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. P

    Equation of Tangent: y = -7x + 5 @ (1,1)

    Determine the equation 1 the tangent to the given function, at the given point. y = (3x^-2 - 2x^3) , @ (1,1)
  2. P

    Determine Point of Tangency & Value of "b" for y=8x+b

    The line y = 8x + b is tangent to the curve y = 2x^2. Determine the point of tangency and the value of "b"
  3. J

    Curvature in terms of the tangent vector

    My teacher wrote an alternative equation on the board for curvature, and I am wondering how it is true: k = | dT/dt / |dR/dt| | where T is the unit tangent vector. I know k = |R' x R''| / |R'|^3 = |dT/ds| but I am not sure about the formula in question. How is it true/derived?
  4. L

    Slope of Tangent Line for g(x)=x^2-4 at (1,-3) | Calculus Homework Solution

    Homework Statement find the slope of the tangent line whose g(x)=x^2-4 at point (1,-3) Homework Equations lim f(x+Δχ) -F(c)/ (Δχ) The Attempt at a Solution g(x)= x^2-4 G(1+Δχ)= (1+Δχ)^2-4 ==> Δχ^2+2Δχ-3 lim (Δχ^2+2Δχ-3) - (-3)/(Δχ) = 0 but in the book the answer is 2 so what...
  5. N

    Tangent plane approximation via parametric equations

    Hi everyone, I'm an 18-year-old from Germany and I'm making use of MIT's OpenCourseWare programme. Currently, I'm watching the Calculus II course, and am having some trouble understanding how to find the equation: z=z0 + a(x-x0) + b(y-y0) by using parametric equations/vectors. a and b being...
  6. M

    The function f(x) = e^3x +6x^2 +1 has a horizontal tangent at x =?

    Homework Statement i can't solve it! im really lost. i know you find f''(x) (i got 9e^3x +12) but i don't know where to go from there what would i do on my calculator?
  7. A

    How do I find tangent lines passing through a given point on a graph?

    Homework Statement A) Find all points on the graph of f(x)=x^2 with tangent lines passing through the point (5,0). B) Show that no line tangent to g(x)=1/x passes through the origin. These are problems from a lab problem solving assignment. Homework Equations y=mx+b f`(x) =...
  8. E

    Tangent Plane Equation for Surface x^2 + y^2 - xyz = 1 at Point (2,3,2)

    Homework Statement Find the equation of the tangent plane at (2,3,2) to the surface below. x^2 + y^2 - xyz = 1 The question asks me to do this in two way, one is to view the surface as the graph of a function of two variables z = g(x,y). and the other one is to view the surface as a...
  9. I

    Here's a theorem about the tangent function

    [b]1. Here's a theorem about the tangent function. tan(x) = c iff 1: x = tan^-1*c or 2: x = tan^-1*c+n(180) for some integer n. Find a solution to tan x = 1 between 270 and 450 degrees. [b]2. [b]3. I am not really sure how to go about starting this problem. I have only two...
  10. A

    I (I must find the equation of the sphere that is tangent to 2 planes.

    Homework Statement My professor gave us 4 points A(2,3,-5) B(-2,5,1) C(9,0,4) D(3,-5,8) So I must find an equation of the sphere that is tangent to 2 planes. These 2 planes are 18x + 39y - z= 158 18 x + 39y -z = -149 Homework Equations equation of sphere C(h,k,l) (x-h)^2 + (y-k)^2...
  11. J

    Equation of Tangent to the Line

    3.) Give the equation of the line tangent to the curve at the given point. a.) (y)(tan^-1 x) = x*y at (sqrt(3),0) b.) ln y = x^2 +(2)*e^x at (0, e^2) Please help me with this problem, I am stuck. Here is some of my work process: i have no idea how to solve for y in...
  12. C

    Parameterized tangent line to a parameterized curve

    Homework Statement I seem to remember that a parameterized a(t) curve in \mathbb{R}^3 that one can construct the tangent from the slope of a'(t) and the curve itself. such that the tangent line L = a(t) + s * a'(t) to a. This is supposedly a straight line in \mathbb{R}^3. To make a...
  13. P

    If the tangent line to y = f (x) at (9,7) passes through the point (0,6)

    Homework Statement If the tangent line to y = f (x) at (9,7) passes through the point (0,6), find the following. (a) f (9) (b) f ' (9) Homework Equations f '(x) = f(x+h) - f(x)/h ? The Attempt at a Solution I really have no idea how to even begin going about this one, so a...
  14. P

    (a) Find the slope of the tangent to the curve at the point where x = a.

    Homework Statement Consider the following curve. y = 4 + 4x2 - 2x3. (a) Find the slope of the tangent to the curve at the point where x = a. Homework Equations m=y(a+h) - y(a)/h The Attempt at a Solution 1. m=y(a+h) - y(a)/h = -a^3-a^2+3ah+h^2-4 2. 6a^2 3. -6a^2+8a I know the derivative is...
  15. M

    I think linear approximation? (square root, tangent, e^x)

    Homework Statement the value of f(x) = (sqrrt e^x +3) at x=0.08 obtained from the tangent to the graph at x=0 is...? Homework Equations The Attempt at a Solution i used linear approximation. (sqrrt e^o +3) + (1/2(sqrrt3+e^0)(0.08) i got an answer but i know its wrong. i...
  16. D

    Equation of a tangent line given x

    I am having a time trying to figure out this problem given only an x value. The equation is "Find the equation of the tangent line to y= Square root of x^4-2337 ...at x=7. If anybody could help me solve this it would be a great help. Thanks
  17. J

    Tangent lines and areas between curves.

    I have a function: y=e^x the tangent line at the point (1,e) would be x*e? in order to find the area between the tangent line, the y-axis and y=e^x i equate these functions and solve for x? I got this far (e^x)/(x)=e how do i solve for x
  18. P

    Graphing Tangents to a Quadratic Curve

    Homework Statement Find an equation for the tangent to the curve at the given point. Then sketch the curve and tangent together Homework Equations y=3-x^2 (-1,2) The Attempt at a Solution slope = f(x + h) + f(x) / h f(-1 + h) + 2 / h 3- (-x + h)^2 + 2 / h 3 - 1 + 2h + H^2 + 2...
  19. A

    An Unknown Conic and Equation of Tangent Line HELP

    An Unknown Conic and Equation of Tangent Line **HELP** Homework Statement (a) Find the positive y-coordinate of the point on the conic whose x-coordinate is 27cos\theta, where 0<\theta<\pi/2 (b) The slope of the tangent line to the conic through the point (27cos\theta, y1) is -1/27cot\theta...
  20. G

    Finding the eq. of all tangent lines on a curve

    Homework Statement Problem: Find the equations of all tangent lines to the curve y = x + 2x so that also go through the point (3, 14). 2. Do not use a derivitive 3. I don't even know where to start. I searched my book there isn't really available. Any help would be much appreciated
  21. S

    Finding the Vertical Tangent Line of x3 + xy - y2 = 10 Curve

    The curve defined by x3 +xy - y2 = 10 has a vertical tangent line when x = ? To find when the tangent line is vertical, could I find when the slope is undefined for the original function? The way I previously tried this problem was by taking the first derivative and then finding when x = 0...
  22. H

    What are the remaining steps to complete this problem?

    Homework Statement Given the function y = x^{2} + 3x - 4 find, a) the equation of tangent + normal at x = -2. b)the equation of the tangent that has slope of -2. c)local min and max. d)the equations of the tangent from the point (2 , -4) to the curve. e) area enclosed by the tangent at...
  23. B

    How to Solve for x or y in a Horizontal Tangent Problem?

    Homework Statement 4x - 2xy = 0 how do i solve for EITHER x or y? Homework Equations The Attempt at a Solution i got y = 2 at first but when i plugged it into the original equation to find x it was impossible to get an answer (the x's canceled out)
  24. C

    Understanding Second Order PDEs: Explaining the Unit Tangent to a Curve

    Hi In my lecturer's notes he describes the unit tangent to a curve y=Y(X) as (i + Y'(X)j)/[(1+[Y'(X)]^2)^(0.5)] in an introduction to second order PDEs I'm a bit confused by this. Where did it come from? Can anyone explain Thanks
  25. C

    Understanding Unit Tangent to a Curve in PDEs

    Hi In my lecturer's notes he describes the unit tangent to a curve y=Y(X) as (i + Y'(X)j)/[(1+[Y'(X)]^2)^(0.5)] in an introduction to second order PDEs I'm a bit confused by this. Where did it come from? Can anyone explain Thanks
  26. G

    Basis for tangent space and cotangent space

    Hello, I'm trying (somewhat haphazardly) to teach myself about differential forms. A question I have which is confusing me at the moment is about the tangent and cotangent spaces. In https://www.physicsforums.com/showthread.php?t=2953" the basis for the tangent space was described in terms of...
  27. M

    Find the equation of the tangent line

    Homework Statement Find the equation of the tangent line to the curve 3xy = x^3 + y^3 at point (2/3,4/3) Homework Equations The Attempt at a Solution I found the deriviative to be 3x^2-3y / 3x - 3y^2 which i simplified to x^2 - y / x - y^2 I found the slope of at those...
  28. M

    Finding the slope of the tangent

    Homework Statement Find the slope(s) of the tangent at x = 1 for 2y^2 - xy -x^2 = 0 Homework Equations The Attempt at a Solution Ive figured the derivative to be 2x+4/4y+x But I don't know how to make this equation work to find a y value so I can find the slope. I tried...
  29. F

    Trigonometric functions - angular measures and tangent curves

    I encountered a few problems for a few questions while doing my homework. 1. Angular measure problem: A Ferris wheel with a radius of 25.3m makes 2 rotations every minute. a) Find the average angular speed of the Ferris wheel in radians per second. b) How far does a rider travel if the ride...
  30. M

    Finding the point on the graph where the tangent is parallel

    Homework Statement Find the point on the graph y = sqrt 2x-1 where the tangent line is parallel to x - 3y = 16 Homework Equations The Attempt at a Solution So what I started off was was trying to find the slope of x - 3y = 16. I rearranged the equation to a y = equation and...
  31. J

    Find the tangent line between two surfaces

    Homework Statement Let C be the intersection of the two surfaces: S1: x^2 + 4y^2 + z^2 = 6; s2: z = x^2 + 2y; Show that the point (1, -1, -1) is on the curve C and find the tangent line to the curve C at the point (1, -1, -1). Homework Equations partial derivates, maybe the gradient...
  32. K

    Equation for Tangent Line to Inverse Function at (3,1) of f(x)=x^3+2x^2-x+1

    Homework Statement Find an equation for the line tangent to to the graph of f^-1 at the pt (3,1) if f(x)=x^3+2x^2-x+1 Homework Equations The Attempt at a Solution I used the Inverse Function Thm 1/(3x^2+4x) Now do I plug in 3 to this to find slope? 1/21 y-1=1/21(x-3)...
  33. G

    Finding parametric equations of a tangent line

    Homework Statement Find parametric equations of the tangent line at the point (-2,2,4) to the curve of intersection of the surface z=2x2-y2 and the plane z=4 Homework Equations Not sure The Attempt at a Solution Not sure quite how to approach this. take the gradient of...
  34. G

    Finding the equation of the tangent plane

    Homework Statement Find the equations of the tangent plane and the normal line to the given surface at the specified point. xy+yz+zx=3 @ (1,1,1) Homework Equations z-z0=fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0) The Attempt at a Solution I feel like this should be really easy, but I'm...
  35. M

    Finding the derivative of the function and the slope of the tangent

    Homework Statement Find the slope of the tangent to y=3x2 - 6x at x = 2 by first determining the derivative of the function from first principles Homework Equations f(x+h)-f(x) / h The Attempt at a Solution For the derivative I got -6x and the slope of the tangent is -12...
  36. M

    Differentiation of Unit Tangent

    So we are given T(t) = c'(t)/||c'(t)|| as well as ||T|| = 1 We also know T(t)dotT(t) = 1 and T'(t)dotT(t) = 0 The problem asks us to find T'(t) I tried differentiating c'(t)/||c'(t)|| treating ||c'(t)|| as the square root of the dot product of c'(t) with itself. I used the product...
  37. W

    Parallel Tangent Planes on a Given Surface

    Homework Statement A surface is given by the equation 2x2+y2-z2=64 Find all points p on the surface where the tangent plane is parallel to the plane 12x+y-3z=0. Homework Equations I know the equation of a plane tangent to a surface at point (x0,y0,z0) is given by the equation...
  38. L

    Tangent to an Ellipse given the slope of the tangent

    Homework Statement Determine the points on the ellipse x^2 a 2y^2=1 where the tangent line has a slope of 1 Homework Equations I'm able to solve problems when given points and asked to find equations of the tangent lines. However, I'm struggling to do the inverse. The Attempt at a Solution...
  39. 1

    Find an equation of the tangent line

    Homework Statement Find an Equation of the tangent line to the curve y=10^x at the point (1,10) Homework Equations Derivative y=mx+b The Attempt at a Solution I am totally lost I don't know what to do with 10^x Please help me this will save me failing the class
  40. gabbagabbahey

    Explicitly summing a series involving hyperbolic tangent

    I recently came across a problem where I was able to show that \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \tanh \left( \frac{n \pi}{2} \right)=\frac{\ln 2- \pi}{4} through numerical approximation...However, I don't have much practice evaluating such summations analytically, and I was wondering if...
  41. S

    Finding tangent lines to an ellipse that pass through a given point

    Homework Statement Find the equations of all the tangent lines to x^2 + 4y^2 = 36 that pass through the point (12,3) Homework Equations the derivative of the ellipse is dy/dx = -2x/8y (I'm not sure if that is correct, i have only recently learned implicit differentiation.) The...
  42. W

    Unit Tangent for a Curve: Finding T(t) using the Product Rule

    Homework Statement r(t) = 〈sin(t) − t cos(t), cos(t) +t sin(t), 5 t2 + 7 〉 Find the unit tangent. T(t)= The Attempt at a Solution r'(t) = <cos(t) + tsin(t), -sin(t) + tcos(t), 10t> T(t) = r'(t)/ |r'(t)| |r(t)| = sqrt( 1 + 101t^2) And so on. Supposably according to the...
  43. pellman

    Simple question about definition of tangent bundle

    So I'm trying to learn about fibre bundles and I am looking at the example of a tangent bundle. Given a differentiable manifold M. Denote the tangent space at p \in M by T_p M. Is the definition of the tangent bundle TM = \lbrace (p, T_p M)|p \in M \rbrace or is it TM = \lbrace...
  44. J

    Tangent to a curve and normal to the surface

    Hi i have a very basic question in calculus. i know that the tangent to the curve y=f(x) is denoted by dy/dx which makes sense to me geometrically. But normal to a surface k(x1,x2,x3) is Grad k. how is this? it doesn't make sense to me geometrically. can anyone throw more light upon this please
  45. M

    Points where tangent line touches 2 circles

    Homework Statement On the circles y^2+x^2=1 and y^2+(x-3)^2=4 There is a line with positive slope that is tangent to both circles. Determine the points at which this tangent touches each circle. Homework Equations the derivative of the first circle i found: y'=-x/y the derivative of...
  46. S

    Create tangent to a 2D spline curve in Pro/E

    How can I create a line tangent to a spline curve interpolated with several points in 2D in pro/engineer wildfire2.
  47. A

    Quick Tangent Line Approximation (Derivatives)

    Homework Statement The tangent line through given points (pi/4,pi/4) m=1 y= cos(y)cos(x)/sin(x)sin(y) The Attempt at a Solution dy/dx= d/dx[cos(y)cos(x)/sin(x)sin(y)] First use quotient rule ? vu'-uv'/v^2 v= sin(x)sin(y) v'= do i need to use product rule? for product rule...
  48. I

    The line tangent to a curve, no derivative

    Homework Statement Let (x0,y0) be a point on the graph f(x) = x2 for x0 not equal to 0. Find the equation of the line tangent to the graph of f at that point by finding a line that intersects the cure in exactly one point. Do not use the derivative to find this line.Homework Equations f(x) =...
  49. mbisCool

    Finding Tangent Line to a Curve at a Given Point

    Homework Statement For what values of a and b is the line 3x+y=b tangent to the curve y=ax^3 when x=5 ? The Attempt at a Solution I believe i would need to find the equation of the tangent to y=ax^3 when x=5 then that should be equal to -3x+b=y if I am not mistaken but I am not sure...
  50. Q

    Proving \alpha^i = v x^i on tangent vector space T_pM"

    Let me first confess this a copy/paste of a question I asked on another forum; I trust it's not against the rules. Let M be a C^{\infty} manifold, and, for some neighbourhood U\ni p \subsetneq M let there be local coordinates x^i such that p=(x^1,\,x^2,...,x^n) Suppose that T_pM is a...
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