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. C

    Demonstrating Tangent Space Independence in Manifolds

    Hello I'm french so sorry for the mistake. If we have a manifold and a point p with a card (U, x) defined on on an open set U which contain p, of the manifold, we can defined the tangent space in p by the following equivalence relation : if we have 2 parametered curve : dfinded from...
  2. T

    MHB Limit of e to the tangent of x

    lim {e}^{tan(x)} x\implies\{(pi/2)}^{+} How do I start? Do I plug in and if so what next?
  3. L

    Parametrics tangent line with point given

    Homework Statement I understand the setup for finding the slope, but always get confused whether I've fully simplified when trig identities get involved. [/B] Homework Equations My dy/dx is [/B] 4sin(θ)cos(θ) -2csc2(θ) which I simplified to just (-2sin(θ)cos(θ))/(csc2(θ) Does that...
  4. G

    Tangent to Parametric Equations

    Homework Statement Consider the curve with parametric equations: x = t - cos t, y = sin t. Determine exactly the equation of the tangent to the curve at the point where t=-0.5pi. Homework EquationsThe Attempt at a Solution The equation of a line is y - y1 = m ( x - x1 ) I substituted t = -pi/2...
  5. T

    MHB At what point on the curve is there a tangent line parallel to the line

    At what point on the curve $$y = e^x$$ is the tangent line parallel to the line $$y = 2x$$ The derivative of y is $$\frac{dy}{dx} = e^x$$ But I'm unsure how to proceed from here.
  6. P

    How much the dielectric loss tangent of 10 nm thick SiO2 ?

    How much the dielectric loss tangent of 10 nm thick SiO2 at 1Ghz ? I am studying the MOSFETs i want to know the the dielectric loss tangent of SiO2 dielectric at such manometer thickness, So if anybody know any reference about it Please let me know..
  7. S

    Finding Tangent, Normal & Osculating Planes of r(t) at t=π/4

    Homework Statement Find the equations of the tangent line, normal plane and osculating plane to the curve r(t) = -2sin(t) i + 2cos(t) j + 3 k at the point corresponding to t = π/4. Homework Equations T[/B]^(t) = r'(t) // ||r'(t)|| u = a i + b j + c k, ||u|| = √(a^2 + b^2 + c^2) N^(t) =...
  8. S

    How to draw tangent from point of inflection on a chart with peaks

    Hi , I have an assignment which requires that I draw tangents from the point of inflection on a peak to the x-axis.I still cannot figure out how to do that using qtiplot or excel.
  9. D

    Tangent Vector for r=sint, theta=t/3

    Homework Statement Find the tangent vector and unit tangent vector for the curve: r=sint, theta=t/3 for 0<=t<=6pi. Homework Equations If the tangent vector is r'(t)e(hat)r + r*theta(t)e(hat)theta, how does the restriction on t affect the answer? The same for the unit tangent vector, they don't...
  10. R

    Calculating div(theta) and tangent curves

    Homework Statement Calculate ∇Θ where Θ(x)=\frac{\vec{p} \cdot \vec{x}}{r^3}. Here \vec{p} is a constant vector and r=|\vec{x}|. In addition, sketch the tangent curves of the vector function ∇Θ for \vec{p}=p\hat{z} (b) Calculate ∇ (cross) A → \vec{A}=\frac{\vec{m}x\vec{X}}{r^3} m is...
  11. Yae Miteo

    Find parametric equations for the tangent line

    Homework Statement Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Homework Equations x = 1+2 \sqrt{t}, \quad y = t^3 - t, \quad z = t^3 + t, \quad (3, 0, 2) The Attempt at a Solution I began by...
  12. D

    Forces on a string applied perpendicular / tangent direction

    Hello guys, I've had the following discussion at work: We are currently using a suture with a nylon thread on a tissue, when the thread of the suture is tightened in a force equal to if we've put a 500 gram weight on both ends of the suture line (meaning both end are tied together to the...
  13. I

    MHB Graph $r(t)$ for t = $\pi/4$: Position and Tangent Vectors

    $r(t)=sin(t)$i $+ 2cos(t)$j $t= \pi/4$ sketch the position vector and the tangent vector $r'(t)=cos(t)$i $- 2sin(t)$j $r(\pi/4)= \frac{\sqrt{2}}{2}$i $+ \sqrt{2}$j $r'(\pi/4)= \frac{\sqrt{2}}{2}$i $- \sqrt{2}$j $\left\langle \frac{\sqrt{2}}{2}, \sqrt{2} \right\rangle$ $\left\langle...
  14. E

    Can You Solve f(x) + g(x,y) = 0 Where g Is Tangent to f?

    Say we have two functions with the following properties: f(x) is negative and monotonically approaches zero as x increases. g(x,y) is a linear function in x and is, for any given y, tangent to f(x) at some point x_0(y) that depends on the choice of y in a known way. Additionally, for any...
  15. M

    Curve tangent is orthogonal to curve at a point

    Homework Statement . Let ##C## be a curve that doesn't pass through the origin and let ##P## be the closest point on the curve to the origin. Prove that the tangent to ##C## at ##P## is orthogonal to the vector ##P##. The attempt at a solution. Suppose ##P=\gamma(t_0)##, I want to...
  16. S

    MHB Find the equation of the tangent

    determine the point (a,F(a)) for which F'(a)=a, given that f(x)= -x^2+3x-7. write the equation of the tangent to f(x) at the point found 1) i have tried putting into first principle but when i do my denominator become 0 so i am kind stuck , i don't know what to do please help me thanks so much
  17. P

    Exploring the Concept of Tangent on a Large Circle

    Theoretically it is said that, tangent touches to a single point on a circle. But If my circle is very big, and large enough, then i think, it should not be a just single point where my tangent is touching, though is will be a very small portion depending on how large is the circle. If i have...
  18. Greg Bernhardt

    Insights What is a Tangent Line? A 5 Minute Introduction

    Continue reading...
  19. Dethrone

    MHB Why Are My Tangent Half-Angle Substitution Answers Different from the Workbook?

    I did two questions from my workbook that involved the tangent half-angle substitution, z = tan (\frac{x}{2}). The answers that I got, for two questions, were different (but correct, I think) in the same way. Can you assist me in how to acquire the workbook answer? 1. \int \frac{dx}{1-2sinx}...
  20. rakeru

    Two values for m for which a line is tangent to a parabola

    Homework Statement Hi! I am doing some problems to practice for a math competition, and I'm wondering if I did this correctly. I don't really have an answer sheet, so I have no way of knowing whether I'm right. If you would please review it, that would be cool! It reads: There are...
  21. I

    MHB Find equation of tangent to curve at $t=-\pi$

    find an equation of the tangent to the curve at the point corresponding to the given value of the parameter $x=t\cos\left({t}\right)$, $y=t\sin\left({t}\right)$, $t=-\pi$ help me
  22. S

    Tangent to the circle at a given point

    Homework Statement I basically have the radius of the circle and its displacement from the origin, so ##(x-p)^2+(y-q)^2=r^2## And now I need to find a tangent to the circle at a given point ##(a,b)##. Or at least the slope of the tangent. How would one do that? Homework Equations...
  23. PsychonautQQ

    Is There a Point of Tangency Between f(x) = x^(1/2) and y = (x/4) + 1?

    Homework Statement Q: Does the graph of f(x) = x^(1/2) have a point of tangency with the line y = (x/4) + 1? Homework Equations lim x->a (f(x) - f(a)) / (x-a) The Attempt at a Solution If the limit exists of the relevant equation than there is a point of tangency. So I'm...
  24. DreamWeaver

    MHB Sum of two inverse tangent functions

    By considering the product of complex numbers: z = (2+i)(3+i) Show that \tan^{-1}\frac{1}{2} + \tan^{-1}\frac{1}{3} = \frac{\pi}{4}
  25. M

    Horizontal Tangent in Parametric Equations: Finding & Understanding

    If the tangent is horizontal, it is where the tangent is zero. In single var. calc. that would be at max. or min. for example. I am confused about what horizontal tangent refers to when I am given a parametric equation. E.g. At what value of t does x=t^2 -t and y=t^2 +t have a horizontal...
  26. S

    MHB Help find eqn of circle given another circle that is tangent

    Please help me find the standard equation of the circle passing through the point (−3,1) and containing the points of intersection of the circles x^2 + y^2 + 5x = 1 and x^2 + y^2 + y = 7 I don't know how to begin, I am used to tangent lines or other points, but I don't know what is visually...
  27. F

    MHB Solving a Logarithmic Curve: Tangent Lines & Approximations

    guys I need loads of help for this question(s)? what do i even do ?!? Sketch the curve y = lnx and find the tangent line to this curve at the point where the curve crosses the x-axis. Deduce that, for small delta, ln(1 + δ) ≈ δ . - I know what the y = Inx curve looks like but what do i do...
  28. S

    MHB Help find eqn of circle given another circle that is tangent

    please help me find the standard equation of the circles that have radius 10 and are tangent to the circle X^2 + y^2 = 25 at the point (3,4). the soln: (x-9)^2 + (y-12)^2 = 100, (x+3)^2 + (y+4)^2 = 100, i found the eqn that intersects the centre of the small circle and the larger one to be...
  29. M

    X intercept of Line Tangent to Curve

    Homework Statement What is the x-intercept of the line tangent to the curve x(t) = 3 + cos(∏t), y(t) = t^2 + t + 1, when t = 1? Homework Equations Derivative, y=mx+b The Attempt at a Solution To find the line tangent to the curve: d/dt = <-∏sin(∏t), 2t+1> at t=1 <-∏, 3> dy/dx = dy/du *...
  30. L

    MHB How long is the tangent that intercepts between the coordinate axes?

    348) given the length of the ellipse to a2 h2 + b2y2 = a2 b2 Find the length of the tangent shorter, that intercepts between? the coordinate axes answer L = a + b The equations a2 h2 + b2y2 = a2 b2 And the line y = mx + b as the objetive function
  31. R

    Finding the Tangent Line at a Point on a Curve

    I need to find the tangent line to the curve xe^Y+ye^x=1 at the point (0,1). I took the derivative and found to be: dy/dx=-(ye^x-e^y)/(xe^y+ye^x) I set that equal to 0 so: 0=-ye^x-e^y I have tried using a natural log to get y on one side and x on the other, but so far no good. How can...
  32. M

    Point of intersection of tangent line with another line

    Homework Statement Find the pt. at which the tangent line to the curve x=3t^2 - t, y=2t+t^3 at t=1 intersects the line y=2-x. Homework Equations Possibly <6t-1, 2+3t^2> if the tangent is not already present The Attempt at a Solution I am confused about how to go about solving...
  33. R

    Slopes of tangent lines of parametric curves.

    1. The problem statement,ll variables and given/known data I have the first and second derivatives of a parametric function and the book is asking for when the slope of the tangent is vertical and horizontal. I get that horizontal is when dy/dx is 0. But what about vertical, is that dy/dx is 1...
  34. C

    Tangent vector to curve - notational confusion.

    Given a curve ##\gamma: I \to M## where ##I\subset \mathbb{R}## and ##M## is a manifold, the tangent vector to the curve at ##\gamma(0) = p \in M## is defined in some modern differential geomtery texts to be the differential operator $$V_{\gamma(0)}= \gamma_* \left(\frac{d}{dt}\right)_{t=0}.$$...
  35. D

    Tangent will not meet the curve again

    Homework Statement can anyone give me hint on how to show the tangent will not meet the curve again? Homework Equations The Attempt at a Solution
  36. M

    What is the slope of the common tangent line between two quadratic curves?

    I don't quite understand the context of common tangent problems. This is one of the problems I am trying to solve: Prove that there is a line that is a common tangent to the parabolas y = x2 and y2 = x. This is how I tried to solve it at least: y2 = x --> y = \pm\sqrt{x} CASE I...
  37. L

    MHB Max and min tangent and triangle

    in what point of the circumference: x2 + y2 = 1 the tangent to this, (to yhe circunference) form with the coordinate axes the triangle of smaller area?. answer ( +/-(sqrt2/2), +/-(sqrt2/2) ) Ok y2= 1-x2 Now I don't know in what point must i get the tangent?? I don't think it is in 0,0 I...
  38. BiGyElLoWhAt

    Tangent plane of a parametric function

    Ok, so I'm really hoping someone can help me logic my way through this. I have a function to the effect of: ##r(u,v)=f(u,v)\hat{i} + g(u,v)\hat{j} +h(u,v)\hat{k}## I need to find an equation of a tangent plane at a point ##(u_{0},v_{0})## and quite frankly I'm at a loss on how to do this. So...
  39. J

    Partial derivatives; Tangent Planes

    Hi guys, Question is: Find the slopes of the curves of intersection of surface z = f(x,y) with the planes perpendicular to the x-axis and y-axis respectively at the given point. z = 2x2y ...at (1,1). fx(x,y) = 4xy ∴ Slope = 4 fy(x,y) = 2x2 ∴ Slope = 2 Is this wrong? Answer...
  40. karush

    MHB How Do You Find a Common Tangent Line to Two Parabolas?

    https://www.physicsforums.com/attachments/2227 The region $R$, is bounded by the graphs of f(x)=x^2 -3 and g(x)=(x-3)^2, and the line T, as is shown in the figure above. T is tangent to the graph of f at point (a,a^2-3) and tangent to the graph of g at point (b,(b-3)^2) a. Show that...
  41. B

    Inverse tangent of a complex number

    Homework Statement I have to find ##\tan^{-1}(2i)##. Homework Equations The Attempt at a Solution So far I have ##\tan^{-1}(2i)=z\iff tan z= 2i\iff \dfrac{sin z}{cos z}=2i ##. From here I get that ##-3=e^{-2zi}##. I do no know how to take it further to get ##z=i\dfrac{\ln...
  42. S

    Calculating Curve Tangent at x=-π/4

    Homework Statement Deside curve tangent in point x=-π/4 Homework Equations f(x)=1/3sin(3x-π/4) y=f(x) The Attempt at a Solution f`(-π/4)=-1 using the tangent equation y=kx+m y=-1*(-π/4)+m y=1/3sin(3(-π/4)-π/4) ≈3.33*10^-14 3.33*10^-14=-1*(-π/4)+m f(x)≈-1*(-π/4)+0,79 is...
  43. L

    MHB Minimizing the slope of the tangent to a curve

    determine the point on the graph of: y = x3 - 4 x2 in which the tangent line has the minimum slope. answer (4/3, -128/27) ok my original idea was yo derive the curve first 3x2-8x But when I equal to 0 I get x= 3/8 The curve would be the main and the constrain y = mx I tried and i couldnot...
  44. Feodalherren

    Tangent Plane to f(x,y) = sin(x)cos(y) at (π/3,π/2)

    Homework Statement 3a) Find the equation of the tangent plane to the function f(x,y) = sin(x)cos(y) at the point (∏/3,∏/2). The Attempt at a Solution There is quite clearly a z in the definition. What's going on?
  45. MarkFL

    MHB Find Tangent Line to Curve (x+2y)^2+2x-y-3=0: Answer at Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  46. G

    I get two different values when calculating the tangent of a curve.

    [b]1. (a) Find the slope of the tangent line to the curve y=x-x^3 at point (1,0) (i) using the 1st definition of a limit: lim(x->a)- (f(x)-f(a))/(x-a) (ii) using the 2nd equation of a limit: lim(h->a)- (f(a+h)-f(a))/hThe Attempt at a Solution In my attempt I got two different values (the same...
  47. S

    Tangent to a Circle Homework Solution

    Homework Statement Given that the line ##y=mx+c## is a tangent to the circle ##(x-a)^{2} +(y-b)^{2} =r^{2}##, show that ##(1+m^{2}) r^{2}=(c-b+ma)^{2}##Homework Equations Quadratic discriminant, sum and product of rootsThe Attempt at a Solution I substituted y=mx+c into the equation of the...
  48. B

    Implications of the tangent of a function being at a maximum

    Does the tangent of a function being at a maximum necessarily mean that the function itself is at a maximum? I am supposed to find whether del is at a maximum at w = (tansig*taneps)^(-1/2) del = arctan(w*(tsig-teps)/(1+(w^2*(tsig*teps)))) tansig and taneps are constants and w is the...
  49. K

    Unit vector tangent to the surface

    I have the following question: Given that ø = (x^2)y + cos(z) find the unit vector n which is both tangent to the surface of constant ø at (1,1,∏/2) and normal to the vector b = x + y - 2z (where x y and z are the unit vectors) I have calculated ∇ø = 2x + y - z (again where x y and z are...
  50. R

    Does Arctanh Go to Infinity When Its Argument Approaches Infinity?

    So, I'm doing a problem where I take arctanh to a limit, and I was wondering if the arctanh function goes to infinity if the argument inside of the function goes to infinity when passing through the limit.
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