:
4406:
4401:
4400:
4395:
4391:
4390:
4389:
4377:
4376:
4358:
4357:
4352:
4348:
4347:
4346:
4334:
4333:
4315:
4314:
4309:
4305:
4304:
4303:
4291:
4290:
4265:
4263:
4262:
4257:
4252:
4251:
4239:
4238:
4219:
4217:
4216:
4211:
4206:
4205:
4193:
4192:
4173:
4171:
4170:
4165:
4163:
4162:
4150:
4149:
4092:
4090:
4089:
4084:
4068:
4066:
4065:
4060:
3953:
3951:
3950:
3945:
3937:
3932:
3908:
3906:
3905:
3900:
3888:
3887:
3875:
3874:
3859:
3858:
3836:
3834:
3833:
3828:
3819:
3818:
3806:
3805:
3793:
3792:
3780:
3779:
3758:
3757:
3745:
3744:
3676:
3674:
3673:
3668:
3645:
3643:
3635:
3627:
3607:
3605:
3597:
3589:
3574:
3572:
3571:
3566:
3511:
3509:
3508:
3503:
3480:
3478:
3470:
3462:
3442:
3440:
3432:
3424:
3397:
3395:
3394:
3389:
3366:
3364:
3356:
3348:
3315:
3313:
3312:
3307:
3305:
3303:
3295:
3287:
3285:
3284:
3229:
3227:
3226:
3221:
3201:
3199:
3191:
3183:
3169:
3167:
3159:
3151:
3136:
3134:
3133:
3128:
3096:
3094:
3086:
3078:
3046:
3044:
3036:
3028:
3016:
3014:
3013:
3008:
2942:
2940:
2939:
2934:
2932:
2930:
2922:
2914:
2912:
2911:
2905:
2903:
2895:
2887:
2882:
2880:
2872:
2864:
2849:
2847:
2846:
2841:
2779:
2777:
2776:
2771:
2742:
2740:
2732:
2724:
2698:
2696:
2688:
2680:
2654:
2652:
2644:
2636:
2617:
2615:
2614:
2609:
2600:
2599:
2590:
2589:
2577:
2576:
2561:
2560:
2539:
2538:
2520:
2519:
2478:
2476:
2475:
2470:
2467:
2466:
2454:
2453:
2435:
2434:
2416:
2415:
2368:
2366:
2365:
2360:
2325:
2323:
2315:
2307:
2275:
2273:
2265:
2257:
2225:
2223:
2215:
2207:
2195:
2193:
2192:
2187:
2143:
2141:
2133:
2125:
2114:
2112:
2104:
2096:
2085:
2083:
2075:
2067:
2013:algebraic curves
2004:) is said to be
1992:
1990:
1989:
1984:
1958:
1956:
1948:
1940:
1920:
1918:
1910:
1902:
1887:
1885:
1884:
1879:
1853:
1851:
1843:
1835:
1808:
1806:
1798:
1790:
1775:
1773:
1772:
1767:
1726:
1724:
1716:
1708:
1670:
1668:
1660:
1652:
1617:
1615:
1614:
1609:
1604:
1602:
1594:
1586:
1584:
1583:
1577:
1575:
1567:
1559:
1551:
1549:
1541:
1533:
1502:
1500:
1499:
1494:
1461:
1459:
1458:
1453:
1432:
1430:
1429:
1424:
1422:
1421:
1393:
1391:
1390:
1385:
1344:
1342:
1341:
1336:
1307:
1305:
1304:
1299:
1270:
1268:
1260:
1252:
1216:
1214:
1213:
1208:
1197:
1064:= 0 is equal to
1013:
1011:
1010:
1005:
975:
917:of the function
860:
858:
857:
852:
770:
768:
767:
762:
757:
752:
717:
703:is equal to the
695:passing through
566:supporting lines
521:inflection point
451:
443:
438:infinitely close
382:. Independently
381:
379:
378:
373:
361:
359:
358:
353:
332:
330:
329:
324:
265:(c. 300 BC). In
255:
254:
238:
224:
155:
148:
142:
131:
115:
105:
89:infinitely close
21:
6954:
6953:
6949:
6948:
6947:
6945:
6944:
6943:
6914:
6913:
6912:
6907:
6903:Steinmetz solid
6888:Integration Bee
6822:
6804:
6730:
6672:Colin Maclaurin
6648:
6616:
6610:
6482:
6476:Tensor calculus
6453:Volume integral
6389:
6364:Basic theorems
6327:Vector calculus
6321:
6202:
6169:Newton's method
6004:
5983:One-sided limit
5959:
5940:Rolle's theorem
5930:Linear function
5881:
5876:
5798:
5795:
5773:
5768:
5767:
5758:
5757:
5753:
5744:
5740:
5735:
5731:
5726:
5722:
5717:
5713:
5708:
5701:
5696:
5692:
5687:
5680:
5669:
5665:
5660:
5653:
5638:10.2307/2301226
5620:
5616:
5599:
5595:
5584:
5580:
5573:
5559:
5555:
5540:10.2307/2695381
5524:
5520:
5513:
5499:
5495:
5485:
5483:
5479:
5473:
5466:
5456:
5454:
5448:
5444:
5435:
5431:
5424:
5410:
5406:
5399:Acta Eruditorum
5391:
5387:
5382:
5339:Supporting line
5304:Newton's method
5300:
5292:Euclidean space
5268:
5262:
5233:
5229:
5220:
5216:
5211:
5208:
5207:
5191:
5188:
5187:
5171:
5168:
5167:
5151:
5148:
5147:
5126:
5118:
5116:
5114:
5111:
5110:
5089:
5081:
5079:
5077:
5074:
5073:
5050:
5046:
5028:
5024:
5015:
5011:
4998:
4990:
4988:
4976:
4972:
4954:
4950:
4941:
4937:
4924:
4916:
4914:
4905:
4901:
4893:
4890:
4889:
4867:
4863:
4854:
4850:
4841:
4837:
4832:
4829:
4828:
4791:
4788:
4787:
4760:
4754:
4744:
4737:
4712:
4701:
4697:
4688:
4684:
4683:
4679:
4678:
4669:
4658:
4654:
4645:
4641:
4640:
4636:
4635:
4626:
4615:
4611:
4602:
4598:
4597:
4593:
4592:
4590:
4587:
4586:
4556:
4545:
4541:
4532:
4528:
4527:
4523:
4522:
4513:
4502:
4498:
4489:
4485:
4484:
4480:
4479:
4470:
4459:
4455:
4446:
4442:
4441:
4437:
4436:
4434:
4431:
4430:
4396:
4385:
4381:
4372:
4368:
4367:
4363:
4362:
4353:
4342:
4338:
4329:
4325:
4324:
4320:
4319:
4310:
4299:
4295:
4286:
4282:
4281:
4277:
4276:
4274:
4271:
4270:
4247:
4243:
4234:
4230:
4225:
4222:
4221:
4201:
4197:
4188:
4184:
4179:
4176:
4175:
4158:
4154:
4145:
4141:
4139:
4136:
4135:
4106:
4104:Tangent circles
4100:
4098:Tangent circles
4095:
4094:
4078:
4075:
4074:
4054:
4051:
4050:
4011:
4003:
3931:
3920:
3917:
3916:
3883:
3879:
3870:
3866:
3854:
3850:
3848:
3845:
3844:
3814:
3810:
3801:
3797:
3788:
3784:
3775:
3771:
3753:
3749:
3740:
3736:
3731:
3728:
3727:
3698:
3689:
3683:
3636:
3628:
3626:
3598:
3590:
3588:
3586:
3583:
3582:
3523:
3520:
3519:
3471:
3463:
3461:
3433:
3425:
3423:
3421:
3418:
3417:
3357:
3349:
3347:
3327:
3324:
3323:
3296:
3288:
3286:
3280:
3279:
3271:
3268:
3267:
3245:
3239:
3192:
3184:
3182:
3160:
3152:
3150:
3148:
3145:
3144:
3087:
3079:
3077:
3037:
3029:
3027:
3025:
3022:
3021:
2951:
2948:
2947:
2923:
2915:
2913:
2907:
2906:
2896:
2888:
2886:
2873:
2865:
2863:
2861:
2858:
2857:
2798:
2795:
2794:
2733:
2725:
2723:
2689:
2681:
2679:
2645:
2637:
2635:
2633:
2630:
2629:
2595:
2591:
2585:
2581:
2566:
2562:
2556:
2552:
2528:
2524:
2515:
2511:
2503:
2500:
2499:
2490:
2462:
2458:
2449:
2445:
2424:
2420:
2411:
2407:
2399:
2396:
2395:
2316:
2308:
2306:
2266:
2258:
2256:
2216:
2208:
2206:
2204:
2201:
2200:
2134:
2126:
2124:
2105:
2097:
2095:
2076:
2068:
2066:
2064:
2061:
2060:
2057:Euler's theorem
1949:
1941:
1939:
1911:
1903:
1901:
1899:
1896:
1895:
1844:
1836:
1834:
1799:
1791:
1789:
1787:
1784:
1783:
1717:
1709:
1707:
1661:
1653:
1651:
1649:
1646:
1645:
1595:
1587:
1585:
1579:
1578:
1568:
1560:
1558:
1542:
1534:
1532:
1530:
1527:
1526:
1470:
1467:
1466:
1438:
1435:
1434:
1417:
1413:
1399:
1396:
1395:
1369:
1366:
1365:
1324:
1321:
1320:
1261:
1253:
1251:
1237:
1234:
1233:
1193:
1185:
1182:
1181:
1166:
1039:
968:
945:
942:
941:
867:
806:
803:
802:
718:
716:
714:
711:
710:
629:
574:
562:convex geometry
468:
458:
449:
441:
367:
364:
363:
338:
335:
334:
303:
300:
299:
245:
202:Similarly, the
197:affine function
174:Euclidean space
153:
140:
136:
117:
107:
92:
41:
30:
23:
22:
18:Surface tangent
15:
12:
11:
5:
6952:
6942:
6941:
6936:
6931:
6926:
6909:
6908:
6906:
6905:
6900:
6895:
6890:
6885:
6883:Gabriel's horn
6880:
6875:
6874:
6873:
6868:
6863:
6858:
6853:
6845:
6844:
6843:
6834:
6832:
6828:
6827:
6824:
6823:
6821:
6820:
6815:
6813:List of limits
6809:
6806:
6805:
6803:
6802:
6801:
6800:
6795:
6790:
6780:
6779:
6778:
6768:
6763:
6758:
6753:
6747:
6745:
6736:
6732:
6731:
6729:
6728:
6721:
6714:
6712:Leonhard Euler
6709:
6704:
6699:
6694:
6689:
6684:
6679:
6674:
6669:
6664:
6658:
6656:
6650:
6649:
6647:
6646:
6641:
6636:
6631:
6626:
6620:
6618:
6612:
6611:
6609:
6608:
6607:
6606:
6601:
6596:
6591:
6586:
6581:
6576:
6571:
6566:
6561:
6553:
6552:
6551:
6546:
6545:
6544:
6539:
6529:
6524:
6519:
6514:
6509:
6504:
6496:
6490:
6488:
6484:
6483:
6481:
6480:
6479:
6478:
6473:
6468:
6463:
6455:
6450:
6445:
6440:
6435:
6430:
6425:
6420:
6415:
6413:Hessian matrix
6410:
6405:
6399:
6397:
6391:
6390:
6388:
6387:
6386:
6385:
6380:
6375:
6370:
6368:Line integrals
6362:
6361:
6360:
6355:
6350:
6345:
6340:
6331:
6329:
6323:
6322:
6320:
6319:
6314:
6309:
6308:
6307:
6302:
6294:
6289:
6288:
6287:
6277:
6276:
6275:
6270:
6265:
6255:
6250:
6249:
6248:
6238:
6233:
6228:
6223:
6218:
6216:Antiderivative
6212:
6210:
6204:
6203:
6201:
6200:
6199:
6198:
6193:
6188:
6178:
6177:
6176:
6171:
6163:
6162:
6161:
6156:
6151:
6146:
6136:
6135:
6134:
6129:
6124:
6119:
6111:
6110:
6109:
6104:
6103:
6102:
6092:
6087:
6082:
6077:
6072:
6062:
6061:
6060:
6055:
6045:
6040:
6035:
6030:
6025:
6020:
6014:
6012:
6006:
6005:
6003:
6002:
5997:
5992:
5987:
5986:
5985:
5975:
5969:
5967:
5961:
5960:
5958:
5957:
5952:
5947:
5942:
5937:
5932:
5927:
5922:
5917:
5912:
5907:
5902:
5897:
5891:
5889:
5883:
5882:
5875:
5874:
5867:
5860:
5852:
5846:
5845:
5839:
5833:
5822:"Tangent Line"
5814:
5800:"Tangent line"
5794:
5793:External links
5791:
5790:
5789:
5772:
5769:
5766:
5765:
5751:
5738:
5729:
5720:
5711:
5699:
5690:
5678:
5663:
5651:
5614:
5593:
5585:Noah Webster,
5578:
5572:978-0321387004
5571:
5553:
5534:(3): 206–216.
5518:
5512:978-0321387004
5511:
5493:
5464:
5442:
5429:
5422:
5404:
5384:
5383:
5381:
5378:
5377:
5376:
5371:
5366:
5364:Tangent vector
5361:
5356:
5351:
5346:
5341:
5336:
5331:
5326:
5321:
5316:
5311:
5306:
5299:
5296:
5264:Main article:
5261:
5258:
5241:
5236:
5232:
5228:
5223:
5219:
5215:
5195:
5175:
5155:
5132:
5129:
5124:
5121:
5095:
5092:
5087:
5084:
5058:
5053:
5049:
5045:
5042:
5039:
5036:
5031:
5027:
5023:
5018:
5014:
5010:
5004:
5001:
4996:
4993:
4987:
4984:
4979:
4975:
4971:
4968:
4965:
4962:
4957:
4953:
4949:
4944:
4940:
4936:
4930:
4927:
4922:
4919:
4913:
4908:
4904:
4900:
4897:
4875:
4870:
4866:
4862:
4857:
4853:
4849:
4844:
4840:
4836:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4736:
4733:
4732:
4731:
4720:
4715:
4710:
4704:
4700:
4696:
4691:
4687:
4682:
4677:
4672:
4667:
4661:
4657:
4653:
4648:
4644:
4639:
4634:
4629:
4624:
4618:
4614:
4610:
4605:
4601:
4596:
4576:
4575:
4564:
4559:
4554:
4548:
4544:
4540:
4535:
4531:
4526:
4521:
4516:
4511:
4505:
4501:
4497:
4492:
4488:
4483:
4478:
4473:
4468:
4462:
4458:
4454:
4449:
4445:
4440:
4416:
4415:
4404:
4399:
4394:
4388:
4384:
4380:
4375:
4371:
4366:
4361:
4356:
4351:
4345:
4341:
4337:
4332:
4328:
4323:
4318:
4313:
4308:
4302:
4298:
4294:
4289:
4285:
4280:
4255:
4250:
4246:
4242:
4237:
4233:
4229:
4209:
4204:
4200:
4196:
4191:
4187:
4183:
4161:
4157:
4153:
4148:
4144:
4102:Main article:
4099:
4096:
4082:
4058:
4019:tangent vector
4012:
4008:Tangent vector
4004:
4002:
3999:
3955:
3954:
3943:
3940:
3935:
3930:
3927:
3924:
3910:
3909:
3897:
3894:
3891:
3886:
3882:
3878:
3873:
3869:
3865:
3862:
3857:
3853:
3838:
3837:
3825:
3822:
3817:
3813:
3809:
3804:
3800:
3796:
3791:
3787:
3783:
3778:
3774:
3770:
3767:
3764:
3761:
3756:
3752:
3748:
3743:
3739:
3735:
3710:singular point
3697:
3694:
3682:
3679:
3678:
3677:
3666:
3663:
3660:
3657:
3654:
3651:
3648:
3642:
3639:
3634:
3631:
3625:
3622:
3619:
3616:
3613:
3610:
3604:
3601:
3596:
3593:
3576:
3575:
3564:
3561:
3558:
3555:
3552:
3549:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3513:
3512:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3477:
3474:
3469:
3466:
3460:
3457:
3454:
3451:
3448:
3445:
3439:
3436:
3431:
3428:
3399:
3398:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3363:
3360:
3355:
3352:
3346:
3343:
3340:
3337:
3334:
3331:
3317:
3316:
3302:
3299:
3294:
3291:
3283:
3278:
3275:
3238:
3235:
3231:
3230:
3219:
3216:
3213:
3210:
3207:
3204:
3198:
3195:
3190:
3187:
3181:
3178:
3175:
3172:
3166:
3163:
3158:
3155:
3138:
3137:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3093:
3090:
3085:
3082:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3043:
3040:
3035:
3032:
3006:
3003:
3000:
2997:
2994:
2991:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2965:
2962:
2959:
2956:
2944:
2943:
2929:
2926:
2921:
2918:
2910:
2902:
2899:
2894:
2891:
2885:
2879:
2876:
2871:
2868:
2851:
2850:
2839:
2836:
2833:
2830:
2827:
2824:
2820:
2817:
2814:
2811:
2808:
2805:
2802:
2788:parametrically
2781:
2780:
2769:
2766:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2739:
2736:
2731:
2728:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2695:
2692:
2687:
2684:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2651:
2648:
2643:
2640:
2619:
2618:
2606:
2603:
2598:
2594:
2588:
2584:
2580:
2575:
2572:
2569:
2565:
2559:
2555:
2551:
2548:
2545:
2542:
2537:
2534:
2531:
2527:
2523:
2518:
2514:
2510:
2507:
2486:
2480:
2479:
2465:
2461:
2457:
2452:
2448:
2444:
2441:
2438:
2433:
2430:
2427:
2423:
2419:
2414:
2410:
2406:
2403:
2370:
2369:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2322:
2319:
2314:
2311:
2305:
2302:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2278:
2272:
2269:
2264:
2261:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2222:
2219:
2214:
2211:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2140:
2137:
2132:
2129:
2123:
2120:
2117:
2111:
2108:
2103:
2100:
2094:
2091:
2088:
2082:
2079:
2074:
2071:
1994:
1993:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1955:
1952:
1947:
1944:
1938:
1935:
1932:
1929:
1926:
1923:
1917:
1914:
1909:
1906:
1889:
1888:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1850:
1847:
1842:
1839:
1832:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1805:
1802:
1797:
1794:
1777:
1776:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1723:
1720:
1715:
1712:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1667:
1664:
1659:
1656:
1641:) = 0 is then
1619:
1618:
1607:
1601:
1598:
1593:
1590:
1582:
1574:
1571:
1566:
1563:
1557:
1554:
1548:
1545:
1540:
1537:
1504:
1503:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1451:
1448:
1445:
1442:
1420:
1416:
1412:
1409:
1406:
1403:
1383:
1380:
1377:
1373:
1334:
1331:
1328:
1309:
1308:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1267:
1264:
1259:
1256:
1250:
1247:
1244:
1241:
1206:
1203:
1200:
1196:
1192:
1189:
1165:
1162:
1153:contrapositive
1134:absolute value
1119:double tangent
1038:
1035:
1019:power function
1015:
1014:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
974:
971:
967:
964:
961:
958:
955:
952:
949:
893:such that, as
866:
863:
862:
861:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
772:
771:
760:
755:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
628:
625:
617:René Descartes
573:
570:
542:cubic function
457:
454:
407:Johannes Hudde
371:
351:
348:
345:
342:
322:
319:
316:
313:
310:
307:
244:
241:
239:, "to touch".
183:is called the
168:and curves in
28:
9:
6:
4:
3:
2:
6951:
6940:
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6921:
6919:
6904:
6901:
6899:
6896:
6894:
6891:
6889:
6886:
6884:
6881:
6879:
6876:
6872:
6869:
6867:
6864:
6862:
6859:
6857:
6854:
6852:
6849:
6848:
6846:
6842:
6839:
6838:
6836:
6835:
6833:
6829:
6819:
6816:
6814:
6811:
6810:
6807:
6799:
6796:
6794:
6791:
6789:
6786:
6785:
6784:
6781:
6777:
6774:
6773:
6772:
6769:
6767:
6764:
6762:
6759:
6757:
6754:
6752:
6749:
6748:
6746:
6744:
6740:
6737:
6733:
6727:
6726:
6722:
6720:
6719:
6715:
6713:
6710:
6708:
6705:
6703:
6700:
6698:
6695:
6693:
6690:
6688:
6687:Infinitesimal
6685:
6683:
6680:
6678:
6675:
6673:
6670:
6668:
6665:
6663:
6660:
6659:
6657:
6655:
6651:
6645:
6642:
6640:
6637:
6635:
6632:
6630:
6627:
6625:
6622:
6621:
6619:
6613:
6605:
6602:
6600:
6597:
6595:
6592:
6590:
6587:
6585:
6582:
6580:
6577:
6575:
6572:
6570:
6567:
6565:
6562:
6560:
6557:
6556:
6554:
6550:
6547:
6543:
6540:
6538:
6535:
6534:
6533:
6530:
6528:
6525:
6523:
6520:
6518:
6515:
6513:
6510:
6508:
6505:
6503:
6500:
6499:
6497:
6495:
6492:
6491:
6489:
6485:
6477:
6474:
6472:
6469:
6467:
6464:
6462:
6459:
6458:
6456:
6454:
6451:
6449:
6446:
6444:
6441:
6439:
6436:
6434:
6431:
6429:
6428:Line integral
6426:
6424:
6421:
6419:
6416:
6414:
6411:
6409:
6406:
6404:
6401:
6400:
6398:
6396:
6392:
6384:
6381:
6379:
6376:
6374:
6371:
6369:
6366:
6365:
6363:
6359:
6356:
6354:
6351:
6349:
6346:
6344:
6341:
6339:
6336:
6335:
6333:
6332:
6330:
6328:
6324:
6318:
6315:
6313:
6310:
6306:
6303:
6301:
6300:Washer method
6298:
6297:
6295:
6293:
6290:
6286:
6283:
6282:
6281:
6278:
6274:
6271:
6269:
6266:
6264:
6263:trigonometric
6261:
6260:
6259:
6256:
6254:
6251:
6247:
6244:
6243:
6242:
6239:
6237:
6234:
6232:
6229:
6227:
6224:
6222:
6219:
6217:
6214:
6213:
6211:
6209:
6205:
6197:
6194:
6192:
6189:
6187:
6184:
6183:
6182:
6179:
6175:
6172:
6170:
6167:
6166:
6164:
6160:
6157:
6155:
6152:
6150:
6147:
6145:
6142:
6141:
6140:
6137:
6133:
6132:Related rates
6130:
6128:
6125:
6123:
6120:
6118:
6115:
6114:
6112:
6108:
6105:
6101:
6098:
6097:
6096:
6093:
6091:
6088:
6086:
6083:
6081:
6078:
6076:
6073:
6071:
6068:
6067:
6066:
6063:
6059:
6056:
6054:
6051:
6050:
6049:
6046:
6044:
6041:
6039:
6036:
6034:
6031:
6029:
6026:
6024:
6021:
6019:
6016:
6015:
6013:
6011:
6007:
6001:
5998:
5996:
5993:
5991:
5988:
5984:
5981:
5980:
5979:
5976:
5974:
5971:
5970:
5968:
5966:
5962:
5956:
5953:
5951:
5948:
5946:
5943:
5941:
5938:
5936:
5933:
5931:
5928:
5926:
5923:
5921:
5918:
5916:
5913:
5911:
5908:
5906:
5903:
5901:
5898:
5896:
5893:
5892:
5890:
5888:
5884:
5880:
5873:
5868:
5866:
5861:
5859:
5854:
5853:
5850:
5843:
5840:
5837:
5834:
5829:
5828:
5823:
5820:
5815:
5811:
5807:
5806:
5801:
5797:
5796:
5786:
5782:
5781:
5775:
5774:
5761:
5755:
5748:
5742:
5733:
5724:
5715:
5706:
5704:
5694:
5685:
5683:
5675:
5674:
5667:
5658:
5656:
5647:
5643:
5639:
5635:
5631:
5627:
5626:
5618:
5610:
5606:
5605:
5597:
5591:
5588:
5582:
5574:
5568:
5564:
5557:
5549:
5545:
5541:
5537:
5533:
5529:
5522:
5514:
5508:
5504:
5497:
5482:. p. 2.8
5478:
5471:
5469:
5453:
5446:
5439:
5433:
5425:
5423:9780521286190
5419:
5415:
5408:
5401:
5400:
5395:
5389:
5385:
5375:
5372:
5370:
5367:
5365:
5362:
5360:
5357:
5355:
5352:
5350:
5347:
5345:
5342:
5340:
5337:
5335:
5332:
5330:
5329:Perpendicular
5327:
5325:
5322:
5320:
5317:
5315:
5312:
5310:
5307:
5305:
5302:
5301:
5295:
5293:
5290:-dimensional
5289:
5285:
5282:-dimensional
5281:
5277:
5276:tangent space
5274:-dimensional
5273:
5267:
5266:Tangent space
5257:
5255:
5234:
5230:
5226:
5221:
5217:
5193:
5173:
5153:
5130:
5122:
5093:
5085:
5070:
5051:
5047:
5043:
5040:
5029:
5025:
5021:
5016:
5012:
5002:
4994:
4985:
4977:
4973:
4969:
4966:
4955:
4951:
4947:
4942:
4938:
4928:
4920:
4911:
4906:
4902:
4898:
4895:
4887:
4868:
4864:
4860:
4855:
4851:
4847:
4842:
4838:
4811:
4808:
4805:
4799:
4796:
4793:
4785:
4781:
4777:
4773:
4769:
4765:
4764:tangent plane
4759:
4753:
4749:
4742:
4718:
4713:
4708:
4702:
4698:
4694:
4689:
4685:
4680:
4675:
4670:
4665:
4659:
4655:
4651:
4646:
4642:
4637:
4632:
4627:
4622:
4616:
4612:
4608:
4603:
4599:
4594:
4585:
4584:
4583:
4581:
4562:
4557:
4552:
4546:
4542:
4538:
4533:
4529:
4524:
4519:
4514:
4509:
4503:
4499:
4495:
4490:
4486:
4481:
4476:
4471:
4466:
4460:
4456:
4452:
4447:
4443:
4438:
4429:
4428:
4427:
4425:
4421:
4402:
4397:
4392:
4386:
4382:
4378:
4373:
4369:
4364:
4359:
4354:
4349:
4343:
4339:
4335:
4330:
4326:
4321:
4316:
4311:
4306:
4300:
4296:
4292:
4287:
4283:
4278:
4269:
4268:
4267:
4248:
4244:
4240:
4235:
4231:
4202:
4198:
4194:
4189:
4185:
4159:
4155:
4151:
4146:
4142:
4134:
4130:
4126:
4121:
4119:
4110:
4105:
4080:
4072:
4056:
4048:
4044:
4040:
4039:tangent space
4036:
4032:
4028:
4024:
4020:
4016:
4009:
3998:
3996:
3992:
3988:
3984:
3979:
3977:
3973:
3969:
3964:
3958:
3941:
3938:
3933:
3928:
3925:
3922:
3915:
3914:
3913:
3895:
3892:
3884:
3880:
3876:
3871:
3867:
3863:
3855:
3851:
3843:
3842:
3841:
3823:
3815:
3811:
3807:
3802:
3798:
3789:
3785:
3781:
3776:
3768:
3765:
3762:
3759:
3754:
3750:
3746:
3741:
3737:
3726:
3725:
3724:
3722:
3717:
3715:
3711:
3702:
3693:
3688:
3664:
3661:
3655:
3652:
3649:
3640:
3637:
3632:
3629:
3623:
3617:
3614:
3611:
3602:
3599:
3594:
3591:
3581:
3580:
3579:
3559:
3553:
3550:
3547:
3543:
3537:
3531:
3528:
3525:
3518:
3517:
3516:
3499:
3496:
3490:
3487:
3484:
3475:
3467:
3458:
3452:
3449:
3446:
3437:
3429:
3416:
3415:
3414:
3412:
3408:
3404:
3385:
3382:
3376:
3373:
3370:
3361:
3358:
3353:
3350:
3344:
3338:
3335:
3332:
3322:
3321:
3320:
3300:
3297:
3292:
3289:
3276:
3273:
3266:
3265:
3264:
3262:
3258:
3254:
3250:
3244:
3234:
3217:
3214:
3211:
3205:
3196:
3193:
3188:
3185:
3179:
3173:
3164:
3161:
3156:
3153:
3143:
3142:
3141:
3124:
3118:
3115:
3112:
3106:
3100:
3091:
3088:
3083:
3080:
3074:
3068:
3065:
3062:
3056:
3050:
3041:
3038:
3033:
3030:
3020:
3019:
3018:
3001:
2995:
2992:
2989:
2985:
2979:
2973:
2970:
2967:
2963:
2960:
2957:
2954:
2927:
2924:
2919:
2916:
2900:
2897:
2892:
2889:
2883:
2877:
2874:
2869:
2866:
2856:
2855:
2854:
2834:
2828:
2825:
2822:
2818:
2812:
2806:
2803:
2800:
2793:
2792:
2791:
2789:
2784:
2767:
2764:
2758:
2755:
2752:
2749:
2746:
2737:
2729:
2720:
2717:
2714:
2708:
2705:
2702:
2693:
2685:
2676:
2673:
2670:
2664:
2661:
2658:
2649:
2641:
2628:
2627:
2626:
2624:
2604:
2601:
2596:
2592:
2586:
2582:
2578:
2573:
2570:
2567:
2563:
2557:
2553:
2549:
2546:
2543:
2540:
2535:
2532:
2529:
2525:
2521:
2516:
2512:
2508:
2505:
2498:
2497:
2496:
2494:
2489:
2485:
2463:
2459:
2455:
2450:
2446:
2442:
2439:
2436:
2431:
2428:
2425:
2421:
2417:
2412:
2408:
2404:
2401:
2394:
2393:
2392:
2390:
2386:
2382:
2377:
2375:
2356:
2353:
2350:
2347:
2341:
2338:
2335:
2332:
2329:
2320:
2312:
2303:
2300:
2297:
2291:
2288:
2285:
2282:
2279:
2270:
2262:
2253:
2250:
2247:
2241:
2238:
2235:
2232:
2229:
2220:
2212:
2199:
2198:
2197:
2183:
2180:
2174:
2171:
2168:
2165:
2162:
2156:
2153:
2150:
2147:
2144:
2138:
2130:
2121:
2118:
2115:
2109:
2101:
2092:
2089:
2086:
2080:
2072:
2058:
2054:
2050:
2046:
2042:
2038:
2034:
2030:
2026:
2022:
2018:
2014:
2009:
2007:
2003:
1999:
1980:
1977:
1974:
1968:
1965:
1962:
1953:
1945:
1936:
1930:
1927:
1924:
1915:
1907:
1894:
1893:
1892:
1875:
1872:
1869:
1863:
1860:
1857:
1848:
1840:
1830:
1827:
1824:
1818:
1815:
1812:
1803:
1795:
1782:
1781:
1780:
1763:
1760:
1754:
1751:
1748:
1742:
1736:
1733:
1730:
1721:
1713:
1704:
1698:
1695:
1692:
1686:
1680:
1677:
1674:
1665:
1657:
1644:
1643:
1642:
1640:
1636:
1632:
1628:
1624:
1605:
1599:
1591:
1572:
1564:
1555:
1552:
1546:
1543:
1538:
1535:
1525:
1524:
1523:
1521:
1517:
1513:
1509:
1490:
1484:
1478:
1475:
1472:
1465:
1464:
1463:
1446:
1440:
1418:
1410:
1407:
1404:
1378:
1371:
1363:
1359:
1355:
1351:
1346:
1332:
1329:
1326:
1318:
1314:
1292:
1289:
1286:
1280:
1274:
1265:
1262:
1257:
1254:
1248:
1245:
1242:
1239:
1232:
1231:
1230:
1228:
1224:
1220:
1204:
1201:
1198:
1194:
1190:
1187:
1179:
1175:
1171:
1161:
1158:
1157:discontinuity
1154:
1149:
1147:
1143:
1139:
1135:
1131:
1127:
1122:
1120:
1116:
1112:
1108:
1104:
1100:
1096:
1095:
1090:
1086:
1081:
1079:
1075:
1071:
1067:
1063:
1059:
1055:
1050:
1048:
1044:
1034:
1032:
1028:
1024:
1020:
1000:
994:
991:
988:
979:
972:
969:
965:
959:
953:
950:
947:
940:
939:
938:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
892:
888:
884:
880:
876:
872:
847:
841:
838:
835:
829:
826:
820:
814:
811:
808:
801:
800:
799:
797:
793:
789:
785:
781:
777:
774:As the point
758:
753:
746:
740:
737:
731:
728:
725:
719:
709:
708:
707:
706:
702:
698:
694:
690:
686:
682:
678:
674:
670:
666:
662:
658:
654:
650:
646:
642:
638:
634:
624:
621:
618:
614:
613:
608:
604:
595:
591:
587:
583:
578:
569:
567:
563:
559:
553:
551:
547:
543:
539:
535:
531:
527:
523:
522:
515:
513:
509:
505:
501:
497:
493:
489:
481:
477:
474:A tangent, a
472:
467:
463:
453:
447:
439:
435:
431:
426:
424:
420:
416:
412:
408:
404:
400:
396:
391:
389:
385:
369:
346:
340:
317:
314:
311:
305:
297:
293:
290:In the 1630s
288:
286:
282:
278:
276:
272:
268:
264:
263:
258:
249:
240:
237:
236:
231:
226:
223:
222:Tangent space
217:
213:
209:
205:
204:tangent plane
200:
198:
194:
193:
188:
187:
182:
177:
175:
172:-dimensional
171:
167:
163:
159:
152:
146:
139:
135:
129:
125:
121:
114:
110:
103:
99:
95:
90:
86:
82:
81:straight line
78:
74:
71:) to a plane
70:
66:
62:
53:
45:
39:
35:
27:
19:
6798:Secant cubed
6723:
6716:
6697:Isaac Newton
6667:Brook Taylor
6334:Derivatives
6305:Shell method
6033:Differential
5954:
5825:
5803:
5779:
5754:
5746:
5741:
5732:
5723:
5714:
5693:
5671:
5666:
5629:
5623:
5617:
5603:
5596:
5586:
5581:
5562:
5556:
5531:
5527:
5521:
5502:
5496:
5484:. Retrieved
5455:. Retrieved
5445:
5432:
5413:
5407:
5397:
5388:
5344:Tangent cone
5287:
5279:
5271:
5269:
5253:
5071:
4888:
4783:
4779:
4775:
4771:
4763:
4761:
4579:
4577:
4419:
4417:
4174:and centers
4122:
4117:
4115:
4069:is a linear
3994:
3990:
3986:
3982:
3980:
3975:
3971:
3967:
3959:
3956:
3911:
3839:
3718:
3707:
3690:
3577:
3514:
3410:
3406:
3402:
3400:
3318:
3260:
3256:
3252:
3248:
3246:
3232:
3139:
2945:
2852:
2785:
2782:
2625:=1 produces
2622:
2620:
2492:
2487:
2483:
2481:
2388:
2384:
2380:
2378:
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2040:
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2032:
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2020:
2010:
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1626:
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511:
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499:
495:
491:
488:secant lines
485:
427:
419:Isaac Newton
415:Isaac Barrow
392:
289:
279:
274:
270:
260:
256:
246:
227:
203:
201:
190:
185:
184:
178:
169:
166:space curves
161:
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112:
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97:
93:
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65:tangent line
64:
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6866:of surfaces
6617:and numbers
6579:Dirichlet's
6549:Telescoping
6502:Alternating
6090:L'Hôpital's
5887:Precalculus
5475:Shenk, Al.
4127:, then two
4015:mathematics
3714:translating
3249:normal line
2482:where each
778:approaches
693:secant line
482:to a circle
411:John Wallis
257:ephaptoménē
106:at a point
75:at a given
67:(or simply
6918:Categories
6662:Adequality
6348:Divergence
6221:Arc length
6018:Derivative
5380:References
5334:Subtangent
4756:See also:
4071:derivation
3685:See also:
1364:to divide
1217:so by the
1124:The graph
1083:The graph
1052:The graph
929:, denoted
915:derivative
586:derivative
534:hyperbolas
296:adequality
281:Archimedes
267:Apollonius
253:ἐφαπτομένη
158:derivative
6861:of curves
6856:Curvature
6743:Integrals
6537:Maclaurin
6517:Geometric
6408:Geometric
6358:Laplacian
6070:linearity
5910:Factorial
5827:MathWorld
5810:EMS Press
5128:∂
5120:∂
5091:∂
5083:∂
5044:−
5000:∂
4992:∂
4970:−
4926:∂
4918:∂
4899:−
4695:−
4652:−
4609:−
4496:−
4453:−
4379:±
4336:−
4293:−
3929:±
3877:−
3760:−
3653:−
3615:−
3488:−
3473:∂
3465:∂
3459:−
3450:−
3435:∂
3427:∂
3374:−
3336:−
3274:−
3116:−
3107:⋅
3066:−
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2735:∂
2727:∂
2715:⋅
2691:∂
2683:∂
2671:⋅
2647:∂
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2107:∂
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2078:∂
2070:∂
1951:∂
1943:∂
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1905:∂
1870:≠
1846:∂
1838:∂
1801:∂
1793:∂
1752:−
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1719:∂
1711:∂
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1663:∂
1655:∂
1597:∂
1589:∂
1570:∂
1562:∂
1556:−
1522:, giving
1408:−
1290:−
1281:⋅
1243:−
1164:Equations
1132:| of the
1031:logarithm
992:−
839:−
812:−
738:−
530:parabolas
386:used his
384:Descartes
181:intersect
6851:Manifold
6584:Integral
6527:Infinite
6522:Harmonic
6507:Binomial
6353:Gradient
6296:Volumes
6107:Quotient
6048:Notation
5879:Calculus
5450:Euclid.
5298:See also
5284:manifold
4424:distance
2059:implies
2006:singular
973:′
933: ′(
633:function
612:Geometry
607:calculus
558:triangle
538:ellipses
478:, and a
399:Roberval
262:Elements
149:, where
61:geometry
6788:inverse
6776:inverse
6702:Fluxion
6512:Fourier
6378:Stokes'
6373:Green's
6095:Product
5955:Tangent
5812:, 2001
5771:Sources
5646:2301226
5548:2695381
5286:in the
4768:surface
4422:if the
4131:, with
4129:circles
4118:tangent
4031:surface
3409:,
2387:,
2051:,
2047:,
2031:,
2027:,
1514:,
1315:,
1311:where (
1225:,
1155:states
1117:, as a
691:of the
548:of the
526:Circles
434:Leibniz
269:' work
243:History
235:tangere
208:surface
156:is the
85:Leibniz
69:tangent
6871:Tensor
6793:Secant
6559:Abel's
6542:Taylor
6433:Matrix
6383:Gauss'
5965:Limits
5945:Secant
5935:Radian
5644:
5569:
5546:
5509:
5486:1 June
5457:1 June
5420:
5072:Here,
4750:, and
4023:vector
1146:corner
875:Cauchy
546:period
480:secant
464:, and
292:Fermat
271:Conics
248:Euclid
63:, the
6735:Lists
6594:Ratio
6532:Power
6268:Euler
6085:Chain
6075:Power
5950:Slope
5642:JSTOR
5544:JSTOR
5480:(PDF)
4766:to a
4133:radii
4047:germs
4041:of a
4027:curve
4021:is a
3997:= 0.
2391:) as
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879:limit
794:. If
689:slope
582:curve
476:chord
446:limit
230:Latin
212:plane
206:to a
154:'
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134:slope
77:point
73:curve
6604:Term
6599:Root
6338:Curl
5567:ISBN
5507:ISBN
5488:2015
5459:2015
5418:ISBN
5392:In "
5186:and
5109:and
4762:The
4220:and
4017:, a
3140:If
2011:For
1094:cusp
699:and
620:said
550:sine
536:and
494:and
421:and
413:and
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333:and
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5787:ff.
5785:143
5634:doi
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651:= (
590:max
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