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. N

    B Why does the Limit Process give an Exact Slope?

    I am taking a summer calculus class now. For years I've been stuck on the question of why the limit process gives us an exact slope of the tangent line instead of just a very close approximation. I don't need to know the reason for this class I'm taking- we are basically just learning rules of...
  2. chwala

    Find the equation of the tangent plane and normal to a surface

    In my line i have, ##\dfrac{∂r}{du} = \vec{i} +\dfrac{1}{2}u \vec{k} = \vec{i} +1.5 \vec{k}## ##\dfrac{∂r}{dv} = \vec{j} -\dfrac{1}{2}v \vec{k} = \vec{j} -0.5\vec{k}## The normal to plane is given by, ##\dfrac{∂r}{du}× \dfrac{∂r}{dv} = -\dfrac{3}{2} \vec{ i} + \dfrac{1}{2}\vec{j}+\vec{k}##...
  3. A

    Find Values of Osculating Circle Tangent to a Parabola

    I'm at a loss as to how they got to certain steps in the solutions manual. Here's how far I got with this: Since the circle is tangent to y = x^2 + 1, the slope at (1, 2) is going to be 2, as is the slope of the 2nd derivative of the circle, so then... The derivative of the circle would be...
  4. A

    Quadratics Having a Common Tangent

    I do know that since they have a common tangent line, that means: 2x+a = c-2x Since they both have the point (1,0), then since both equations should equal 0 when x = 1: c(1)-(1)^2 = 0 --> c = 1 So now, I replace c with 1 to solve for a in the two derivatives that are equal (common...
  5. chwala

    At what ##x## value is the tangent equally inclined to the given curve?

    I had to look this up; will need to read on it. from my research, https://byjus.com/question-answer/the-equation-of-straight-line-equally-inclined-to-the-axes-and-equidistant-from-the-points-1-2-and-3-4-is-ax-by-c-0-where/ ... I have noted that at equally inclined; the slope value is ##1##...
  6. P

    Separable first order ODE involving tangent

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  7. A

    A About computing the tangent space at 1 of certain lie groups

    Hello :), I am wondering of the right and direct method to calculate the following tangent spaces at ##1##: ##T_ISL_n(R)##, ##T_IU(n)## and ##T_ISU(n)##. Definitions I know: Given a smooth curve ##γ : (− ,) → R^n## with ##γ(0) = x##, a tangent vector ##˙γ(0)## is a vector with components...
  8. chwala

    Find the equation of the tangent in the given problem

    The textbook solution is indicated below; My approach on this question is as follows; ##\dfrac{dy}{dx} = -\dfrac {8}{x^3}## ##\dfrac{dy}{dx} (x=a) = -\dfrac {8}{a^3}## The tangent equation is given by; ##y= -\dfrac {8}{a^3}x+\dfrac{12+a^2}{a^2}## when ##x=0##, ##y=\dfrac{12+a^2}{a^2}##...
  9. 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...
  10. 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...
  11. C

    I Carroll GR: Tangent Space & Partial Derivatives

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  12. 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.
  13. karush

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

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  14. chwala

    Solve the attached problem that involves circle and tangent

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  15. chwala

    Find the radius of the smaller circle in the tangent problem

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  16. K

    I What is the construction of charts for the tangent bundle on the unit circle?

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  17. chwala

    Determine the unit tangent vector

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  18. A

    I Tangent Bundle of Product is diffeomorphic to Product of Tangent Bundles

    My apologies if this question is trivial. I have searched the forum and haven't found an existing answer to this question. I've been working through differential geometry problem sets I found online (associated with MATH 481 at UIUC) and am struggling to show that T(MxN) is diffeomorphic to TM...
  19. brotherbobby

    Multiple angles : Reducing the sum

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  20. brochesspro

    Solving for Tangent Lines: Analytical and Graphical Approaches

    I did it graphically by using GeoGebra. My question is that what can I do to solve it analytically/algebraically. I used the point-slope formula and obtained $$\frac {y - (a^2-4)} {x - a} = 2a$$, which implies that ##y = (2a)x + (-a^2-4)##. I am not sure how to proceed from here onwards...
  21. graphking

    What is a tangent? (multiple meanings)

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  22. S

    Finding value of k so two curves are tangent

    I tried to equate the derivative of the two equations: $$\cos x=-ke^{-k}$$ Then how to continue? Is this question can be solved? Thanks
  23. J

    MHB Equation for tangent of the curve

    Can anyone help me to find the equation of the tangent to the curve x = 2 cos t, y= 2 sin t where t= pi/3??
  24. Greg Bernhardt

    I What is the true meaning of a tangent in mathematics?

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  25. N

    What is the angle needed to solve this right triangle?

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  26. Strand9202

    Concavity and Tangent Functions

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  27. anemone

    MHB Trigonometric of tangent and sine functions

    Simplify $\left(\tan \dfrac{2\pi}{7}-4\sin \dfrac{\pi}{7}\right)\left(\tan \dfrac{3\pi}{7}-4\sin \dfrac{2\pi}{7}\right)\left(\tan \dfrac{6\pi}{7}-4\sin \dfrac{3\pi}{7}\right)$.
  28. brotherbobby

    To prove a trigonometric identity with tan() and cot()

    Attempt : I could not progress far, but the following is what I could do. $$\begin{align*} \mathbf{\text{LHS}} & = (\tan A+\tan B+\tan C)(\cot A+\cot B+\cot C) \\ & = 3+\tan A \cot B+\tan B \cot A+\tan A \cot C+\tan C \cot A+\tan B \cot C+\tan C \cot B\\ & = 3+\frac{\tan^2A+\tan^2B}{\tan A \tan...
  29. Y

    A geometry question in Comsol (draw a semi-circle tangent to a line)

    I am new to Comsol. I want to draw my model which is a Tesla valve. The geometry is little complicated and I don't know how to draw a semi-circle tangent to a line. Is it possible? I draw it in Solidworks and imported it into Comsol but it gives error and I think it is better to draw inside...
  30. D

    Tangent vector fields and covariant derivatives of the 3-sphere

    This week, I've been assigned a problem about a 3-sphere. I am confused how to approach this problem and any comments would be greatly appreciated. (a) - would I be correct to assume the metric G is simply the dot product of two vector fields with dx^2 dy^2 du^2 and dv^2 next to their...
  31. D

    A multivariate function of Toruses - tangent vectors

    Thank you to all those who helped me solve my last question. This week, I've been assigned an interesting problem about toruses. I think I've solved most of this problem on my own, but I'd like to hear a few suggestions for part c. I think this map multiplies tangent vectors by a factor of...
  32. E

    B Just a question about the tangent basis

    I was reading these notes and then on page 23 I saw something a bit weird. Back in this thread I learned that ##\{ \partial_i \}## form a basis of ##T_p M##, and that a tangent vector can be written ##X = X^i \partial_i##, and it's not too difficult to show that components transform like...
  33. K

    Calculate the dual basis and tangent basis vectors

    a) Since ##tan(x/x_0)## is not defined for ##x=\pm\pi/2\cdot x_0## I assume x must be in between those values therefore ##-\pi/2\cdot x_0 < x < \pi/2\cdot x_0## and y can be any real number. Is this the correct answer on a)? b) I can solve x and y for s and t which gives me ##y=y_0\cdot s## and...
  34. U

    I Fixing orientation by fixing a frame in a tangent space

    I would like to show that fixing the orientation of k-manifold smooth connected ##S## in ##\mathbb {R} ^ n ## is equivalent to fixing a frame for one of its tangent spaces. What I know is that different orientations correspond to orienting atlases containing maps that cannot be consistent with...
  35. xyz_1965

    MHB Why is the Law of Tangent rarely used or taught in typical trigonometry courses?

    The next chapter in my personal studies is Law of Sines and Law of Cosines. I am not there yet. I know there is a Law of Tangent but it is rarely used and never taught in a typical trig course anywhere. Why is the Law of Tangent ignored?
  36. qbar

    A How Can I Differentiate Curves Where the Real Part of \( Y(t) \) Vanishes?

    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...
  37. ElectronicTeaCup

    Intersection of a tangent of a hyperbola with asymptotes

    Summary:: Question: Show that the segment of a tangent to a hyperbola which lies between the asymptotes is bisected at the point of tangency. From what I understand of the solution, I should be getting two values of x for the intersection that should be equivalent but with different signs...
  38. S

    Solve this differential equation for the curve & tangent diagram

    Here is my attempt at a solution: y = f(x) yp - ym = dy/dx(xp-xm) ym = 0 yp = dy/dx(xp-xm) xm=ypdy/dx + xm xm is midpoint of OT xm = (ypdy/dx + xm) /2 Not sure where to go from there because the solution from the link uses with the midpoint of the points A and B intersecting the x-axis...
  39. ElectronicTeaCup

    I Product of distances from foci to any tangent of an ellipse

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  40. S

    B Components of Tangent Space Vector on Parametrized Curve

    I'm studying 'A Most Incomprehensible Thing - Notes towards a very gentle introduction to the mathematics of relativity' by Collier, specifically the section 'More detail - contravariant vectors'. To give some background, I'm aware that basis vectors in tangent space are given by...
  41. S

    I Tangent space basis vectors under a coordinate change

    I'm studying 'Core Principles of Special and General Relativity' by Luscombe - the chapter on tensors. Quoting: The book goes on to talk about a switch to the spherical coordinate system, in which ##\mathbf{r}## is specified as: $$\mathbf{r}=r\sin\theta\cos\phi\ \mathbf{\hat...
  42. ttpp1124

    Parallel Tangent Line on y=2-e^x+4x and 2x+y=5?

  43. ttpp1124

    At what point(s) on the given curve is the tangent line horizontal?

  44. ttpp1124

    Determine the equation of the tangent line to the function given

  45. ttpp1124

    Determine the exact value of the slope of the tangent line

  46. ttpp1124

    Finding tangent lines for 𝑓(𝑥) = 𝑥^3 − 𝑥 + 6

    not quite sure if this is right.. can someone confirm?
  47. archaic

    Tangent to a parametrized curve

    ##x(3)=9-6=3##, ##y(3)=27+9=36##. ##\frac{y'(3)}{x'(3)}=\frac{3\times9+3}{2\times3}=\frac{30}{6}=5##. ##y=5(x-3)+36=5x+21##.
  48. F

    Deriving the Adjoint / Tangent Linear Model for Nonlinear PDE

    I am trying to derive the adjoint / tangent linear model matrix for this partial differential equation, but cannot follow the book's steps as I do not know the math. This technique will be used to solve another homework question. Rather than posting the homework question, I would like to...
  49. G

    I Visualizing the cotangent space to a sphere

    Is it correct to say that: the cotangent is given by the gradients (*) to all the curves passing through a point and it actually spans the same tangent space to a point of a sphere? If you visualize them as geometric planes (**), the cotangent and the tangent spaces are more than isomorphic...
  50. karush

    MHB 3.2.15 mvt - Mean value theorem: graphing the secant and tangent lines

    $\tiny{3.2.15}$ Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function the secant line through the endpoints, and the tangent line at $(c,f(c))$. $f(x)=\sqrt{x} \quad [0,4]$ Are the secant line and the tangent line parallel...
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