What is Tangent: Definition and 1000 Discussions

In geometry, the tangent line (or simply tangent) to a plane curve at a given point is 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 said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space.
As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "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 the graph of the affine function that best approximates the original function at the given point.Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space.
The word "tangent" comes from the Latin tangere, "to touch".

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  1. A

    Find Values of Osculating Circle Tangent to a Parabola

    I'm at a loss as to how they got to certain steps in the solutions manual. Here's how far I got with this: Since the circle is tangent to y = x^2 + 1, the slope at (1, 2) is going to be 2, as is the slope of the 2nd derivative of the circle, so then... The derivative of the circle would be...
  2. A

    Quadratics Having a Common Tangent

    I do know that since they have a common tangent line, that means: 2x+a = c-2x Since they both have the point (1,0), then since both equations should equal 0 when x = 1: c(1)-(1)^2 = 0 --> c = 1 So now, I replace c with 1 to solve for a in the two derivatives that are equal (common...
  3. chwala

    At what ##x## value is the tangent equally inclined to the given curve?

    I had to look this up; will need to read on it. from my research, https://byjus.com/question-answer/the-equation-of-straight-line-equally-inclined-to-the-axes-and-equidistant-from-the-points-1-2-and-3-4-is-ax-by-c-0-where/ ... I have noted that at equally inclined; the slope value is ##1##...
  4. P

    Separable first order ODE involving tangent

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  5. A

    A About computing the tangent space at 1 of certain lie groups

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  6. chwala

    Find the equation of the tangent in the given problem

    The textbook solution is indicated below; My approach on this question is as follows; ##\dfrac{dy}{dx} = -\dfrac {8}{x^3}## ##\dfrac{dy}{dx} (x=a) = -\dfrac {8}{a^3}## The tangent equation is given by; ##y= -\dfrac {8}{a^3}x+\dfrac{12+a^2}{a^2}## when ##x=0##, ##y=\dfrac{12+a^2}{a^2}##...
  7. G

    Find the two points on the curve that share a tangent line

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  8. M

    Precise definition of tangent line to a curve

    How do we define tangent line to curve accurately ? I cannot say it is a straight line who intersect the curve in one point because if we draw y = x^2 & make any vertical line, it will intersect the curve and still not the tangent we know. Moreover, tangent line may intersect the curve at other...
  9. C

    I Carroll GR: Tangent Space & Partial Derivatives

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  10. H

    How to find the straight tangent line?

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  11. karush

    MHB Tikz for f(x)=5x^2-2x and tangent line at 1,3

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  12. chwala

    Solve the attached problem that involves circle and tangent

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  13. chwala

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  14. K

    I What is the construction of charts for the tangent bundle on the unit circle?

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  15. chwala

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  16. A

    I Tangent Bundle of Product is diffeomorphic to Product of Tangent Bundles

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  17. brotherbobby

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  18. brochesspro

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  19. graphking

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  20. S

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  21. J

    MHB Equation for tangent of the curve

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  22. Greg Bernhardt

    I What is the true meaning of a tangent in mathematics?

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  23. N

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  24. Strand9202

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  25. anemone

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  26. brotherbobby

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  27. Y

    A geometry question in Comsol (draw a semi-circle tangent to a line)

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  28. docnet

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  29. docnet

    A multivariate function of Toruses - tangent vectors

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  30. E

    B Just a question about the tangent basis

    I was reading these notes and then on page 23 I saw something a bit weird. Back in this thread I learned that ##\{ \partial_i \}## form a basis of ##T_p M##, and that a tangent vector can be written ##X = X^i \partial_i##, and it's not too difficult to show that components transform like...
  31. K

    Calculate the dual basis and tangent basis vectors

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  32. U

    I Fixing orientation by fixing a frame in a tangent space

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  33. xyz_1965

    MHB Why is the Law of Tangent rarely used or taught in typical trigonometry courses?

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  34. qbar

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  35. ElectronicTeaCup

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  36. S

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  37. ElectronicTeaCup

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  38. S

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  39. S

    I Tangent space basis vectors under a coordinate change

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  40. ttpp1124

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  41. ttpp1124

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  42. ttpp1124

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  43. ttpp1124

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  44. ttpp1124

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  45. archaic

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  46. F

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  47. G

    I Visualizing the cotangent space to a sphere

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  48. karush

    MHB 3.2.15 mvt - Mean value theorem: graphing the secant and tangent lines

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  49. Decimal

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  50. Cheesycheese213

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