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".

View More On Wikipedia.org
Recent contents Top users
  1. Z

    Points on two ellipses with identical tangent lines

    Hi, I'm trying to get this working for a program I'm making. I've been working on this for a while, but I can't seem to figure it out. I have multiple rotated ellipses. Imagine you took a rubber band and stretched it around the ellipses. The rubber band would follow the curve of the outside...
  2. O

    Finding Self-Intersection and Unit Tangent Vectors of γ(t)

    Homework Statement Show that the curve γ(t)=(t²-t+1,t³-t) has exactly one self-intersection point and finnd the two unit tangent vectors (in the direction of increasing t) at this point. I have found the self intersection. I know that a unit tangent vector is the derivative of each...
  3. K

    Find equation of line that is perpendicular to the tangent line to the curve

    Homework Statement Find the equation of the line that is perpendicular to the tangent line to the curve, y=(3x+1)/(4x-2) at the point (1,2) Homework Equations The Attempt at a Solution I am absolutely confused with this problem. I tried taking a derivative of the equation. And I...
  4. T

    Derivatives,First Principle andEquation of a Tangent to a Curve

    Homework Statement ok so this is going to be divided into 2 parts, 1st is related to First Principle to obtain the derivative,the 2nd part is about obtaining equations of the tangent. I'd like to apologize for not being able to write the equations and my work neatly as other threads seem to...
  5. jegues

    Tangent unit vector of a curve

    Homework Statement See first figure. Homework Equations The Attempt at a Solution See second figure. When I set t = 0 in \vec{r(t)} I get 0\hat{i} +2\hat{j} + 1\hat{k}. I know this is a vector and not a point but it has the same "coordinates" as the point they are asking us...
  6. R

    Calculus - Tangent lines and radial lines

    Homework Statement Let P be any point (except the origin) on the curve r=f(θ). If ψ is the angle between the tangent line at P and the radial line OP, show that tan(ψ)= (r/(dr/dθ)) Hint: Observe that ψ = φ - θ in the figure. Homework Equations Very few equations come to mind except y =...
  7. U

    How to Prove tan3A + tan2A + tanA Equals tan3Atan2AtanA?

    Hello Everyone, I'm doing my math in advance so I came across a Trigonometry question I came across in my textbook. I did make some progress but I do not know how to go about it further. Homework Statement Prove that, tan3A + tan2A + tanA = tan3Atan2AtanA The Attempt at a...
  8. B

    Implicit Differentiation; Tangent Line

    Homework Statement Find the equation of a tangent line at the curve at point (-3√3, 1) x^(1/3) + y^(1/3) = 4Homework Equations Point-slope: y-1=m(x-1) The Attempt at a Solution I took the derivative of that equation and resulted in -y^(2/3)/x^(2/3) When I tried plugging in x and y to...
  9. A

    Natural Log Composed with Hyperbolic Tangent & this Ratio

    Hello, Consider x \in (0,1) , that is x between 0 and 1. Can someone explain why the following is true: \frac{x-1}{x+1} = \tanh \left( \ln \left( \frac{x}{2} \right) \right)
  10. J

    Triangle and tangent line circle

    A triangle ABC, where ,<A = 60 degrees. Let O be the inscribed circle of triangle ABC, as shown in the figure. Let D, E, and F be the points at which O is tangent to the sides AB, BC and CA. And let G be the point of intersection of the line segment AE and the circle O. (but AE line not cross...
  11. Char. Limit

    So, for tanh(ax),y = tanh(ax)x = tanh-1yx = (tanh-1y)/a

    Homework Statement Now, I decided for no real reason to derive a formula for the hyperbolic tangent using only what I know about the derivative of the inverse hyperbolic tangent. However, what I have looks wrong, and I'd like to check it here. Homework Equations \frac{d...
  12. A

    Find the equation of the tangent at the given point

    Homework Statement For each function, find the equation of the tangent at the given point a) f(x)= 1/ squareroot(2x+1) at x=4 The Attempt at a Solution I'm fairly lost, I understand I'm looking for an equation of a tangent, but I don't understand where to begin, find the derivative...
  13. W

    Why the basis of the tangent space of a manifold is some partials?

    it is quite peculiar i know you do not want to embed the manifold into a R^n Euclidean space but still it is too peculiar it is hard to develop some intuition
  14. R

    Finding the Tangent Vector of a Space Curve at a Given Point

    Homework Statement Here's a worked problem, I can't understand how they have evaluated T at the given point (in part c): [PLAIN]http://img31.imageshack.us/img31/3725/97856984.gif The Attempt at a Solution I just substituted (0,1, \pi/2) into r'(s) but \frac{1}{\sqrt{2}} cos...
  15. P

    Why field lines are tangent to direction of electric forces

    can anybody tell me why electric field of lines are tangent to direction of electric force please be fast need it urgently...
  16. M

    Find the curve given the tangent

    Homework Statement Given that the tangent to the curve c(t) at any point on the curve is T(t) = (-sin(t), cos(t) ), find c(t) if the curve passes through the point (0,0) .The Attempt at a Solution I try to let c(t) = ( x(t), y(t) ) Then c'(t) = ( x'(t), y'(t) ) | c'(t) | =...
  17. mnb96

    Deriving the atan2 Function from Tangent Half-Angle Formulas

    Hello, it is mentioned http://en.wikipedia.org/wiki/Atan2#Definition" that using the tangent half-angle formulas it is possible to express the function atan2 as: \mathrm{atan2}(y,x)=2\mathrm{arctan}\frac{y}{\sqrt{x^2+y^2}+x} How can I derive this result?
  18. J

    Finding a tangent in a Cubic Function

    Homework Statement My cubic function is y=(x-6)(x-1)(x-9) or y=x^3-16x^2+69x-54 I need to find the tangent at the point x=2.5Homework Equations The Attempt at a Solution All that I have managed to do is work out the y value for x=2.5, that is y=34.125 Please help someone!
  19. F

    The mystery of the tangent and the radius fluid in spiral pipe on rotating disk

    Here's something simple but also a bit puzzling, let me know if you have any ideas... For clarity, I'll describe three cases before asking the final question (skip ahead if you like). 1. Mounted on a disk are two curved fences, just a few centimetres apart from each other. Between the fences...
  20. S

    Equation of tangent line (rec. form) to a polar curve

    Equation of tangent line (rec. form) to a polar curve! Homework Statement Quesiton: Find the rectangular form of the equation of the tangent line to the polar curve r=cos^3(theta) at the point corresponding to theta=pi/4 Homework Equations The Attempt at a Solution How to...
  21. S

    Partial Derivatives - Finding tangent in a volume?

    Not sure I understand exactly what this question is asking. This is obviously a volume in R3 and so how do you get a tangent inside a volume? Or is it just along the plane y = 2 intersecting the volume? Also, what is a parametric equation...? Thanks for the help: Question: The ellipsoid 4x^2...
  22. M

    Slope of Tangent Line at (0,-10) for y^3+1004=(e^x+1)^2

    Homework Statement A curve is given by the equation: y^3+1004=(e^x+1)^2 Find the slope of the tangent line at the point (0,-10). Homework Equations The Attempt at a Solution I took the derivative of ((e^x+1)^2-1004)^(1/3) and that is (2e^x(1+e^x))/(3((1+e^x)^2-1004)^(2/3)) but...
  23. S

    Find Equations of Tangents to C1 and C2 | Area Enclosed | Derivatives

    Homework Statement let C1 : y = x - 1/2 x2 and C2 : x = y - 1/2 y2 be curves on the xy plane. 1. find the equation of the tangent to the curve C1 at x = k 2. suppose the line obtained in 1) is also tangent to the curve C2. find all values of k and the equations of the tangents. 3...
  24. B

    How to Find the Equation of a Tangent and Normal Line to a Curve?

    Homework Statement Determine the equation of the tangent line and the equation of the normal line to the curve y at the point (-2,-5) y=1+x+x^2 The Attempt at a Solution y=1+x-x^2 point (-2,-5) y=1+x-x^2 y=1-2x sub in -2 for x...
  25. L

    Photoelectric effect, discrete values of the tangent

    It puzzles me. In Einstein's paper on the photoelectric effect he proposed that photons with E = nhf were the explanation. Wouldn't a more elegant explanation be that the tangent of the electromagentic wave must take on discrete values because of the boundary conditions between the emitter...
  26. B

    Shape operator and The Gauss tangent map

    Let M be a surface in R3 oriented by a unit normal vector field U=g1U1+g2U2+g3U3 Then, the Gauss Map G: M to E, of M sends each point p of M to the point (g1(p),g2(p),g3(p)) of the unit sphere E. Show that the shape operator of M is (minus) the tangent map of its Gauss map: If S and G are...
  27. R

    Slope of the tangent line of an intersection - Directional Derivatives

    Homework Statement Find the slope of the tangent line to the curve of intersection of the vertical plane x - y + 1 =0 and the surface z = x2+y2 at the point (1, 2, 5) Homework Equations Gradients, Cross products The Attempt at a Solution I'm pretty lost here. I think I have to...
  28. B

    Tangent map, gauss map, and shape operator

    Can anyone help me with this problem?? Let M be a surface in R^3 oriented by a unit normal vector field U=g1U1+g2U2+g3U3 Then the Gauss map G:M\rightarrow\Sigma of M sends each point p of M to the point (g1(p),g2(p),g3(p)) of the unit sphere \Sigma. Show that the shape operator of M is...
  29. J

    Find Tangent Slope with Polar coordinates

    Homework Statement Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. r = 9sin(θ) θ = pi/6 Homework Equations dy/dx = (dy/dθ) / (dx/dθ) x=rcosθ y=rsinθ (sinx)^2 = (1/2)(1-cos2x) (cosx)^2 = (1/2)(1+cos2x) 2sinxcosx = sin(2x) The Attempt at...
  30. S

    How Do You Find a Common Tangent Line to Two Different Functions?

    I was working on a problem, and in my solution I came across a situation which I will try and state in the following question: Given two functions, f(x) and g(x), how would you find a line such that the line is tangent to f(x) at some point x=a, and tangent to g(x) at some point x=b, assuming...
  31. L

    Finding f'(1) with Tangent Line at (1,7)

    Homework Statement If the line tangent to the graph of the function f at the point (1,7) passes through the point (-2,-2), then f'(1) is? Answer: 3 Homework Equations The Attempt at a Solution I got the right answer, but I think it was by pure luck. I found the slope...
  32. L

    Write an equation of each horizontal tangent line to the curve.

    Homework Statement Write an equation of each horizontal tangent line to the curve. Homework Equations y = 2y^3 + 6x^2y - 12x + 6y = 1 y' = (4x - 2xy) / x^2 + y^2 + 1) The Attempt at a Solution Well, horizontal tangent line means the derivative equals zero. Thus, 4x - 2xy = 0...
  33. L

    Equation for the line tangent to the graph and use it to approx. f(1.2)

    Homework Statement Write an equation for the line tangent to the graph of f at x = 1 and use it to approximate f(2.1). Homework Equations y = mx+b f(1) = 4 f'(x) = (3x^2 + 1) / 2y m = 1/2 when x = 1 The Attempt at a Solution Well, if the line is tangent to the graph of f at x...
  34. L

    What value of x do the graphs of f and g have parallel tangent lines?

    Homework Statement Let f be the function given by f(x) = 3e^2x and let g be the function given by g(x) = 6x^3. At what value of x do the graphs of f and g have parallel tangent lines? a. -0.701 b. -0.567 c. -0.391 d. -0.302 e. -0.258 Correct answer is c. -0.391 Homework...
  35. F

    How can I find the equation of the tangent line at x=4 for y=2x?

    We know that the slope of the tangent line at a point on a curve is found by evaluating the derivative of the curve at that point. Say we have the curve y=2x. Say I wanted to find the tangent line at x=4 dy/dx=2 The first derivative is a constant, which is not surprising since the curve is...
  36. G

    Tangent Line and Coordinates of Trigonometric Function

    Homework Statement There are infinitely many points on the curve y = \frac{sin x}{\sqrt{2}- cos x} at which the tangent line to this curve is horizontal. Find the x- and y-coordinates of one such point. Homework Equations y' = slope of the tangent line Etc., etc. The Attempt at a...
  37. A

    Find which value of x horizontal Tangent Line

    Find which value of x...horizontal Tangent Line Homework Statement What is given is F(x)= -4/(x-3)(x+4) and the problem asks for to find the value of x where f(x) has a horizontal tangent line.Homework Equations I read somewhere else on these forms that using the quotient rule is the key, and...
  38. r-soy

    Do We Use Radians or Degrees for Calculating Inverse Tangent Functions?

    Hi here in like thi9s queation we use ( Rad ) or (deg ) in the calcolator ?
  39. T

    Finding points on curve where tangent line given

    Homework Statement Find all points on the curve x^2 * y^2 + xy = 2 where the slope of the tangent line is -1. Homework Equations y' = -1 The Attempt at a Solution I got the y' = [ -2xy^2 - y ] / [ 2x^(2)y + x ] to be the gradient which I am sure is right. Then I subbed in y' =...
  40. T

    Horizontal tangent points (implicit differentiation)

    Homework Statement Find the points on the lemniscate: 2( x^2 + y^2 )^2 = 25( x^2 - y^2 ) where the tangent is horizontal Homework Equations Horizontal tangent: y' = 0 The Attempt at a Solution I got the correct gradient of y' = [ 50x - 8x^3 - 8y^2 ] / [ (8x^2)y + 50y + 8y^3 ]...
  41. K

    What is the Tangent Space to the Unitary Group?

    This may seem like an easy question, but my differential geometry is a little rusty. I'm trying to find the tangent space to the Lie group U(n) ; that is, for an arbitrary X \in U(n) I'm trying to find an expression for T_X U(n) . I can't quite remember how to do this. I've been playing...
  42. C

    Equations for tangent & normal at P2 of circle P1 P2 P3?

    Greetings, Given three points P1 P2 P3 on a circle in x,y,z coordinates, I am trying to figure out how to get the tangent and normal at P2. Anyone? Thanks
  43. M

    Something about tangent vector

    hey there, i got stuck on an question here: Parameterise the following paths, in the dirction stated, and hence find a tagent vector(in the same dirction) to each point on the paths. (a)The upper part of the circled centred at (0,0) containing the points (-2,0) and (2,0) going anticlockwise...
  44. Char. Limit

    Tangent Planes: Existence & Extensibility

    Most functions y=f(x) have tangent lines for any point x. Does a function z=f(x,y) have a tangent plane for any point x,y? And could you extend this to higher dimensions if necessary? (Tangent cubes? Tangent hypercubes?) Edit: Sorry, I thought I was posting in the General Math forum...
  45. A

    Find an equation for the tangent line

    suppose the function F is define by F(x)=[integral from 1 to √x] (2t-1)/(t+2)dt for all real numbers x≥0 A.) evaluate F(1) B.) Evaluate F'(1) C.)Find an equation for the tangent line to the graph of F at the point where x=1 D.) on what intervals is the function F increasing? justify your...
  46. G

    Tangent/Normal planes, intersections, vector tangent

    Homework Statement Surface (s) given by x^3*+5*y^2*z-z^2+x*y=0 question ask to find normal and tangent plane to S at the point P=(1,-1,0) then find vector tangential to curve of intersection of S and the plane x+z=1 at point P. Homework Equations The Attempt at a Solution So I started off...
  47. S

    Equation for the tangent of a circle

    Homework Statement If the equation of one tangent to the circle with center at (2, -1) from the origin is 3x + y = 0, then the equation of the other tangent through the origin is: (a) 3x - y = 0 (b) x + 3y = 0 (c) x - 3y = 0 (d) x + 2y = 0Homework Equations An equation of the tangent to the...
  48. C

    Setting a tangent plane parallel to another plane-Cal III

    At what point on the paraboloid y=x^2+z^2 is the tangent plane parallel to the plane x+2y+3z=1 ? Tangent plane equation is... Fx(X,Y,Z,)(x-X)+Fy(X,Y,Z)(y-Y)+Fz(X,Y,Z)(z-Z)=0; for x^2+z^2-y=0 My attempt at the problem... First I found the unit normal for the plane I'm trying to...
  49. A

    Solving Arc Tangent Squared: tan(2x) - 3cot(2x) = 0

    Homework Statement tan(2x) - 3 cot (2x) = 0 Homework Equations Trigonometry Knowledge. The Attempt at a Solution tan(2x) - 3cot(2x) = 0 tan(2x) - 3/tan(2x) = 0 [tan(2x)]^2 - 3 = 0 [tan(2x)]^2 = 3 Is there such thing as Arc Tangent that's squared??
  50. A

    Find the gradient of the tangent

    Homework Statement For every x>-4 where x\in \Re applies sinx+x\leqf(x)\leq8\sqrt{x+4}-16 Find the gradient of the tangent to the curve of f at x_{0}=0 Please help me I am trying to solve this exercise for more than two hours! I'm desperate.
Back
Top