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. moe darklight

    Hard time visualizing gradient vector vs. tangent vector.

    OK, this is really confusing me. Mostly because i suck at spatial stuff. If the gradient vector at a given point points in the direction in which a function is increasing, then how can it be perpendicular to the tangent plane at that point? If it's perpendicular to the tangent plane...
  2. I

    How Do You Calculate the Equations for Tangent and Normal Lines at a Point?

    Homework Statement Find the equation of the tangent and normals of y=x+\frac{1}{\sqrt{x}} at x = 4. Homework Equations \lim_{h \rightarrow 0}\frac{f(x_{0} + h) - f(x_{0})}{h} = m Also: slope of normal = \frac{-1}{slope of the tangent}The Attempt at a Solution m=\lim_{h \rightarrow...
  3. J

    Surface level of a curve, does it influence the tangent plane?

    Homework Statement Does the surface level of a curve influence the tangent plane of that curve? If so, how do I find the tangent plane specific to that level?
  4. M

    Tangent Line Equation for Ellipse: Parametric Equations at (1,2,2)

    Homework Statement The ellipsoid 4x^2 + 2y^2 + z^2 = 16 intersects the plane y = 2 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1,2,2)Homework Equations x = x0 + at y = y0 + bt z = z0 + ct The Attempt at a Solution Well i know that x0,y0 and z0...
  5. S

    Point on a parabola where the tangent intersects a point

    1. A car is traveling on a highway shaped like a parabola with its vertex at the origin. The car begins at a point 100m west and 100m north of the origin and is traveling easterly. There is a statue 100m east and 50m north of the origin. At what point on the highway will the car`s headlights...
  6. U

    Implicit Diff: Tangent Line at (-3*31/2,1)

    Homework Statement Implicit Differentiation:Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2/3 + y2/3 = 4 at (-3*31/2,1) Homework Equations ? None? The Attempt at a Solution 2/3 * x-1/3 + 2/3y-1/3*y' = 0 then after a few...
  7. L

    Proof of a focus point on parabola and tangent line equal angles

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  8. Q

    Equation of a line that has to be tangent

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  9. Q

    Equation for tangent to a curve and parallel to another line

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  10. R

    Curve and tangent of a surface intersected by a plane

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  11. G

    What Are the Tangent Lines to an Ellipse Passing Through a Given Point?

    Homework Statement Find the equations of both the tangent lines to the ellipse x2 + 9y2 = 81 that pass through the point (27, 3). One is horizontal the other is not. Homework Equations The Attempt at a Solution horizontal, easy: y = 3 x^2+9y^2=81 derivative: 2x +...
  12. K

    Finding Tangent Plane and Normal Line Equations for a Given Surface

    Homework Statement Find equations of the following. x2-2y2+z2+yz=7, (5,3,-3) (a) the tangent plane (b) the normal line to the given surface at the point Homework Equations I know it involves fx, fy, fz The Attempt at a Solution I got 10x-15y-3z=7. Is this correct? Because its not true at...
  13. F

    Determining perpendicular tangent line

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  14. S

    Computing tangent spaces of implicitly defined manifolds

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  15. U

    Find parametric equations for the tangent line at the point

    Homework Statement Find parametric equations for the tangent line at the point (\cos(\frac{4 \pi}{6}) ,\sin(\frac{4 \pi}{6}) ,\frac{4 \pi}{6}) ) on the curve x=\cos t,\ y=\sin t, \ z=t Homework Equations The Attempt at a Solution Took the derivative r'(t)=( -sint...
  16. M

    Unit tangent vector to a curve at a point

    Homework Statement Find the unit tangent vector T(t) to the curve r(t) at the point with the given value of the parameter, t. r(t)=<e^(2t), t^(-2), 1/(3t)> t=1 Homework Equations none The Attempt at a Solution So first I took the derevative to get r'(t) which I got to be...
  17. B

    The hyperbolic tangent function and physics

    Kind of an odd question, but here goes: can anyone offer any examples of physical systems that include the use of the hyperbolic tangent function in their mathematical solution?
  18. F

    Determining slope of the tangent line.

    Homework Statement I've never seen such a problem before: Find the two straight lines that are perpendicular to y=0.25x and tanget to the curve f(x)=1/xHomework Equations y=0.25x ; f(x)=1/x The Attempt at a Solution Using the power rule, I found the derivatives of: f(x)=-x^{-2} and y=0.25...
  19. G

    Limit of an inverse tangent function

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  20. F

    Tangent Vector for Vector Function?

    Tangent function vector? Homework Statement i know the formula for this problem. My homework was to like 5 of these and got the other 4. now stuck on the algebra part. please take a look and let me know where to go from here. problem is consider the vector function given below: r(t) =...
  21. M

    Solving Tangent Plane Problem: x2 + 2y2 + 3z2 = 21 and x+4y+6z=0

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  22. X

    Help needed in finding Tangent to a graph

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  23. N

    Maximal Exterior product of Tangent space

    I am working through a book on Kahler manifolds and for one of the proofs it states that the maximum exterior power of TM is m (where M has complex dimension). Could you explain why this is the case rather than the maximum exterior power being 2m.
  24. J

    Integration by Parts of Inverse Tangent

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  25. M

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  26. A

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  27. R

    Parametric Equations - Finding Tangent Lines

    So I'm looking at this example in my textbook where they're trying to find the vert. & horizontal tangent lines of r = sin Θ, 0 < Θ < pi. they say to first change it to parametric equations where x = r cos Θ = sin Θ cos Θ and y = r sin Θ = sin Θ sin Θ = sin^2 Θ But now I'm just really...
  28. S

    Two points on a curve with a common Tangent

    Homework Statement Find the two points on the curve y=x^4-2x^2-x that have a common tangent line.Homework Equations See below.The Attempt at a Solution y=f(x)=x^4-2x^2-x \frac{d}{dx}\right((x^4-2x^2-x)=4x^3-4x-1 To my understanding, two points A,B who have a common Tangent such that T_A=T_B...
  29. G

    Help with f ' (a) and tangent lines

    I'm taking calculus online and I need help figuring out how to solve the following problem. My book and notes given from my teacher do not show how to solve problems like this and I keep ending up with answers that don't work. The question is; Find f '(a) when f(t)=(3t+3)/(t+6)...
  30. S

    Behavior of a tangent light beam to an event horizon?

    Given present theory, how would a laser photon stream behave at a tangent point to an event horizon (EH)? Is it possible for a photon stream to orbit a black hole? Could the beam be split at the EH with one branch spiraling into the black hole while one branch follows some geodesic (perhaps not...
  31. A

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    Homework Statement I've been working on this problem for several hours with no luck. Any help would be appreciated. (It's not homework) A thin rod of mass M and length L is hinged at the bottom, and almost balanced vertically. It starts to fall. Find the tangential component of the force...
  32. S

    Equation for the tangent line

    Homework Statement Find the equation for the tangent line to the curve f(x) = 2x3 - 5x - 3 at x = 2 Homework Equations How do i start this? I'm really confused and have an assignment with a bunch of these questions due tomorrow by midnight... if someone could help me that would be...
  33. S

    Find the equation of the line tangent to the graph

    Find the equation of the line tangent to the graph of y=x^2 at the point (1/2,1/4) and how to find that?
  34. P

    Differential geometry:2-form computation on a pair of tangent vectors

    Homework Statement I am given a one form Psi = zdx -xydy and two vectors v(1,1,-2) and w(-2,1,1) both tangent vectors of R3 at point P(2,-1,0). I am asked to find dPsi(v,w). Homework Equations Lie bracket? The Attempt at a Solution I know how to computer Psi(v) at p but this...
  35. M

    Can Every Derivation Be Expressed as a Directional Derivative?

    I've encountered a definition of the tangent vector via the notion of derivations on the manifold and I have some problems with it. I would actually like to show that every derivation can be expressed as a directional derivative, but I'm not ver successful in doing so. I have this definition...
  36. T

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    Find an equation of the tangent to the curve at the given point by both eliminating the parameter and without eliminating the parameter. x = tan(θ) y = sec(θ) (1 , √2) y = ? attempt ; y - y1 = m(x-x1) y = √2 x = 1 y1 = sec(θ) x1 = tan(θ) substituting and solving it...
  37. Y

    Calculus 3, dealing with tangent planes and surfaces.

    okay i came up with doing the gradient of the ellipsoid. Then set that equal to the vector, <4,-4,6>. I solved and got x,y,z = 1,-2,1 I looked at the answer key and it said (1/3) (1,-2,1) Does anyone know where the 1/3 came from?
  38. M

    Position vector and tangent vector in Riemannian spaces

    In Euclidean vector spaces the derivative of the position vector of a running point of a curve is the tangent vector of the curve. In thehttp://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf" , on page 78 appears a vector which can be regarded as position vector in a Riemann space...
  39. C

    Calc Tangent Problem: Find Lines Tangent to f(x) at x=-1

    Homework Statement Let f(x)=a(7-x^2) find in terms of a, the equations of the lines tangent to these curves at x=-1. Homework Equations ? The Attempt at a Solution So I took the derivative of f(x). f'(x)=a(-2x)+7-x^2 then i plugged in -1 into f'(x) and i got f'(-1)=2ax+7+1=2ax+8...
  40. B

    Why is the tangent space of a lie group manifold at the origin the lie algebra?

    Question is in the title. Seems a lot of people throw that statement around as if its obvious, but it isn't obvious to me. I can kind of see how it might be true. If you take a group element, differentiate it wrt the group parameters to pull down the generators, and then evaluate this...
  41. A

    Finding Horizontal Tangent Lines on a Parabola

    Homework Statement I have a homework that i couldn't do :( can you explain it to me please ? the problem is : find all values of x=c so that the tangent line to the graph of f(x) af (c , f(c)) will be horizontal http://img13.imageshack.us/img13/4222/scan0002izs.jpg The Attempt...
  42. Y

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  43. E

    What mistake did I make in proving that all tangent bundles are trivial?

    In trying to understand why not all tangent bundles are trivial, I've attempted to prove that they are all trivial and see where things go wrong. Unfortunately, I finished the proof and cannot find my mistake. Here it is: Let M be an n-manifold with coordinate charts (U_\alpha...
  44. R

    Finding t in the Two Tangent Lines at (0,2)

    Homework Statement [/b] My book talks about find the two tangent lines at the point (0,2) for http://mathbin.net/equations/7402_0.png and http://mathbin.net/equations/7402_1.png .[/URL] It says that t then is equal to pi/2 and -pi/2. I do not know how to they solved for this t. Any help?
  45. G

    Very basic precal question about cotangent and tangent

    So, I just learned in class that to get inverse cot-1(x) I have to do tan-1(1/x). Then add either pi, or 180 depending on wheather we are using radians or degrees. And I don't understand this. If x is cot, then when we do 1/x, shouldn't we get tan? And after than isn't it enough to just do...
  46. S

    Dot product: normal and tangent

    Homework Statement Tangent plane goes through point P=(a,b,f(a,b)). Any point on the plane is then Q=(x,y,z)=(x,y,f(a,b)+fx(a,b)(x-a)+fy(a,b)(y-b)) (fx and fy are partial derivatives) and the vector \overline{PQ} is on tangent plane. Calculate dot product n.\overline{PQ} and show...
  47. H

    Finding the Equation of a Tangent Line Perpendicular to a Given Line

    Homework Statement Find the equation of the tangent line on the function, f(x) = X^{2}-4x+1, which is perpendicular to the line, x+2y=10. Homework Equations The Attempt at a Solution x+2y=10 2y = 10-x y=-1/2x+5 Slope of perpendicular is -1/2, so slope of tangent is 2. The...
  48. A

    Is this correct. Tangent lines.

    Homework Statement r(t) = sin t (i) + cos t (j) + t k Find the equation of the line tangent to r(t) at the point (0,1,0) If you plug in 0 into to for r(t), you get (0,1,0). Thus t must equal o. To find the vector of r' (t) or the derivative of r(t) this equals = < cos t, -...
  49. Ƒ

    Equation of the tangent line at (0, pi/2)

    Find the equation for the line that is tangent to the given formula if y = pi/2 when x = 0 Homework Equations (x+1)dy - [(1/2)secycscy]dx = 0 The Attempt at a Solution I tried to do this, and I got that dy/dx = 1 / (0*0), which is infinity. So... y = \infty?
  50. P

    Implicit Differentiation - Tangent Line & Horizontal Tangents

    Homework Statement Find an equation of the tangent line to this curve at the point (1, -2). Homework Equations The Attempt at a Solution 2y' = 3x^2+6x y' = 3x^2+6x y'=3/2x^2+3x y+2=3(x-1) y+2=3x-3 y=3x-5
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