What is Tangent: Definition and 1000 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. F

    Find an equation to the tangent line to a curve and parrallel

    Homework Statement Find an equation of the tangent line to the curve y = x√x that is parallel to the line y = 1+3x. Homework Equations m = 3 The Attempt at a Solution Here is my attempt: dy/dx(x) * dy/dx(x^(1/2)) = (1) * (1/2x^(-1/2)) = (1/2x^(-1/2)) (1/2x^(-1/2)) = 3 →...
  2. D

    Equation of tangent - Implicit or Partial DifferentiatioN?

    Homework Statement Need to find the tangent to the curve at: e^(xy) + x^2*y - (y-x)^2 + 3 I just implicitly differentiate the expression to find the gradient and then use the points given to find the equation, right? Or does this involve partial differentiation? Homework Equations...
  3. S

    Variant of inverse tangent derivative

    Homework Statement 1/(u^2+4) Homework Equations The Attempt at a Solution I know that 1/(x^2+1) is the derivative of the inverse tangent function, and that is proved by using tany = x, derivative of both sides with secx=(1+tan^2x) and tan^2x = x^2. I don't know how to use the...
  4. 6

    Simple vector function problem, find slope of tangent?

    Homework Statement Show that the curve r = (t2,t3-t) Intersects itself at (1,0), and find the slopes of the tangents at this point. Homework Equations The Attempt at a Solution Okay I can show it intersects itself there, but what I am having trouble with is when they say slopes...
  5. B

    Finding Tangent Lines to f(x)=x/(x-1) Passing through (-1,5)

    Find equations of the tangent lines to the graph of f(x)=\frac{x}{x-1} that pass through the point (-1, 5). Well, first I took the derivative, and afterwards, I made the connection that the derivative was a slope at any instant on the graph. By this, I inferred that f'(x) = m. I knew that the...
  6. B

    Need help with solving the slope of a tangent line

    1. The question is What is the slope of the line tangent to f(x)=2x^2-x-7 at x=-1 2. Must be solved using derivatives 3. So basically i know the equation (f(x+deltax)-f(x)/deltax) So i plug in the problem and i get (2(x+deltax)^2-(x+deltax)-7)-(2x^2-x-7) Now i know what the...
  7. I

    Is the School Definition of a Tangent Line Always Accurate in Calculus?

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  8. J

    Using integration to find the total change in angle between two tangent lines

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  9. K

    Calculating the length of a tangent curve

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  10. N

    Integrals involving Secant & Tangent Derivation

    Homework Statement If the power of the secand is even and positive.. \int sec^{2k} x tan^{n} x dx = \int (sec^2 x)^{k-1} tan ^n x sec^2 x dx The Attempt at a Solution The way I see it, sec^{2k} x = sec^2 x dx * sec^k x dx the next step seems to be to break down sec^k, but on closer...
  11. M

    How is the tangent and area inverse?

    I've never been able to visualize how the tangent to a curve and the area under a curve are inverses of each other, can anyone give some intuitiveness to this?
  12. B

    Inverse tangent function in real and complex domain

    Homework Statement See attached file. Homework Equations The Attempt at a Solution I've only been able to do part (a) of this question. I ended up with: tanz= i ({\frac{1-e^{(2iz)}}{1+e^{(2iz)}}}) I'm not sure how to approach the next two parts. If anyone could give me any...
  13. S

    Please help determine the tangent of the parametric equation.

    Homework Statement A curve C is defined by the parametric equation x=t^2, y=t^3-3t. a) show that C has two tangents at the point (3,0) and find their equations. b) find the points on C where the tangent is horizontal Homework Equations y-y1=m(x1-x), (dy/dt)/(dx/dt)=m, when dy/dt=0...
  14. M

    Determining Tangent Slope w/Point Not On Curve

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  15. Seydlitz

    Finding Tangent and Intersection for Parametric Curve

    Homework Statement A curve is defined by the parametric equations: x = 2t^3 y = 2t^2 t =/ 0 1)Prove that the equation of the tangent at the point with parameter t is 2x - 3ty + 2t^3 = 0. Proven, and I've no problem with this part. 2.)The tangent at the point t = 2 meets the curve...
  16. H

    Find the equation of the tangent - Please help trying for 2 hours now.

    Homework Statement Ok so this is a question from last years past paper of my course: X= 1/2 intersects the circle that is centered at origin at two points, one of which is in the lower half plane y<0; what is the equation of the tangent tot the same circle at this point? Homework Equations...
  17. C

    Unit tangent vectors, Normal vectors, and Gradients

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  18. H

    How is T the tangent to a vector

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  19. J

    Polar Tangent Lines: Finding Slopes at the Pole

    Homework Statement r=2-3cosθ Find the tangent line at any point, and at the point (2,∏) Find the tangent line(s) at the pole Homework Equations Do I have to use x=rcosθ and y=rsinθ to convert it to rectangular to find slopes? The Attempt at a Solution Is the point 2∏ even a...
  20. K

    Finding the tangent plane and normal line

    D is a set of all points (x,y) in R2 distinct from (0,0). I have the funtion f: D --> R which is defined by: f(x,y) = (2xy)/(x2+y2 Im asked to find the equations (in plural) of the tangelnt plane and the normal line to the graph of the function at (1,2) My attempt: I use the tangent plane...
  21. S

    Solving Intersection Curve at (1,1,1): Derivatives & Tangent Line

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  22. P

    Show line through point that is tangent to f(x)does not exist

    1. show that there is no line through the point (2,7) that is tangent to the parabola y =x^2 +x 2. y-y1=m(x-x1) 3. y'=f'(x)=2x+1 m1=2x +1 m1=2(2) +1 =5 m2=((x^2 +x)-7)/(x-2) m1=m2? ((x^2 +x)-7)/(x-2)=5 x^2-4x+3=0 2^2-4(2)+3=/=0 -1=0 I'm thinking that i would...
  23. T

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  24. B

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    Homework Statement Find derivative of tan^{-1}(\frac{3sinx}{4+5cosx}) Homework Equations deriviative of tan^{-1}=\frac{U'}{1+U^{2}} The Attempt at a Solution I found U'= \frac{12cosx+15}{(4+5cosx)^{2}} 1+U^{2}=1+\frac{9sin^{2}x}{(4+5cosx)^{2}} I think my components are correct but my...
  25. L

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    Homework Statement Find the equations of the lines that pass through (0,0) and are tangent to x^2 - 4x + y^2 + 1 = 0 My confusion I've been given a problem of this sort recently, except now it involves implicit differentiation. I know "how" to get to the correct answer. I just...
  26. L

    Find f`(x) for the tangent line of the graph

    Homework Statement Suppose line tangent to graph of y=f(x) at x =3 passes through (-3, 7) & (2,-1). Find f'(3), what is the equation of the tangent line to f at 3? Homework Equations I found the slope of which equals -8/5 Im not sure how to find the equation... do I do...
  27. M

    Given the plane curve, find tangent vector

    Homework Statement Consider the plane curve \overrightarrow{r(t)}=e^tcost(t)\hat{i}+e^tsin(t) \hat{j} Find the following when t= ∏/2 Part A: \hat{T}(t) Part B: \hat{B}(t) Part C: \hat{N}(t) Homework Equations \hat{N}(t)=\frac{\hat{T}(t)}{||\hat{T}(t)||}...
  28. Z

    Find two points on an ellpise that have horizontal tangent

    Homework Statement The equation 5x^2 - 6xy + 5y^2 = 16 represents an ellipse. Determine two points on the ellipse at which the tangent is horizontal. Homework Equations The Attempt at a Solution I find the derivative of the equation: (-10x + 6y) / (-6x + 10y) = 0 iff...
  29. S

    Unit tangent, unit normal, unit binormal, curvature

    Homework Statement Question: "Find the unit tangent, normal and binormal vectors T, N, B, and the curvature of the curve x = 4t, y = -3t^2, z = -4t^3 at t = 1." Answer: T = 0.285714285714286 i - 0.428571428571429 j - 0.857142857142857 k N = -0.75644794981871 i + 0.448265451744421 -...
  30. Y

    Vector Calculus - Equations for planes tangent to given equation

    Homework Statement My problem is one pertaining to my Vector Calculus course. The assignment is asking us to "Find equations for the planes tangent to z = x2 + 6x + y3 that are parallel to the plane 4x − 12y + z = 7." The problem I'm having with the problem is the plural aspect. It states...
  31. G

    Find the equation of the line with slope -1 that is tangent to the curve y=1/x-1

    Homework Statement Find the equation of the line with slope -1 that is tangent to the curve y=1/x-1 Homework Equations y=1/x-1 The Attempt at a Solution Slope of -1 means y=-1x+k So... -1x+k = 1/x-1 I don't know how to rearrange this into a quadratic equation so that I...
  32. ArcanaNoir

    Integral of unit tangent vector equals arc length?

    Homework Statement Let c(t) be a path and T the unit tangent vector. What is \int_c \mathbf{T} \cdot d\mathbf{s} Homework Equations The unit tangent vector of c(t) is c'(t) over the magnitude of c'(t) : \mathbf{T} = \frac{c'(t)}{||c'(t)||} The length of c(t) can be represented by ...
  33. 1

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  34. T

    Calculating a limit with tangent as denominator

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  35. L

    Equations of lines through a point and tangent to a function?

    Homework Statement Find the equations of all the lines through the origin that are tangent to the curve y = (some complicated cubic function) I looked up the question in google and found a much simpler example, y = x^2 passing through (1,-1). However, I don't even get what's going on...
  36. B

    Center of circle from two points and a tangent angle

    So my problem is this: I need to figure out the center of a circle given two points. At one of the points, I know the tangent angle. So I know (x1, y1, θ1) and (x2, y2) and need to find (xc, yc). I also need to do this on a computer so I need some sort of closed-form solution. The way I...
  37. H

    Tangent Lines to an Ellipse Passing Through an Outside Point

    Homework Statement Find equations of both the tangent lines to the ellipse x2 + 4y2 = 36 that pass through the point (12, 3). Homework Equations The equation of an ellipse is x2/a2 + y2/b2 = 1. I converted the given equation to x2/36 + y2/9 = 1 by dividing each value by 36. The...
  38. O

    Find unit tangent vector at indicated point

    Homework Statement Find the unit tangent vector at the indicated point of the vector function r(t) = e(19t)costi + e(19t)sintj + e(19t) kT(pi/2) = <___i+___j+___k>Homework Equations r'(t) / |r'(t)| The Attempt at a SolutionAnswers: 19e(19*∏/2)(cos(∏/2)-sin(∏/2)) /...
  39. C

    Help with a limit equation involving tangent

    Homework Statement How do I solve this equation? lim x>0 (x^3 - 2x^2 + x)/tanx I don't know what to do here, please help. Thank you
  40. 9

    Find equation of tangent line given only x. Please help

    Homework Statement Find equation of tangent line, given x = -1. Not given y. I am used to having this when I am given both y and x. Homework Equations (x^3 - 4x + 2)(x^4 + 3x - 5)The Attempt at a Solution Differentiate (3x^2 - 4)(4x^3+3) Multiply 12x^5 - 9x^2 - 8x^3 - 12 Plug in -1, find...
  41. S

    Find a vector tangent to the curve of intersection of two cylinders

    I have attached both the question and the solution. I just have questions as to why the solution is the way it is (sorry if they seem stupid but, while I get how to do it mechanically, I don't understand the fundamental reasoning as to why anything is being done): 1) Why are we taking the...
  42. M

    Finding tangent lines that pass through given points

    Homework Statement Find the co-ordinates of all points on the curve f(x)= x3 whose tangent lines pass through the point (a,0) Homework Equations f '(x) = nxn-1 The Attempt at a Solution I am really not sure how to attack this question. My initial thoughts are to find f '(x) then...
  43. H

    How to find the point a tangent line hits when given a point off of the graph.

    Homework Statement (a) Draw a diagram to show that there are two tangent lines to the parabola y = x^2 that pass through the point (0, -4). (Do this on paper. Your teacher may ask you to turn in this work.) (b) Find the coordinates of the points where these tangent lines intersect the...
  44. K

    Tangent space and tangent plane

    Homework Statement So I'm a little confused about what a tangent space is. Is it the same as the equation of the tangent plane in lower dimensions? My notes define the tangent space as follows. Let M be a hypersurface of Rd. Let x(s) be a differentiable curve in M such that x(0)=x0 is in...
  45. grgrsanjay

    MHB Solve Vertical Tangent: y-e^(xy) + x = 0

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  46. ArcanaNoir

    Find direction of tangent line to equation

    Homework Statement Find the direction of the line tangent to the curve x^4+y^4=32 at the point (2, -2) Homework Equations Anything goes, we're in vector calculus now. The Attempt at a Solution So, to find the tangent line, I was thinking of taking the gradient, but I'm not...
  47. Z

    Finding Tangent and Perpendicular Vectors on a 2D Graph

    Homework Statement f (x) = e^(3x) + sin(2x) + 3x +1 (a) Find a vector V that is tangent to the graph of y = f(x) at the point ( 0, 2). (b) Find a vector N that is perpendicular to the graph of y = f(x) at the point ( 0, 2). 2. The attempt at a solution The first step I took is to...
  48. B

    Find an equation of the tangent line to the curve at the given point.

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  49. B

    Why don't the secants cancel in this equation?

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  50. B

    Question about directional derivatives and tangent planes ?

    Is it possible to find a directional derivative for a point on z = f(x,y) at a point (x,y) in a direction (u1,u2) using the plane tangent to z at (x,y)? If so, how? Thanks!
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