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. Titan97

    Question on intersection of tangent and chord

    Homework Statement Show that The tangent at (c,ec) on the curve y=ex intersects the chord joining the points (c-1,ec-1) and (c+1,ec+1) at the left of x=c Homework Equations Legrange's mean value theorem The Attempt at a Solution f'(c)=ec Applying LMVT at c-1, c+1...
  2. modularmonads

    [Euclidean Geometry] Kiselev's Plainimetry Question 242

    Homework Statement Two lines passing through a point Μ are tangent to a circle at the points A and B. Through a point С taken on the smaller of the arcs AB, a third tangent is drawn up to its intersection points D and Ε with MA and MB respectively. Prove that (1) the perimeter of ▲DME, and (2)...
  3. auditt241

    Unit Tangent Vector in a Scalar Field

    Hello, I am attempting to calculate unit normal and tangent vectors for a scalar field I have, Φ(x,y). For my unit normal, I simply used: \hat{n}=\frac{\nabla \phi}{|\nabla \phi|} However, I'm struggling with using this approach to calculate the unit tangent. I need to express it in terms of the...
  4. A

    Moment of inertia about tangent of a hoop

    The hoop has radius R. I used the same way to plot the axis for the hoop: ##l^2 = r^2= 4R^2cos\theta## since: ##r=2Rcos\theta## 2\rho \int_{0}^{\frac{\pi}{2}} r^2 r d\theta 2\rho \int_{0}^{\frac{\pi}{2}} 8R^3 cos^3\theta d\theta answer is \frac{32}{3}R^3\rho , and its wrong using the...
  5. L

    Find an Equation of the Tangent Lines Passing Through a Poin

    Homework Statement Find an equation of the tangent lines [in point-slope form] to the hyperbola x^2 + y^2 = 16 that pass through the point (2, -2). Homework Equations Point-slope form: y - y_1 = m(x - x_1) Slope: m = \frac {y_1 - y_0} {x_1 - x_0} The Attempt at a Solution First, take the...
  6. F

    Exploring Tangents on Parabolas: A Comparison of y^2=4ax and x^2=4ay

    I know a tangent drawn on parabola having equation like y^2=4ax is y=mx+(a/m) which provides c=a/m Then how is it going to turn for equation like x^2=4ay? From my derivation it will be like -c=am^2 when the equation of tangent is y=mx+c. The derivation comes from the following: y=mx+c or...
  7. X

    Complete the square and find the tangent line

    I'm having a hard time solving this problem and was wondering if someone could explain to me how to solve it. Math isn't my strong point so be as noob friendly as possible...Thanks! So the function is f(x)=4x2+2x , a=-2 First I'm supposed to complete the square and graph it(Yes I looked online...
  8. T

    MHB Differentiation and Tangent Lines.

    Given f(x) = \arctan\left({\frac{\sqrt{1+x}}{\sqrt{1-x}}}\right) I differentiated and this was my answer. \d{y}{x} = \frac{1}{2\sqrt{1+x}\sqrt{1-x}{(1-x)}^{2}} I used implicit differentiation on the elliptic curve {x}^{2}+4{y}^{2} = 36 and it wants two horizontal tangents through (12,3)...
  9. U

    Are there any real-life examples of tangent waves in the universe?

    I'm in high school, just finished Grade 11 and I have learned about sine, cosine, and tangent waves in my math & physics classes. The question is more of where are tangent waves found in nature/this universe? I have thought that maybe electrons experience some sort of tangent wavelike behavior...
  10. orion

    Tangent vector as derivation question

    I have a question concerning the tangent space. Consider a manifold Mn and take Mn to be ℝn with the Euclidean metric for the purposes of this question. The directional derivative of a function in the direction of a vector v is (a) vf = ∑ vi(∂f/∂xi) where the sum runs from 1 to n. The...
  11. A

    What is the Complex Tangent Formula Proof for Homework?

    Homework Statement This is an easy one, but keep in mind I'm kind of a newbie, anyway I can't figure out how to get the next formula... tan(z) = (tan(a)+i tanh(b))/(1 - i tan(a)tan(b)) Homework Equations This is the third part of an excercise, previous I proof the follow, -all using the...
  12. C

    Tangent to a a curve, something seems wrong (Calculus)

    I'm studying Calculus and i can see that the definition of the tangent to a point on a curve is y= f'(a)(x-a)+b this must mean that f'(a) = (y-b)/(x-a) But that to me seems troubeling, because f'(a) is the slope at ONE point, while (y-b)/(x-a) is a quotient with the difference between...
  13. vktsn0303

    Understanding Tangent Vectors at Points on a Curve

    I was reading about the tangent vector at a point on a curve. It is formulated as r' = Lim Δt→0 [r(t+Δt) - r(t)] / Δt (sorry for the misrepresentation of the 'Lim Δt→0 ') where r(t) is a position vector to the curve and t is a parameter and r' is the derivative of r(t). All I can...
  14. S

    What Is the Equation of the Tangent to the Cissoid of Diocles at (1,1)?

    Homework Statement The cissoid of Diocles is given by the relation y2(2-x) = x3. Find the equation to the tangent line to the curve at the point (1,1). Homework EquationsThe Attempt at a Solution Solution d/dx [ y2(2-x) ] = d/dx [ x3 ] 2y dy/dx (2-x) + y2(-1) = 3x2 Therefore, dy/dx = 3x2+y2...
  15. N

    Graphical meaning of tangent in optimization problem

    In a trivial optimization problem, when seeking the value of x2 that minimizes y(x2)/(x2-x1), the solution is graphically given by the tangent line shown in the figure. I'm having a lot of difficulty understanding why this is true, i.e., the logical steps behind the equivalence supporting the...
  16. D

    Differential map between tangent spaces

    I've been struggling since starting to study differential geometry to justify the definition of a one-form as a differential of a function and how this is equal to a tangent vector acting on this function, i.e. given f:M\rightarrow\mathbb{R} we can define the differential map...
  17. S

    Calculating Tangent Vector to Curve: $\varphi$

    Homework Statement Calculate the tangent vector to a curve $$r=R(1-\varepsilon \sin ^2 \varphi)$$ as a function of ##\varphi## Homework EquationsThe Attempt at a Solution Ok, I tried like this: I defined a vector $$(f(\varphi),\varphi)=(R(1-\varepsilon \sin ^2 \varphi),\varphi)$$ and than I...
  18. hideelo

    Proof of dimension of the tangent space

    I am attaching a picture of a proof from the book "general relativity" by wald. This is supposed to show that the tangent space of an n dimensional manifold is also n dimensional. I have two questions. In equation 2.2.3 couldn't the function be anything at a since the (x-a) term is 0? How is...
  19. B

    MHB Plane that is tangent to two curves at an intersection

    Can someone please help me with how to approach/solve this question? construct a plane that is tangent to both curves at the point of intersection. 1st curve: x(v)=3 y(v)=4 z(v)=v 0<v<2 2nd curve: x(u)=3+sin(u) y(u)=4−u z(u)=1−u −1<u<1 My first approach was to find a point of intersection...
  20. AdityaDev

    Finding y=f(x) with Tangents and Equal Abscissae Intersection

    Homework Statement Given two curves y=f(x) passing through (0,1) and ##g(x)=\int\limits_{-\infty}^xf(t)dt## passing through (0,1/n). The tangents drawn to both curves at the points with equal abscissae intersect on the x-axis. Find y=f(x). Homework Equations None The Attempt at a Solution...
  21. anemone

    MHB Floor Function and Tangent function

    Solve the equation $x=\pi\left\lfloor\tan\dfrac{\pi}{x}\right\rfloor$.
  22. binbagsss

    Product of Tangent Vectors & Affine Parameter

    If ##\sigma## is an affine paramter, then the only freedom of choice we have to specify another affine parameter is ##a\sigma+b##, a,b constants. [1] For the tangent vector, ##\xi^{a}=dx^{a}/du##, along some curve parameterized by ##u## My book says that ' if ##\xi^{a}\xi_{a}\neq 0##, then by...
  23. D

    Solving the equation of a line, tangent to a curve

    Having some trouble with this.. Need to find equation of a line with a slope of -1 that is tangent to the curve y=1/(x-1). So, rearranging slope formula as y=-1x+k and setting the equations equal, y=-1x+k=1/(x-1) y=(-1x+k)(x-1)=1 Here is where my multiplication is either totally wrong or I am...
  24. Drakkith

    Equations of two Lines that are Tangent to a Parabola

    Homework Statement [/B] Find equations of both lines through the point (2,-3) that are tangent to the parabola y=x2+x. Homework Equations Slope formula: (Y2-Y1)/(X2-X1) = M The Attempt at a Solution Here's what I think I need to do. First I think I need to find the derivative of the...
  25. C

    Curves and tangent vectors in a manifold setting

    Consider the following definition: (##M## denotes a manifold structure, ##U## are subsets of the manifold and ##\phi## the transition functions) Def: A smooth curve in ##M## is a map ##\gamma: I \rightarrow M,## where ##I \subset \mathbb{R}## is an open interval, such that for any chart...
  26. S

    Point of contact of circle an tangent

    I want to find out the co-ordinate of point of contact of tangent to a circle from external point when its center and radius are known. Please Help me . . . Thank you in advance . . .
  27. Drakkith

    Equation of the Tangent Line to a Curve at a Given Point

    Homework Statement Find an equation of the tangent line to the curve at the given point. Homework Equations y=x¼ Point = (1,1) The Attempt at a Solution [/B] Derivative of y is y' = ¼x-¾ Plugging in the derivative to the equation for a line: y-1=¼(x-1)-¾. My book's answer is Y=¼x+¾, but I...
  28. B

    Why do we need the limit to exist for the slope of the tangent line?

    Homework Statement My textbook says that the slope of the tangent line at a point can be expressed as a limit of secant lines: m = \underset{x \rightarrow a}{\lim} \, \frac{f(x) - f(a)}{x - a} \, . If x > a and we approach a from the right, why do we have to insist that this limit exists...
  29. T

    Understanding Tangent and Cotangent

    I believe I understand both, for instance tangent is equal to y/x and so when you have something like root3/3 it would be equal to the points (root3/2 (x coordinate), and 1/2(y coordinate.) For Cotangent equal to zero, would be (0,1) and (0, -1) am I off base here? I think also that Cot is...
  30. D

    Tangent vectors as directional derivatives

    I have a few conceptual questions that I'd like to clear up if possible. The first is about directional derivatives in general. If one has a function f defined in some region and one wishes to know the rate of change of that function (i.e. its derivative) along a particular direction in that...
  31. B

    Hyperbolic tangent function for terminal velocity with Vo>Vt

    Hi! First post on this forum, though not the first time visiting :) I am working on a model of an object falling from one layer of air density into another layer with a higher density, effectively changing the acceleration from positive to negative instantly. (Somehow I am thinking of positive...
  32. Calpalned

    Find the equation of the tangent plane

    Homework Statement Find the equation of the plane tangent to ##x^2+3y^2+6z^2=67## at the point ##(1, 2,3)## Homework Equations ##w-w_0 = F_x(x-x_0) + F_y(y-y_0) + F_z(z-z_0) ## The Attempt at a Solution Using the above formula, I get ##w-67 = 2x(x-1) + 6y(y-2) + 12z (z-3)## = ##w-67 = 2(x-1)...
  33. T

    Tangent zeros with a calculator?

    I'm tasked with findings he solutions to a particular tangent equation, and when I did it before with sine it was just a matter of plugging in the equation to the calculator, then getting all the zeros. Since it is tangent now and not sine, do I do something special? Is it no longer the x value...
  34. Calpalned

    How Do Equations for Tangent Planes Differ in Usage?

    Homework Statement What is the difference between the two given equations below? When would you use one or the either? Homework Equations ## z - z_1 = \frac{∂z}{∂x}(x_1, y_1, z_1)(x - x_1) + \frac{∂z}{∂y}(x_1, y_1, z_1)(y - y_1) ## ## \frac{∂f}{∂x}(x_0, y_0, z_0)(x - x_0) +...
  35. Calpalned

    How Do General Plane Equations Differ from Tangent Surface Equations?

    Homework Statement What's the difference between the two equations for a plane? This question is somewhat related to my other, overarching question here: https://www.physicsforums.com/threads/i-am-confused-about-how-multivariable-calc-works.798798/ Homework Equations ## a(x - x_0) + b(y -...
  36. D

    Equation of the tangent plane in R^4

    Let f: \mathbb R^2 \to \mathbb R^2 given by f=(sin(x-y),cos(x+y)) : find the equation of the tangent plane to the graph of the function in \mathbb R^4 at (\frac{\pi}{4}, \frac{\pi}{4}, 0 ,0 ) and then find a parametric representation of the equation of the tangent plane What I did: the...
  37. C

    How to Sketch a v-t Graph from a p-t Graph Using Tangents?

    Hi! I'm having troubles drawing a velocity-time graph from a position-time graph. I know parabolic p-t graphs have diagonal lines for their v-t graphs, but I'm not sure why. I also know tangents are important to use, but again, I don't understand why. Any clarification as to why this happens (in...
  38. R

    A confusion with tangent galvanometer

    When no current is passed through the coil it shows no deflection i.e. B is in the direction shown. But according to theory both fields should be perpendicular (Earth's horizontal field and the one due to coil). But I am confused because the coil would ideally create a magnetic field that is...
  39. D

    Understanding the notion of a tangent bundle

    I've been reading up on the definition of a tangent bundle, partially with an aim of gaining a deeper understanding of the formulation of Lagrangian mechanics, and there are a few things that I'm a little unclear about. From what I've read the tangent bundle is defined as the disjoint union of...
  40. G

    When you press braking pedal -- wheel road surface tangent F

    - You are driving car with 100 mile/hour - You are pressing brake pedal. - The speed of car is suddenly gone down. Decelerated. - However, Wheel is rolling on road without sliding and car stopped smoothly finally.In this case, - the tangent force acting on road surface is related with...
  41. D

    Questions about tangent spaces & the tangent bundle

    This is a slightly physics oriented question, so apologies for that. Basically, having started studying differential geometry it has started to become a little clearer to me why one can consider the Lagrangian as a function of position and velocity, but I don't feel I'm quite there yet. My...
  42. Calpalned

    Find the slope of the tangent at the given angle theta

    Homework Statement Find the slope of the tangent line to the give polar curve at the point specified by the value of theta R = 1/θ, θ = π Homework Equations Slope of a polar equation is (dR/dθsinθ+Rcosθ/dR/dθcosθ-Rsinθ) The Attempt at a Solution Using my calculator, I plugged π for θ, -.101...
  43. Wayland Bugg

    Rope tangent angle over pully given position of offset load

    I am trying to symbolically resolve the angle of rope that suspends a load (rectangle) between two pulleys given the length of rope that is let out over the pulleys. See attached image. the load is not intended to move horizontally, only vertically. so you can imagine each pulley must have an...
  44. S

    Tangent to ellipse also tangent to circle

    Homework Statement if the tangent at a point P("theta") on the ellipse 16 (x^2) + 11 (y^2) = 256 is also tangent to the circle (x^2) + (y^2) + 2(x) = 15 then ("theta") = ?? 2. The attempt at a solution {{{{ i have taken "theta" as "d" }}}} P [4 cos d , (16/(sqrt11)) sin d] equation of...
  45. M

    MHB Finding the equation of a curve given the tangent equation

    I have a calculus question and I am not sure where to get started. The question states:"Find the value of c>0 such that the line y=x+1 is the tangent line to the curve c√x (i.e. intersects the curve at one point and shares the same slope at that point)." Thanks
  46. Calpalned

    How Can a Graph Have Both Vertical and Horizontal Tangents at (2, -4)?

    For the parametric equations x = t^3 - 3t and y = t^3 - 3t^2 I got that the graph has a vertical tangent when t is = to postive or negative one. And it is horizontal at t = 2. However, this implies that at the point (x,y) = (2, -4) the graph has both a vertical and horizontal tangent. How...
  47. D

    Vectors in Tangent Space to a Manifold Independent of Coordinate Chart

    In Nakahara's book, "Geometry, Topology and Physics" he states that it is, by construction, clear from the definition of a vector as a differential operator [itex] X[\itex] acting on some function [itex]f:M\rightarrow\mathbb{R}[\itex] at a point [itex]p\in M[\itex] (where [itex]M[\itex] is an...
  48. AdityaDev

    3D geometry: parametric equation and tangents

    I have a doubt in 3d geometry. I calculus and I know how to do partial derivatives.(but I don't know what it means). If you have a parametric equation ##x=t, y=t^2,z=t^3## (the equation is randomn) What does ##\vec{r}=t\hat{i}+t^2\hat{j}+t^3\hat{k}## represent? now if it represents the position...
  49. Randall

    How do I find the tangent to this parametric curve?

    Homework Statement Let C be the curve given parametrically by x = (t^3) - 3t; y = (t^2) - 5t a) Find an equation for the line tangent to C at the point corresponding to t = 4 b) Determine the values of t where the tangent line is horizontal or vertical. Homework Equations dy/dx =...
  50. quantumdude

    A Puzzle: Find an Ellipse Tangent to a Circle

    I was asked to look at a problem (not homework) in which a tangent ellipse is to be found for a circle. This puzzle is turning out to be more than I bargained for. See the attached image because hey, a picture's worth a thousand words. The givens in this problem are to be the radius ##R_i## of...
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