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. Cheesycheese213

    Propagation of uncertainty with tangent function?

    For a lab, I needed to calculate the uncertainty of a refractive index that was found using Snell's law. I found an equation online for propagation of error for any general function, which was I thought that since my equation was I could just get rid of the variable y, and have After...
  2. mnb96

    I Question on tangent space and jet spaces

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  3. Addez123

    Find all points where the level surface tangent plane is parallel

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  4. G

    MHB What Is ⌊θ13⌋ for the Recursive Function fn(x)?

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  5. Like Tony Stark

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  6. B

    MHB Find the Angle of Parable Tangent from Human Running Step

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  7. P

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  9. Y

    MHB How Can I Find the Equation of the Dotted Tangent Line of a Circle?

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  10. F

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  11. Math Amateur

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  13. S

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  14. G

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  15. A

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  16. A

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  17. S

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  18. A

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  19. karush

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  20. K

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  21. A

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  22. K

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  23. Math Amateur

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  24. T

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  25. J

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  26. C

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  27. opus

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  28. opus

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  29. opus

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  30. N

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  31. Jozefina Gramatikova

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  32. K

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  33. isukatphysics69

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  34. C

    MHB Secant and Tangent Angles in Circles, finding an arc length.

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  35. L

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  36. K

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  37. L

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  38. T

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  39. Olinguito

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  40. Monoxdifly

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  41. Monoxdifly

    MHB What Is the Correct Equation for the Tangent of a Circle at a Given Point?

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  42. Physics345

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  43. opus

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  44. F

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  45. N

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  46. A

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  47. F

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  48. Velo

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  49. J

    A Is tangent bundle TM the product manifold of M and T_pM?

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  50. H

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