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. K

    MHB Find Tangent Vector & Vector Equation for Curve r(t)

    For the curve defined by r(t) = 3*t*i + 2*t^2*j − 3*t^4*k Find the tangent vector r′(t0) at the point P(4,8,−16), given that the position vector of P is r(t0). and Find the vector equation of the tangent line to the trajectory through P. Im unsure as to how to go about solving this. I've...
  2. MarkFL

    MHB Find Where Two Tangent Lines Intersect on a Circle | Math Help

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  3. J

    Finding the Slope of Tangent Line for Polar Curves

    Homework Statement Calculate a function of theta that gives the slope of the tangent line (dy/dx) to the polar curve r at some angle theta. Homework Equations dy/dx= f'(theta)sin(theta) + f(theta)cos(theta)/[f'(theta)cos(theta) - f(theta) sin(theta)] The Attempt at a Solution I am...
  4. MarkFL

    MHB Tangent Line & Log Diff EQs: Jawairia's Qs at Yahoo Answers

    Here are the questions: I have posted a link there to this topic so the OP can see my work.
  5. B

    Trigonometric equation with tangent

    Homework Statement Solve: 2 tan x (tan x - 1) = 3. Homework Equations Pythagorean identities? The Attempt at a Solution I tried the following: 2 tan^2 x - 2 tan x = 3 2 (sec^2 x - 1) - 2 tan x = 3 2 (1 - cos^2 x) - 2 sin x cos x = 3 cos^2 x (multiplying through by cos^2 x)...
  6. G

    MHB How to Calculate the Radius of a Circle Tangent to Two Lines?

    I'm having some difficulty with this problem and any help would be appreciated. What is the radius of a circle tangent to the lines y = 3x + 7 and y = .5x - 3 and containing the point (8,1)? I've determined that the given point (8,1) is the point of tangency of the line y = .5x - 3 and the...
  7. L

    First Fundamental Form from tangent vectors

    Hi Appologies for formatting issues this is the first time I have submitted something to the forum. I have a pretty simple problem, I am just going through the derivation of the First Fundamental Form and I think I am missing something in the derivation. If we have a point x = (x1,x2)...
  8. alyafey22

    MHB Do we have identities for the inverse tangent function for complex numbers?

    Do we have identities for the following \arctan(x+y) = \arctan(x)-\arctan(y) =
  9. MarkFL

    MHB Ann's questions at Yahoo Answers regarding finding tangent lines

    Here are the questions: I have posted a link there to this topic so the OP can see my work.
  10. V

    What is the Tangent Weight of a Spline?

    What is meant by the tangent weight of a spline?
  11. A

    Why Does Tangent Go To Infinity?

    Why does tangent go to infinity when it increases 0 to 90 degrees?
  12. J

    Finding the Tangent Line at y = x * e^(2x) | (2, 2*e^(4)) - Homework Solution

    Homework Statement Find the equation of the tangent line at the curve y= x * e^(2x) at the point (2, 2*e^(4)) Homework Equations The Attempt at a Solution f'(x)= e^(2x) * (2x+ 1) (e^4)(5) = 5e^4 y- 2*e^4 = 5*e^4 (x-2) y= 5(e^4)*x - 8*(e^4) Is this right? It seems too easy...
  13. C

    Proving formula for approximation of a plane tangent to Z

    I think I've got the basics of forum notation now, thanks to Fredrick from my other thread. Here goes: Show: Z = z_0 + a(x-x_0) + b(y-y_0) where a = f_x = \frac{\partial f}{\partial x} and b = f_y = \frac{\partial f}{\partial y} I'm attempting this using the coordinate method, but how...
  14. D

    Find points where tangent line is horizontal

    Homework Statement Find the points on the graph of x3-y3=3xy-3 where the tangent line is horizontal Homework Equations y = f(x) so implicit differentiation must be used when taking the derivative of y (xy)' = xy' + y The Attempt at a Solution So if the tangent line is...
  15. A

    A tangent line to both functions

    Homework Statement Determine a line that is tangent to both f(x)=x2 and g(x)=x2-2x Homework Equations The Attempt at a Solution f(x)=x2 => f'(x)=2x g(x)=x2-2x => g'(x)=2x-2 f'(a) = f'(b) 2a = 2b-2 I don't know how to continue. Thanks for help.
  16. J

    Equation of Tangent Line for y = arctan(x) at (sqrt3,0)

    Homework Statement Give the equation of the line tangent to the curve at the given point. ytan^-1x = xy at (sqrt3,0) Homework Equations The Attempt at a Solution Would it be right to do an implicit differentiation or to isolate for y here? I isolated for y and got...
  17. D

    Find the equation of tangent line for derivatives of functions

    Homework Statement Find an equation of the tangent line at the point indicated f(x) = 5x2-2x+9 , x = 1 Homework Equations (d/dx) bx = ln(b)bx General Power Rule which states: (d/dx) g(x)n = n(g(x))n-1 * g'(x) The Attempt at a Solution So looking at a previous problem...
  18. Petrus

    MHB Horizontal Asymptote of Inverse Tangent Function

    Hello MHB, I got one question, I am currently working with an old exam and I am suposed to draw it with vertican/horizontal lines (and those that are oblique). f(x)=\frac{x}{2}+\tan^{-1}(\frac{1}{x}) for the horizontel line \lim_{x->\infty^{\pm}}\frac{x}{2}+\tan^{-1}(\frac{x}{2}) Is it enough...
  19. MarkFL

    MHB Paul's questions at Yahoo Answers regarding tangent and normal lines

    Here are the questions: Here is a link to the questions: Equation of tangent and normal? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  20. Petrus

    MHB Tangent equation for two points.

    Hello MHB, I am doing some old exam and got less knowledge for this problem. consider the graph of the function g(x)=\frac{1}{x}, \ x\neq 0 the point \left(3,-1 \right) lies on two tangents to the graph. Decide tangents equation. My progress well I derivate and find the slope g'(x)=\ln(x) so...
  21. MarkFL

    MHB Find Tangent Line w/ Definite Integral - Amy's Question

    Here is the question: Here is a link to the question: Finding tangent line with definite integral? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  22. N

    Curve and tangent line problem, finding the area of enclosed region

    Homework Statement Given that the curve y = x^3 has a tangent line that passes through point (0, 2), find the area of the region enclosed by the curve and the line by the following steps. Homework Equations The Attempt at a Solution Let f(x) = x^3 and let the coordinates of the...
  23. S

    How Is the Value of f(5) Determined Given Parallel Tangent Lines?

    Homework Statement The line x-2y+9=0 is tangent to the graph of y=f(x) at (3,6) and is also parallel to the line through (1,f(1)) and (5,f(5)). If f is differentiable on the closed interval [1,5] and f(1)=2, find f(5) A) 2 B) 3 C) 4 D) 5 E) None of these The correct answer is (C) 4The Attempt...
  24. E

    Is angular momentum conserved if you move off at a Tangent ?

    Lets imagine a binary system of two astronauts in space connected to one another via light rope. The rope is taut and they're spinning round and round with their axis of rotation being the the axis perpendicular to the their centre of mass. Now, my question is this. Let's say they each let go...
  25. M

    Finding a tangent line to a level curve - Don't understand solution

    Homework Statement If f(x,y) = xy, find the gradient vector \nabla f(3,2) and use it to find the tangent line to the level curve f(x,y) = 6 at the point (3,2) Homework Equations The Attempt at a Solution f(x,y)=xy \Rightarrow\nabla f(x,y)=<y,x>,\nabla f(3,2)=<2,3> \nabla...
  26. Petrus

    MHB How Do You Find Vertical and Horizontal Tangents in Parametric Equations?

    I am currently working with parametric equation and trying to solve finding points on the curve where the tangent is horizontal or vertical. When I do with trigometry I get problem... And I need help to understand this. I know what vertical and horizontal means. exemple this one i am working...
  27. E

    Searching for a Function with x=Slope of Tangent

    Okay, so I am in grade 12 calc and I was learning about e today, how the slope of the tangent at any point is also the y value at that point. What I was wondering is if there is a function that has the x value equal to the slope at any given point. I think it would look something like a...
  28. L

    Tangent space in manifolds, how do we exactly define?

    If we have a manifold with a chart projected onto ##R^n## cartesian space and define a curve ##f(x^\mu(\lambda))=g(\lambda)## then we can write the identity \frac{dg}{d\lambda} = \frac{dx^\mu}{d\lambda} \frac{\partial f}{\partial x^\mu} in the operator form: \frac{d}{d\lambda} =...
  29. P

    Tangent plane equation question.

    Homework Statement Consider a surface ω with equation: x^2 + y^2 + 4z^2 = 16 Find an equation for the tangent plane to ω at point (a,b,c). Homework Equations Tangent plane, 3 variables: f_{1}(a,b,c)(x-a) + f_{2}(a,b,c)(y-b) + f_{3}(a,b,c)(z-c)= 0 The Attempt at a Solution I get at the...
  30. Petrus

    MHB MHBFind Tangent Equation to Curve: (2sin(2t), 2sin(t)) at (√3,1)

    Hello MHB, Find and an equation of the tangent(s) to the curve at the given point x=2\sin(2t), y=2\sin(t) \left(\sqrt{3},1 \right) first we need to find the slope so we derivate \frac{dy}{dt}=2\cos(t), \frac{dx}{dt}=4\cos(2t) so we got \frac{dy}{dx}= \frac{2\cos(t)}{4cos(2t)} we need to solve...
  31. N

    MHB Find the Tangent lines of the slopes of the three zeroes

    1+50sinx/x^2+3 -5 < x < 5 3 zeroes: 0.02, 3.16, -3.12 Find the derivative and the slopes of the tangent lines. I need help with the last part. I found out the three zeroes by adding and subtracting pi from the equation at top by setting it to zero. Thank you!
  32. A

    MHB Find the Radius of 4th Circle When All are Tangent: Hint d/2

    The centers of three circles are situated on a line. The center of the fourth circle is situated at given distance d from that line. What is the radius of the fourth circle if we know that each circle is tangent to other three. Please give me a hint, if you can. Answer: d/2.
  33. S

    Problem about tangent plane to surface

    Homework Statement find the points on the surface x^2 + y^2 + z^2 = 7 where its tangent plane is parallel to 2x + 4y + 6z = 1 Homework Equations Equation of a tangent plane: fx(x - x0) + fy(y - y0) + fz(z - z0) = 0, where fx means partial derivative of f respect to x n1 X n2 = 0 The Attempt...
  34. C

    Finding the X-coordinate of a normal to the tangent

    Homework Statement Tangent to C at point P(2,7) has an equation of y=3x+1. Point Q also lies on C and is perpendicular to the tangent, show that the X-coordinate is [1/3(2+√6)] Homework Equations curve C has equation y= x3-2x2-x+9 dy/dx = 3x2-4x-1 The Attempt at a Solution gradient of...
  35. Feodalherren

    Points on ellipse where tangent slope=1

    Homework Statement 7 Find the points on the ellipse x^2 + 2y^2 = 1 where the tangent line has slope 1 Homework Equations The Attempt at a Solution I got the correct X and Y values but this gives me four possibilities and the answer key says there are two points. I got x...
  36. S

    Find all points where f(x) has a horiztonal tangent line

    1. Given. F(x) = 4x/(x^2+1) 2. Problem Find all points (x,y) where the function has a horiztonal tangent line 3. Attempt I took the derivative of F(x) and came to (-4x^2+4)/(x^2+1)^2 I set it equal to zero and found an x value of 1. I used that x value and plugged it into...
  37. Vahsek

    Finite sum formula for tangent (trigonometry)

    Hi everyone, I've been looking for the finite sum formulae of trig functions. I've found the easiest ones (sine and cosine). But the one for the tangent seems to be very hard. No mathematical tricks work. Plus I've looked it up on the internet. Nothing. I will greatly appreciate your help...
  38. T

    Finding the Equation of a Tangent at a Given Point on a Cubic Curve

    Homework Statement The equation of the tangent to the curve f(x)=ax^3+bx at the point of contact (-1;3) is y-x-4=0. Calculate the values of a and b Homework Equations y-y1=m(x-x1) The Attempt at a Solution I am totally stuck, here is what I could derive: Equation of tangent...
  39. MarkFL

    MHB Angle Sum/Difference Identities: Billy's Pre-calc Math Problem

    Here is the question: Here is a link to the question: Pre-calc math problem? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  40. P

    At what points on this curve is the tangent line horizontal?

    Homework Statement At what points on the curve y = (x^2)/(2x+5) is the tangent line horizontal? Homework Equations Quotient rule The Attempt at a Solution I figured out the derivative which is 2x(2x+5) - 2x^2 ----------------- (2x+5)^2 I also know that for the equation of...
  41. Petrus

    MHB Tangent lines of trigonometric functions

    Hello, I got problem with A homework "find an equation of the tangent line to curve at the given point. $y=sec(x)$. $(pi/3,2)$ progress: $y'=sec(x)tan(x)$. So basicly that sec(x) don't say me much so i rewrite it as $1/cos(x)$ $y'=1/cos(x)•tan(x)$ now i can put $pi/3$ on the function to...
  42. B

    How is the Chain Rule Applied in Geometric Tangent Vectors?

    So let ℝ^{n}_{a}={(a,v) : a \in ℝ^{n}, v \in ℝ^{n}} so any geometric tangent vector, which is an element of ℝ^{n}_{a} yields a map Dv|af = Dvf(a) = \frac{d}{dt}|_{t=0}f(a+tv) this operation is linear over ℝ and satisfies the product rule Dv|a(fg) = f(a)Dvg + g(a)Dvf if v|a =...
  43. B

    Tangent Bundles, T(MxN) is Diffeomorphic to TM x TN

    Homework Statement If M and N are smooth manifolds, then T(MxN) is diffeomorphic to TM x TN Homework Equations The Attempt at a Solution So I'm here let ((p,q),v) \in T(MxN) then p \in M and q \in N and v \in T(p,q)(MxN). so T(p,q)(MxN) v = \sum_{i=1}^{m+n}...
  44. MarkFL

    MHB Find Equation of Tangent Line at (4π, f(4π)) - Graffer's Question

    Here is the question: Here is a link to the question: An equation of the tangent line to the curve y=f(x)=x(10cos x- 2sinx) at the point (4pi, f(4pi)) is y=? - Yahoo! Answers I have posted a link there to this topic, so the OP can find my response.
  45. GreenGoblin

    MHB Unit tangent vector and equation of tangent line to curve

    "find a unit tangent vector and the equation of the tangent line to the curve r(t) = (t, t^2, cost), t>=0 at the point r(pi/2)." NOW, what I don't get is, how is that a curve? This is not like the example I have studied and I don't really get the question. So I don't know where to start. Once I...
  46. K

    Finding tangent line that passes through a point not on curve

    Homework Statement Find the number of tangent lines to the curve: y=\frac{3x}{x-2} which pass through the point (-1,9). Find also the points of contact of these tangent lines with the curve.The Attempt at a Solution 1. I found the equation of lines passing through (-1,9) -> y=(x+1)m+9 2. I...
  47. F

    Find the values where the tangent line is horizontal

    1. Find the values for x at which the tangent line is horizontal 2. f(x) = x + 2sinx 3. I found the derivative to be f'(x) = 1 + 2cosx I then set the derivative equal to zero and it came out to be 2cosx = -1, cosx = -\frac{1}{2} So the values of the horizontal tangent are 2∏/3 ± 2∏
  48. D

    Intersection of surface and tangent plane

    Homework Statement I have a surface given by z=x^2 - y^2 and its tangent plane at the point (x,y)=(1,1) given by z = 2x-2y. I am asked to compute the intersection of the tangent plane with the surface. The Attempt at a Solution I did the obvious and set x^2-y^2 = 2x -2y to find the x,y...
  49. K

    Question about finding a tangent

    Homework Statement Show that, for each real number t \in the interval (0, 1], the curve given by: y=ln(\frac{x+\sqrt{1+x^{2}}}{1+\sqrt{2}}) has a tangent line with slope t. Find the points on the curve at which the tangent line has slope 2/3. The Attempt at a Solution I found the first...
  50. C

    Solving Tangent to Ellipse: x^2+5y^2=5, y=mx+c

    An ellipse has the equation x^2+5y^2=5 a line has the equation y=mx+c a) show that if the line is a tangent to the ellipse then c^2=5m^2+1 b) hence find the equation of the tangent parallel to the line x-2y+1=0 I tried to find the gradient of x^2+5y^2=5 at a point (x1,y1) and then put it...
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