Tangent Definition and 1000 Threads

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. C

    Tangent line, to a surface, through a point, parallel to a plane

    Hey y'all, this is my first post. I am currently stuck on a multivariable question. Please let me know if you can help. Homework Statement The point, P = (1, 2, 2) lies on the surface z = x^2 + y^2 -3x. Find parametric equations for the tangent line to the surface through the point P parallel...
  2. MarkFL

    MHB Brandon's questions at Yahoo Answers regarding tangent lines

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  3. H

    Is the tangent line really a point when the slope is zero?

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  4. F

    Calculating Basis of Tangent Plane

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  5. K

    How is the equation given for v_b tangent to it?

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  6. Ascendant78

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  7. H

    MHB Radius of Sphere Tangent to Two Lines

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  8. R

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  9. J

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  10. MarkFL

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  11. orion

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  12. J

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  13. J

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  14. H

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  15. K

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  16. Lebombo

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  17. S

    MHB Find the equation of the tangent line pt 3 (Need someone to check my work)

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  18. mnb96

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    Hello, if we consider a diffeomorphism f:M-->N between two manifolds, we can easily obtain a basis for the tangent space of N at p from the differential of f. I was wondering, why should we always express tangent vectors as linear combinations of tangent basis vectors? Could it be useful in...
  19. S

    MHB Find the equation of the tangent line (Can someone check my work?)

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  20. S

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  21. M

    Need help qualitatively understanding the concept of a tangent bundle

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  22. MarkFL

    MHB Find Tangent Lines Through Origin to a Circle

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  23. U

    What is the minimum distance between a tangent and normal of a parabola?

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  24. DreamWeaver

    MHB Finite Tangent product / quotient

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  25. C

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  26. mnb96

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  27. mnb96

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  28. S

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  29. D

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  30. D

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  31. B

    How to deal with INVERSE TANGENT?

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  32. mesa

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  33. MarkFL

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  34. P

    Find Tangent Line: f(2) & f'(2) at y=4x-5

    1. If an equation of the tangent line to the curve y=f(x) at the point a=2 where is y=4x-5, find f(2) and f'(2). Homework Equations m=\frac{f(x)-f(a)}{x-a} The Attempt at a Solution To be honest, I really don't know where to start. Here's what I have so far...
  35. M

    Why does the tangent function behave like the angle itself on a unit circle?

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  36. anemone

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  37. O

    Need help showing important tangent plane property:

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  38. C

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  39. MarkFL

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  40. MarkFL

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  41. A

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  42. MarkFL

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  43. P

    Tangent Lines/Instaneous Velocity

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  44. C

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  45. O

    Calculus Review - Tangent of function and find y-intercept

    Homework Statement Consider g(ξ) = [(2H)/π] arctan(ξ). Plot a graph of the function g(ξ). Imagine a line that passes through the point on the curve at ξ0 = 1.30, and which is tangent to the curve at that point. Where does the tangent line intersect the vertical axis? [DATA: H = 2.00 ; ξ0 =...
  46. C

    MHB How Do You Find the Equation of a Tangent Line at a Given Point?

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  47. PsychonautQQ

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  48. PsychonautQQ

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  49. B

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  50. C

    MHB Finding the equation of a tangent line at a given point (tan and e^x questions)

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