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. L

    How using a mirror to find the tangent at a point on the curve works

    Hi, I recently learned that to find the tangent at a point on any curve, you can simply place a mirror on that point and reflect the part of the curve on one side of that point such that the reflection flows smoothly into the other part of the curve on the other side. Once this is done, draw a...
  2. Y

    Inverse Tangent Question: Impact of Positive and Negative Delta on Psi with Time

    \Psi=\tan^{-1}\left(\frac{\cos\omega t}{\cos(\omega t+\delta)}\right) I want to find out whether ##\Psi## increase or decrease with time t, if ##\delta## is positive and if ##\delta## is negative. \Psi=\tan^{-1}\left(\frac{\cos\omega t}{\cos(\cos\omega t \cos \delta+\sin\omega t...
  3. A

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  4. F

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  5. H

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  6. A

    Solving for x in y = Tan(x): A Last Resort

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  7. T

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  8. MarkFL

    MHB What Are the Values of a and b for a Cubic Curve's Tangent Line?

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  9. C

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  10. L

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  11. M

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  12. A

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  13. S

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  14. A

    MHB Equation of a line tangent to a circle

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  15. A

    Deriving the equation of a tangent plane

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  16. S

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    Find the equation of the line tangent to f(x)= 3x^3 + 2 at x = 1. a) y = 9x-4 b) y = 9x+5 c) y = 3x 2 d) y = 3x+1 e) Not enough information given. Im confused on this one, but I am thinking about E, because it doesn't specify if the equation should be parallel or perpendicular to the...
  17. M

    Confusion regarding differential forms and tangent space (Spivak,Calc. on Manifolds)

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  18. B

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  19. M

    MHB Approximation Problems (Finding an equation of a Tangent Line)

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  20. R

    Finding the tangent line using implicit differentiation.

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  21. R

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  22. M

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  23. T

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  24. T

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  25. M

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  26. V

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  27. B

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  28. D

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  29. D

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  30. B

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    Homework Statement Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the space curve at point P r(t)= <2sin(t), 2cos(t), 4sin2(t)>, P(1, √3, 1) The Attempt at a Solution I found T(t) using the formula T(t)= r'(t)/||r'(t)|| r'(t)= <2cos(t)...
  31. J

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  32. T

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  33. P

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  34. S

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  35. B

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  36. P

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  37. Gliese123

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  38. M

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  39. Q

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  40. J

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  41. J

    Tangent Line to f(x) Without Specified Point

    Homework Statement Hello again. The question asks me to find an equation of the tangent to the graph: f(x)= - sin^2 x + 1/2, x \epsilon [0, \frac{\pi}{2} which makes an angle of 135° with the x-axis (measure anti-clockwise from the positive x-axis). Assume that the scales along the...
  42. V

    Do all four-vectors live in a tangent space?

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  43. X

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  44. F

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  45. P

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  46. G

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  47. P

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  48. N

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  49. M

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  50. G

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