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. E

    Finding the tangent and normal of a trigonometric equation?

    Homework Statement a. find the tangent and normal line at P (0,0) b. find when the tangent is horizontal Equation: 3x+sin3x at (0,0) The Attempt at a Solution Find the derivative- y'=3 (?) or it it 3x+cos(3x)? But from there, I don't know how to find the rate of change to create a...
  2. A

    A silly question about unit tangent and unit normal

    I know how to obtain the unit tangent, it is very easy. But for the case of unit normal, I am confused. Usually we need to know \vec{T'}(t) and |\vec{T'}(t)| in order the find \vec{N}(t). However are there any quicker method to do it? Since I saw the textbook do it without step, it seems...
  3. N

    Integrating Tangent by parts; 0 = -1

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  4. R

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  5. D

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  6. D

    Showing that a function does not have a strong tangent

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  7. S

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  8. N

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  9. S

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  10. T

    Tangent line to the curve f(x)= x + 1/x

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  11. S

    Finding where two curves share tangent lines

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  12. J

    Questions about finding values of a tangent

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  13. N

    How can I find two points where a function has the same tangent line?

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  14. U

    Triangle 90<x<135 such that B is tangent

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  15. D

    Tangent vectors as equivalence classes of curves

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  16. F

    Tangent plane, why is it orthongonal and not parallel?

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  17. M

    Hamiltonian for a particle moving on a plane tangent to the surface of the easth

    Homework Statement A point of mass m is placed on a frictionless plane that is tangent to the Earth’s surface. Determine Hamilton’s equations taking: (a) the distance x (b) the angle q as the generalized coordinate. Homework Equations The Attempt at a Solution Take the...
  18. 5

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  19. N

    Difference between Tangent Plane and Linearization

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  20. Z

    Equation of a Tangent Line to a Polar Curve

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  21. W

    Finding unit tangent and unit normal vectors

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  22. S

    Parametrizing a Hyperbola to Find Unit Tangent & Normal Vectors

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  23. F

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  24. F

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  25. A

    How to Find the Tangent Line to a Curve in R3 Using Vector Functions?

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  26. 1

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  27. E

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  28. D

    General Relativity math, Tangent vectors on manifolds

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  29. M

    Discover the Cosine and Tangent Demonstration for cos(x)² = 1/1+tan(x)²

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  30. K

    Determine equations of the lines tangent to

    The problem is: Tangent Lines: Determine equations of the lines tangent to the graph of Y = x√(5-x²) at the points (1,2) and (-2,-2). Graph the function and the tangent lines. I have no IDEA where to go with this. I am taking calculus over the summer and we are in week 2 and I'm...
  31. H

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  32. estro

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  33. I

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  34. M

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  35. D

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  36. D

    Finding the points on an ellipse where the slope of the tangent line equals 1

    If there is an an ellipse x^2/9 + y^2/16 = 1, and the slope of the tangent is dx/dy = -16x/9y, how do you find what points at which the slope of the tangent is 1? I have no idea how to answer this and I've been trying for like an hour. Can anyone help me?
  37. T

    Tangent at self-intersection point

    Homework Statement Find the slopes of the two tangent lines of x^3-y^2+x^2=0 at 0,0. Homework Equations Differentiating implicitly we get (dy(x))/(dx) = (x (2+3 x))/(2 y). The Attempt at a Solution I'm not sure how to deal with the derivative being undefined at 0,0 when there are...
  38. P

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    Homework Statement Find equations of the tangent plant and the normal line to x-z=4arctan(yz) at (1+∏, 1, 1) Homework Equations The Attempt at a Solution I took the partials and got fx=0 fy=0 fz=(-4y)/((yz)2+1) so for the plane i got -2z-2=0 and for the normal line I got .5z-2=0...
  39. nukeman

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    Homework Statement Center of circle is: (3,2) Tangent point: (8,4) Question: What is the equation of the tangent line? Homework Equations The Attempt at a Solution I am just not getting it. So, would the radius be 5? Now, would i go: (x-3)^2 + (y-2)^2 = 5 ...
  40. D

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  41. D

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  42. R

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    Given: let f (x,y,z) = x2 + y2 + z2 = 2y - 3x and g(x,y,z) = 3x + y2 - z2 A. Find an equation for the tangent plane for the surface g(x,y,z) = 9 at the point (3, -1, 1) B. Find the line tangent to the intersection of the surfaces f(x,y,z) = 0 and g(x,y,z) =9 at the point (3,-1,1) A. I...
  43. F

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  44. T

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  45. S

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  46. H

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  47. G

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  48. A

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  49. P

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  50. A

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