What is Tangent line: Definition and 279 Discussions

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. Anachronist

    B Circle tangent to two lines and another circle

    I'm trying to solve this for a model I'm making in OpenSCAD. Given a circle of radius r centered on the origin, and two perpendicular lines at x=a and y=b, where is the center (x1,y1) of a circle that is tangent to both lines and the centered circle? Here's a picture: I thought it would be...
  2. C

    Parametric curve question (determining unknown point)

    My work so far: I am stuck because when I inputted the two possible values of t and k, neither solution worked. Where did I go wrong? Pointers would be appreciated! :)
  3. G

    Find the two points on the curve that share a tangent line

    IMPORTANT: NO CALCULATORS I assumed two points, (a, f(a)) and (b, f(b)) where b is greater than a. Since the tangent line is shared, I did f'(a) = f'(b): 1) 4a^3 - 4a - 1 = 4b^3 - 4b - 1 2) 4a^3 - 4a = 4b^3 - 4b 3) 4(a^3 - a) = 4(b^3 - b) 4) a^3 - a = b^3 - b 5) a^3 - b^3 = a - b 6) (a...
  4. M

    Precise definition of tangent line to a curve

    How do we define tangent line to curve accurately ? I cannot say it is a straight line who intersect the curve in one point because if we draw y = x^2 & make any vertical line, it will intersect the curve and still not the tangent we know. Moreover, tangent line may intersect the curve at other...
  5. H

    How to find the straight tangent line?

    I have solved the gradient: gradf(2,-1)=(4,2) and have the tangent plane: 4x+2y+3=0 Somehow the answer is: 3=2x+y And i really don´t understand why.
  6. karush

    MHB Tikz for f(x)=5x^2-2x and tangent line at 1,3

    ok I am actually trying to plot $f(x)=5x^2-2x$ with the tangent line going thru $(1,3)$ which is $8\left(x-1\right)+3$ I thot I could just change this from an example but does seem to like it stack exchange had some samples but they got very complex with other features added anyway mahalo...
  7. U

    Finding the horizontal tan() lines of this equation

    I've been able to find the tangent line with most equations, but I don't have any idea how to do it with a range of values instead of being given a singular value.
  8. qbar

    A Implicitly differentiating the vanishing real part of the hyperbolic tangent of one plus the square of the Hardy Z function

    Let $$Y(t)=tanh(ln(1+Z(t)^2))$$ where Z is the Hardy Z function; I'm trying to calculate the pedal coordinates of the curve defined by $$L = \{ (t (u), s (u)) : {Re} (Y (t (u) + i s (u)))_{} = 0 \}$$ and $$H = \{ (t (u), s (u)) : {Im} (Y (t (u) + i s (u)))_{} = 0 \}$$ , and for that I need to...
  9. fulton33

    Calculating the equations for the tangent/normal lines

    9. When I do this problem I know my slope is -3 because f'(2)=-3. I then went and substituted and got y+5=-3(x-2) which simplified to y=-3x+1 10. I get lost here because the tangent slope would be 0, which would give me the equation y=-2. The normal means perpendicular and the perpendicular...
  10. karush

    MHB 210 AP Calculus Exam problem tangent line to curve

    Find the slope of the tangent line to the graph of $$f(x)=-x^2+4\sqrt{x}$$ at $x=4$ (A) $8-$ (B) $-10$ (C) $-9$ (D) $-5$ (E) $-7$
  11. Y

    MHB Tangent line of a Circle

    Dear all, Attached is a picture of a circle. The lower tangent line is y=0.5x. The center of the circle is M(4,7) while the point A is (3,6). I found the equation of the circle, it is: $(x-4)^{2}+(y-7)^{2}=20$ and I wish to find the dotted tangent line. I know that it is parallel to the...
  12. G

    MHB Tangent line to circle making 30º with a second circle

    Hi all, this is my first thread! I am having problems trying to find the way of drawing a line which is tangent to a circle and intersects another circle making a 30º intersection. Let´s say I have circle A with coordinates 479183.87, 4365099.87 (x1,y1) and a radius of 27780m. I have a second...
  13. karush

    MHB 205.2.6.1-2 another implicit problem with tangent line.

    Find the slope of the curve at the given point} $2y^8 + 7x^5 = 3y +6x \quad (1,1)$ Separate the variables $2y^8-3y=-7x^5+6x$ d/dx $16y^7y'-3y'=-35x^4+6$ isolate y' $\displaystyle y'=\frac{-35x^4+6}{16y^7-3}$ plug in (1,1)...
  14. C

    MHB Calculus III: Proving Tangent Line = 1 for Every Point on Curve

    Hi, I'm stuck on a homework problem in my Calculus III class. I solved 3a really easily, but 3b is giving me a lot of trouble. I know that to find the tangent line, I first have to find the slope, which is represented by the vector: <3cos^2(t)(-sin(t)), 3sin^2(t)(cos(t))>. I know the formula...
  15. opus

    Find eqn of Tangent Line to graph- Implicit Differentiation

    Homework Statement Find the equation of the tangent line to the graph of the given equation at the indicated point. ##xy^2+sin(πy)-2x^2=10## at point ##(2,-3)## Homework EquationsThe Attempt at a Solution Please see attached image so you can see my thought process. I think it would make more...
  16. opus

    Find the slope of the tangent line to the function's inverse

    Homework Statement For ##y=f(x)##, find the slope of the tangent line to its inverse function ##f^{-1}## at the indicated point P. ##f(x) = -x^3-x+2## , ##P(-8,2)## Homework Equations The Inverse Function Theorem: ##(f^{-1})'(x) = \frac{1}{f'(f^{-1}(x))}## The Attempt at a Solution So...
  17. N

    Find the center of a circle given a tangent line & point

    Homework Statement "Find the center and radius of the circle that passes through A(1,1) and is tangent to the line y=2x-3 at the point B(3,3). (Picture of the graph: https://imgur.com/a/0wAnqcU) Homework Equations Here's a link: https://imgur.com/a/y71Z9GY The Attempt at a Solution Soo, I've...
  18. isukatphysics69

    Find the equation of a tangent line at each given point

    Homework Statement in title Homework Equations x=2cotθ y=2sin2θ dy/dx = 2sin3θcosθ y-y1=m(x-x1) point = (-2/√3,(3/2) The Attempt at a Solution Have been stuck for hours I solved for the dy/dx above, now I need to figure out how to get rid of the θ to get my equation in terms of x so I was...
  19. A

    B Ellipse tangent line using projections

    Hi :) The question is in dutch so i'l translate it. on an ellipse E with vertex P and P' on the major axis and vertex Q and Q' on the minor axis. chose R(x1,y1), the projection of R on the major axis is R' and on the minor axis is R''. Define the perpendicular projection of the intrersection...
  20. Velo

    MHB Equation of the Tangent Line? (Derivatives)

    So, I can't wrap around my head of why the Equation of the Tangent Line is: y = f(a) + f'(a)(x - a) I get it that it's the equation of a line, and so it should be something like y = mx + b. I also understand why f(a) = b (since it's a point in that line) and why f'(a) = m (since it's the slope)...
  21. H

    MHB Implicit Differentiation to find equation of a tangent line

    I need urgent help. I have this question: Use implicit differentiation to find an equation of the tangent line to the curve at the given point. \begin{equation} {x}^{2/3}+{y}^{2/3}=4 \\ \left(-3\sqrt{3}, 1\right)\end{equation} (astroid) x^{\frac{2}{3}}+y^{\frac{2}{3}}=4 My answer is...
  22. H

    Find the equation of a tangent line to y = x^2?

    Homework Statement the line goes through (0, 3/2) and is orthogonal to a tangent line to the part of parabola y = x^2, x > 0 Homework EquationsThe Attempt at a Solution I have problems regarding finding the equation of tangent line to the part of parabola because the question not specifically...
  23. H

    Find the equation of a cubic graph

    Homework Statement ##x^3 - 4x^2 + ax + b## tangent to x-axis at x = 3 Homework EquationsThe Attempt at a Solution if the graph tangent at x = 3, means at x =3, y = 0 my questions is, is at x = 3 the graph's gradient (slope) = 0 ? if yes why? if yes then means dy/dx = 0 ##3x^2 - 8x + a = 0##...
  24. Schaus

    Find the equation of the tangent line of the curve

    Homework Statement Find the equation of the tangent line to the curve ##\ xy^2 + \frac 2 y = 4## at the point (2,1). Answer says ##\ y-1 = -\frac 1 2(x-2)## And with implicit differentiation I should have gotten ##\frac {dy} {dx}= -\frac {y^2} {2xy-\frac {2} {y^2}}## Homework Equations ##\...
  25. M

    Verify Unit length to y-axis from Tractrix Curve

    Homework Statement The problem is described in the picture I've attached. It is problem number 6. Homework Equations Tangent line of a curve Length of a curve The Attempt at a Solution I don't know why I'm so confused on what seems like it should be a relatively straightforward problem, but I...
  26. D

    Finding equation of tangent line

    Homework Statement The following point (x0,y0), is on the curve sqrtx +sqrty = 1Show that line equation of the tangent line in the point. (x0,y0) Is x/sqrtx0 + y/sqrty0 = 1 I've found the slope which is -sqrty/sqrtx. So slope of the point is -sqrty0/sqrtx0 Homework EquationsThe...
  27. T

    MHB Finding Slope of Tangent Line: How-To Guide

    My professor shown us in class how to find the slope but I still don't understand
  28. C

    Slope of tangent line to curves cut from surface

    Homework Statement find the slope of tangent line to curves cut from surface z = (3x^2) +(4y^2) - 6 by planes thru the point (1,1,1) and parallel to xz planes and yz planes ... Homework EquationsThe Attempt at a Solution slope of tnagent that parallel to xz planes is dz/dy , while the slope of...
  29. J

    MHB Slope of Tangent line to Polar curve

    I am trying to find the slope of the tangent line of this polar equation: r = 4 + sin theta, (4,0) I put the equation into wolfram alpha and it gives me a 3D plot. If someone could help me find the slope of the tangent line, I would really appreciate it. Thank you.
  30. S

    Tangent line and normal on a parabola

    Homework Statement If the normal at P(ap^2 ,2ap) to the parabola y^2 = 4ax meets the curve again at Q(aq^2, 2aq), show that p^2 +pq+2=0 Homework Equations Point-slope form The Attempt at a Solution I tried putting y=2aq and x=aq^2 but I can seem to simplify the whole thing other than...
  31. Eclair_de_XII

    Finding the equation of a tangent line in polar coordinates?

    Homework Statement "Slopes of tangent lines Find the slope of the line tangent to the following polar curves at the given points. At the points where the curve intersects the origin (when this occurs), find the equation of the tangent line in polar coordinates." ##7.##...
  32. N

    Parametric Equations of Tangent Line

    Homework Statement z = 2x^2 + 5y^2 +2 C is cut by the plane x = 2 Find parametric eqns of the line tangent to C @ P(2, 1, 15) Homework Equations z = 5y^2 + 10 dz/dx = 10y dz/dx (1) = 10 The Attempt at a Solution z = 10y + 15 y = t + 1 if the slope is 10/1 then delta z = 10 and delta y = 1...
  33. A

    Equation of the tangent line in the direction of a vector

    I am having issues figuring out how to do the "in the direction of the vector" part of my problem I have found the equation of the tangent line but i do not know how to the the next part. My question asks: Find the equation of the tangent line to the surface defined by the function f(x,y) =...
  34. R

    MHB Finding all points where tangent line is perpendicular

    Hello! I've encountered a problem of find all points (x,y) on $f(x)=\frac{x-\sqrt{\pi}}{x+1}$ where there are tangent lines perpendicular to $y=-(1+\sqrt{\pi}x+7\pi e^{e^{{\pi}^{110}}})$ So I first found derivative and ended up with $f'(x)=\frac{1(x+1)-(x-\sqrt{\pi})(1)}{x^2+2x+1}$ and then...
  35. J

    Calc 1 practice exam question -- finding the y intercept of the tangent line

    I was given the equation of a polynomial told to find the derivative. easy enough. Then asked to give the equation of the tangent line which I've only learned how to get in the form of the question: "find the equation of the tangent line at x=" They gave me the equation of a line parallel to...
  36. J

    Tangent Line on Projectile

    My professor did this question in class and I am a little confused. I wrote it down in my notes but I kind of don't understand it. The question is: Find theta 1/4 of the way through the flight of a projectile in time He does not give us numbers. Everything has to be solved algebraically. My...
  37. karush

    MHB Implicit, tangent line at pt

    Find the equation of the line tangent to $$\sin\left({xy}\right)=y$$ At point $$\left(\frac{\pi}{2 },1\right)$$ Answer $y=1$ I didn't know how to deal with xy. No example given
  38. TyroneTheDino

    Finding the distance from origin to a tangent line

    Homework Statement Find the distance between the origin and the line tangent to ##x^\frac{2}{3}+y^{\frac{2}{3}}=a^{\frac{2}{3}}## at the point P(x,y) Homework Equations [/B] Distance= ##\frac{\left |a_{0}+b_{0}+c \right |}{\sqrt{a^{2}+b^{2}}}## The Attempt at a Solution To begin I find...
  39. TyroneTheDino

    Curvature and Tangent Line Distance Relationship

    Homework Statement Let T be the tangent line at the point P(x,y) to the graph of the curve ##x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}, a>0##. Show that the radius of curvature at P is three times the distance from the origin to the tangent line T.Homework Equations R=1/K ##R=\frac{\left...
  40. 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...
  41. 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...
  42. B

    Definition of 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...
  43. 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...
  44. 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.
  45. 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...
  46. E

    Function + tangent line = 0

    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...
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