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. T

    Tangent to both pieces of a piecewise defined function

    Homework Statement Find the line tangent to the curve f(x) = { (x+3)^2 | x < 0} , {-x^ + 8x -4 | x≥ 0} at two distinct points Homework Equations slope = Δy/Δx The Attempt at a Solution I only got as far as finding the slope of the line to be: a= -50p^2 + 50p. where p is a point...
  2. H

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  3. J

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  4. J

    Find the equations for the two tangent lines.

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  5. M

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  6. L

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  7. M

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  8. L

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  9. B

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  10. E

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  11. J

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  12. r-soy

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  13. T

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  14. U

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  15. J

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  16. D

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  17. J

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  18. J

    Finding Slopes and Equations of Secant and Tangent Lines for a Given Curve

    Homework Statement Given a point P (3, 10) and the equation of a curve as x^2 -5x-4, find the slope of the secant and the equation of the tangent line to the curve Homework Equations The Attempt at a Solution I tried using y = f(x + h) -f(x) all divided by h and got (x + h)^2 -...
  19. H

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  20. L

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  21. H

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  22. R

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  23. M

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  24. K

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  25. K

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  26. M

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  27. C

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  28. E

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  29. C

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  30. J

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  31. J

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  32. A

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  33. M

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  34. L

    Multivariable Calculus - Tangent Line

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  35. A

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  36. N

    Two curves in a cut point would have a same tangent

    Ola, If we have two curves, and they cut them selves in a way like these two: y=x^2 and x^2+(y-1)^2=1. Does it always mean that those two curves in a cut point would have a same tangent, in other words do they need to have the same derivative in that spot? Thanks!
  37. M

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    Homework Statement Find the equations of the tangents to the following graphs for the given values of x. (a) y = ln x, where x = 1/2 Homework Equations The Attempt at a Solution I know ln x differentiated is 1/x but I cannot see when the rest fall into the place. The book I'm...
  38. Jonnyb42

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  39. W

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  40. J

    Second Derivative: What Does it Represent? - James

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  41. M

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  42. Z

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  43. Z

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  44. S

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  45. C

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  46. B

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  47. T

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  48. P

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  49. jegues

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  50. T

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