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. A

    Tangent line of curve of intersection

    Homework Statement Find the parametric equations of the line tangent to the curve of intersection of the paraboloid z = x² + y² and the ellipsoid 4x² + y² + z² = 9 at the point ( -1, 1, 2 ). Homework Equations Probable use of the gradient vector (as this is the chapter we are in)...
  2. I

    Tangent, Horizontal and Vertical Lines at the Pole

    Help me understand exactly what is going on here. I'll put up an attempt at my solution: Find the tangent lines at the pole r = sin5o, [0,pi] (note: o represents theta) Equation: dy/dx = [f'(o)sino + f(o)coso]/[f'(o)coso - f(o)sino] f(o) = sin5o f'(o) = 5cos5o plugging everything...
  3. M

    Exploring Tangent Lines in Cubed Functions

    Homework Statement is it possible to have a tangent line in a cubed function Homework Equations The Attempt at a Solution
  4. M

    Slope of Tangent Line for f(x)=x^3+x at (2,10) | Algebraic Method

    Homework Statement find the slope of the tangent line of f at the given point. Homework Equations f(x)= x^3 + x at (2,10) The Attempt at a Solution I know how to get the answer using the power rule, but I want to know the algebraic way of doing it I get stuck at x^3 + x - 10/ x...
  5. B

    Tangent lines to Parametric Equation

    HALP! I have already killed a forest trying to work this one out on paper. Homework Statement a. Use Implicit differentiation to find the equations of the horizontal tangent lines to the parametric equation: yx^3-2x^2y^2+xy^3=192. b. Use Implicit differentiation to find the equations...
  6. D

    Solving for b: Find the Tangent Line of y = e^{bx} at y = 10x

    For what value of b is the line y = 10x tangent to the curve y = e^{bx} at some point in the xy-plane? y' = b*e^{bx} = 10x How can I solve for b here?
  7. R

    Finding the line tangent to a curve

    Homework Statement Find an equation for the line tangent to the curve at the point defined by the given value of t. x = 2 cos t y = 2 sin t t = pi/4 Homework Equations sin (pi/4) = sqrt(2)/2 cos (pi/4) = sqrt(2)/2 The Attempt at a Solution Determining the slope: [dy/dt]/[dx/dt]...
  8. B

    Calculating the Exact Value of tan(15)

    I am trying to find the exact value for tan(15). I figure my equation as 40 - 30 to give the 15. when deriving my equation is where I have the problem. can anyone help please. (1-√(3)/3)/(1 + 1 * √(3)/3)
  9. P

    Torsion of space curves, why dB/ds is perpendicular to tangent

    Hi, I'm reading this piece from George Cain & James Harod's multivariable calculus material. Section 4.3, which is about Torsion, says this: I don't understand how he deduces dB/ds is perpendicular to T? Where did I get lost? Following the paragraph, it seems to me that T and N...
  10. B

    Finding the tangent acceleration of a car going around a loop.

    Alright, I'm having trouble with this problem, but I know that it's easy and simple. If the car starts at height h= 3.00 R and the radius is R_1 = 25.0 m, compute the tangential acceleration of the passengers when the car is at point C, which is at the end of a horizontal diameter...
  11. K

    What points is the tangent line of curve paralell to a vector?

    Homework Statement 4x^2 - xy + y^2 = 4 At what points is the tangent line to the above curve vector [1 1]T? (T means I transposed the vector) Homework Equations The Attempt at a Solution I have the gradient vector, but I'm conceptually lost about what to do after that.
  12. quasar987

    Does trivial cotangent bundle implies trivial tangent bundle?

    If the tangent bundle is trivial, then the cotangent bundle is trivial. To see this, consider (X_i) a global frame for TM. Then define a global frame (\alpha^i) for T*M by setting \alpha^i(X_j)=\delta_{ij} and extend by linearity. Does trivial cotangent bundle implies trivial tangent bundle? A...
  13. W

    Find Tangent Line to Cardioid at (0, 0.5) Using Implicit Diff.

    Homework Statement Use implicit differentiation to find an equation of the tangent line to the cardioid at the point (0, 0.5). x2 + y2 = (2x2 + 2y2 - x)2Homework Equations Derivative rules point slope formula The Attempt at a Solution I got y' = [16x3-4x2+16xy2-4y2-4y2-8x2+2x] / [2y -...
  14. F

    Equation of both lines that are tangent to the graph y=x^2

    Homework Statement Question: Determine the equations of both lines that are tangent to the graph of f(x) = x2 and pass through point (1,-3). Homework Equations Some of the equations that I could use for this problem are: y-y1=m(x-x1) (Point-slope Equation) the derivative of the...
  15. T

    Forgetting my inverse tangent, polar form of compelx number ASTC

    Homework Statement Find the polar form of 2i − 1 Finding polar form is easy r(cosx + isinx) call the real part a and imaginary part b r = sqrt(a+b) theta = arctan (-2) = - 63.43 This is the wrong angle for theta as it's 116.57 (which is 180 - 64.43), and I guess this if...
  16. C

    How can the common tangent of two curves be found without using derivatives?

    Homework Statement Hi guys, need some help finding the common tangent of two curves. Two curves y = 3X3+6X2+6X+3 and y = 3X3+6X+3 touch each other. Find the common tangent. Homework Equations y = 3X3+6X2+6X+3 and y = 3X3+6X+3 The Attempt at a Solution well i made the two equations equal...
  17. D

    Find points on curve with a tangent line

    Homework Statement Find points on curve y=sinhx where the tangent line has slope 2 Homework Equations The Attempt at a Solution y' = 1/(sqrt(1-x^2)) = 2 4 = 1/(1-x^2) 4-4x^2=1 x=sqrt(3/4) the actual answer should be x = +or- ln(2+sqrt(3))
  18. D

    Equation of Tangent Line to y=(lnx)^cosx at (pi/2,1)

    Find the equation of the tangent line to the curve y=(lnx)^cosx at the point (pi/2, 1)? The Attempt at a Solution lny = cosx(ln(ln(x))) d/dx = -sinx(ln(ln(x))) + cosx/(ln(x)(x)) y' = y(-sinx(ln(ln(x))) + cosx/(ln(x)(x))) this is the part where I get stuck
  19. K

    Normal vectors on tangent spaces.

    Homework Statement Given a surface parameterized by the function f(x) and a point p on that surface, assume that P is a point on the tangent space of f at p. Find the normal vector to the hyperplane at P . The Attempt at a Solution The tangent hyperplane to f at p is given by the...
  20. S

    Finding the equation of the circle given two points and a tangent line

    Homework Statement Find the equation of the circle that passes through the point (3,-2) and tangent to the line y=3x+5 at (-1,2). Answer in standard form. Homework Equations d= |mx0+b-y0| ____________ sqrt(1+m2 is needed to find the radius of the circle (x-h)2+(y-k)2=r2...
  21. F

    Parametric equation for a tangent line (multi vars)

    Homework Statement Find the parametric equation for the line tangent to the curve: x=t^3-1, y=t^4+1, z=t at the point (26, 82, 3). Use the variable t for your parameter. Homework Equations The Attempt at a Solution dx/dt=3t^2, dy/dt=4t^3, dz/dt=1 I got then that...
  22. J

    What Are the Tangent Lines to the Parabola y = x² Through (-1, 3)?

    Homework Statement Give the equation of the two lines through the point (-1, 3) that are tangent to the parabola y= x^2 Homework Equations The Attempt at a Solution
  23. F

    Proving Multiple Tangents for y=(x^2+x-2)^2+3 at x=1

    Homework Statement Show that the tangent to the curve y=(x^2+x-2)^2+3 at the point where x =1 is also tangent to the curve at another point. Homework Equations y=(x^2+x-2)^2+3 The Attempt at a Solution y'=2(x^2+x-2)(2x+1) y'(1)=0 0=2(2x+1)(x+2)(x-1) x= -1/2, -2, 1 Would this be correct?
  24. A

    Solve Unit Tangent Vector at Point P: Find T, N, B

    Homework Statement Find the vectors T, N, and B at the given point. r(t) = (sin(t), cos(t), ln(cos(t))), P = (0,1,0) Homework Equations T(t) = r'(t) / | r'(t) | N(t) = T'(t) / | T'(t) | B(t) = T(t) x N(t) The Attempt at a Solution I am stuck on how to solve for t. I am not...
  25. F

    How Do You Find a Function Whose Integral is x^3 and Has a Tangent Line x+y=0?

    Homework Statement Find the equation f(x) who's integral is \int x^{3} and has a tangent x+y=0 Homework Equations The Attempt at a Solution I know that f(x) is 1/4x4+c because of the integral. The tangent is the derivative of f(x) at some point i have the equations y=x3+c y=-x but...
  26. C

    Does y=x^3+(x-3)^(1/3) Have a Vertical Tangent, Cusp, Corner, or None at x=3?

    Homework Statement At x=3 does the function y=x^3 + (x-3)^(1/3) have a vertical tangent, cusp, corner or none? The Attempt at a Solution I took the derivative y'=3x^2 + 1 / 3x^(2/3) then i replaced 3 in which gave 27 + 1 / 0 I don't understand how to come to the correct...
  27. J

    No use of the tangent in Riemann sum?

    Where is the use of the "tangents at every point on the curve" in the Riemann sum? Riemann sum allows us to find the area of under the curve, and this involves only the height of each rectangle (i.e. the function f(x) at each x), and the width (i.e. the x), and the two are multiplied together...
  28. D

    Recurrence relation of tangent

    Is it possible to find a recurrence relation of tan(nx) where n is a positive integer and x is a real variable? My friend said that it is possible. I don't see how to do it. Does anyone have a way to do it?
  29. J

    Using differentials in the equation of tangent line to the curve x^2 at point 3

    To understand differentials better, I'm trying to use differentials dy and dx in the equation of the tangent line to the curve x^2 at point 3. Here is the equation of the tangent line to the curve x^2 at point 3: y=f'(3)(x-3)+f(3)=2(3)(x-3)+9=6(x-3)+9 But since we are dealing with the...
  30. F

    Finding Perpendicular Lines to y=0.25x and Tangent to f(x)=1/x

    Homework Statement Find two straight lines that are perpendicular to y=0.25x and tangent to the curve f(x) = 1/x. Homework Equations y=0.25x f(x) = 1/x. The Attempt at a Solution What I did was equate y and f(x) and determined when they equal which is 2 and -2. The points are (2, 1/2) and...
  31. J

    How to graph the tangent to a curve at some point?

    Using a graphing software, I'm trying to graph three things: 1. The function x^2. 2. It's derivative 2x. 3. The tangent to the curve at point 3. Now I know that that if I want to find the slope of the curve at point 3, I should substitute 3 into the derivative 2x, which will give 6. What...
  32. S

    Tangent Line on a Parametric Curve

    Homework Statement A curve is defined parametrically by x=sin3t, y=cos3t, 0≤ t ≤ 2pi. Find the equation of the line tangent to the curve at the point defined by t=2pi/9. Homework Equations The Attempt at a Solution ?
  33. T

    Optimization - find two points on a curve with a common tangent line?

    Homework Statement Find two points on curve y=x4-2x2-x that have a common tangent line. Homework Equations *the one stated above dy/dx = 4x3-4x-1 The Attempt at a Solution equation of a tangent line: y=mx+b (4x3-4x-1) = m at two different points? So there are two points for which...
  34. B

    Circle to Circle Tangent Equation

    I am trying to write an Excel template which will chart two circles and their external tangent lines - similar to a belt and two pulley system. I have the formulas to calculate and chart the circles. I am looking for a formula to calculate the tangent points between the circles, given the...
  35. C

    Find the equation of the tangent of the parabola y^2= 4px

    hello everyone, :confused:i'm having can't seem to solve these two questions... 1) Find the equation of the tangent of the parabola y^2= 4px, perpendicular to the lines 4Y- X + 3=0, and find the point of contact? 2) Find the equation of the tangent to the parabola Y^2= 10x, at the...
  36. C

    Find the slope of the tangent line using a specific formula

    Homework Statement Find the slope of the tangent line using a specific formula g(x)=3t-t2 at (0,0) Homework Equations Im told to use this equation by the book f(c+deltax) - f(c) Deltax The Attempt at a Solution Everytime i plug it in by way of the books style i...
  37. C

    Paramentric eq. of tangent line

    Homework Statement Find the equation of the tangent line to f(t)=<cot(t),csc(t)> at the point (1/sq.rt of 3, 2/sq.rt of 3) Homework Equations n/a The Attempt at a Solution I started by finding the slope, y'/x', so I got csc(t)cot(t)/csc^2(t). I then used the equation of a line...
  38. J

    Equation of a tangent line to a given circle from an external point - Stuck

    Hello - I've been stuck on this for a while now and I really need some help. Homework Statement Find the equations for all lines that are tangent to the circle x^2+y^2=2y and pass through the point (0,4).Homework Equations y=mx+b ax^2+bx+c=0The Attempt at a Solution From the given equation of...
  39. J

    Volume of tetrahedra formed from coordinate and tangent planes

    I have that P is the tangent plane to the surface xyz=a^{3} at the point (r,s,t). I need to show that the volume of the tetrahedron, T, formed by the coordinate planes and the tangent plane to P is indepedent of the point (r,s,t). I have found that P is; \frac{x}{r} + \frac{y}{s} +...
  40. M

    Slope of a tangent From first principles

    Homework Statement # 23 http://img26.imageshack.us/img26/1008/questionsh.jpg Homework Equations m1 * m2 = -1 The Attempt at a Solution i get -4 not sure what's wrong: http://img4.imageshack.us/img4/6092/15608741.jpg
  41. P

    Tangent vector to curve of intersection of 2 surfaces

    Homework Statement Find the tangent vector at the point (1, 1, 2) to the curve of intersection of the surfaces z = x2 + y2 and z = x + y. Homework Equations The Attempt at a Solution I haven't started the problem, because I'm not sure what the first thing to do is. Do I have to parametrize...
  42. P

    Tangent line of the curve of intersection of two surfaces

    How do I find this?
  43. B

    Using the inverse hyperbolic tangent function to solve ODE

    Homework Statement Hi all. I have to solve the differential equation \frac{dv}{dt} = g(1 - \frac{\rho}{g}v^2). The Attempt at a Solution Apparently the solution should involve the inverse hyperbolic tangent function - with the equation in this form it should just be separable, correct...
  44. M

    Trying to get my head around tangent bundles

    Hello, Say you have a function f on the domain R^n, and an integral transform P which integrates f over all possible straight lines in R^n. I am lead to believe that the range of this is R^(2n), or a tangent bundle, which I am having MASSIVE problems visualising! Am I right in saying the...
  45. C

    Equation of tangent line w/ sqrt.

    Homework Statement Find the equation of the tangent line to y=sqrt(25-x^2) at the point (4,3) Homework Equations The Attempt at a Solution (25-x^2)^1/2 1/2(25-x^2)^-1/2 (2x)1/2(25-x^2)^-1/2 x(25-x^2)^-1/2 I have no idea what to do or where to go after this... Please Help!
  46. N

    Finding Points of Parallel Tangent Plane for z=(x^2)(e^y)

    Homework Statement Find the points on the graph of z=(x^2)(e^y) at which the tangent plane is parallel to 5x-2y-.5z=0 Homework Equations An equation of the tangent plane to z=f(x,y) at (a,b) is: z = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b) The Attempt at a Solution Partial derivatives...
  47. Y

    Where Does the Tangent Half-Angle Identity Come From?

    I saw this identity somewhere and have been looking for a derivation but I can't seem to find one. It would be of great help if someone can show me where this comes from. tan\frac{\theta}{2} = csc\theta - cot\theta
  48. F

    Angle of inclincation between tangent and plane

    Homework Statement Find the angle of inclination of the tangent and plane to the surface at the given point. (I we are comparing the tangent plane to the XY-plane) Homework Equations 2xy-z3=0 Point: (2,2,2) The Attempt at a Solution This one was thrown into the textbook just to piss...
  49. F

    Finding the equation of this tangent

    I have this graph The blue line is a tangent to the black The blue line starts at (0,0) I need to show that the blue line y=Hx is a tangent to the black line y= x(x-L)(1-x), (0<L<1) By deducing that H = (1-L)2/4 I took the example L=0.25 in the pic, where H = 9/64 where it's a...
  50. D

    Finding points on a curve where the tangent slope = 6

    Homework Statement Find the x values of the points on the curve f(x) = (x^3) + (3x^2) + 3x + 6 where the tangent has a slope equal to m = 6 Homework Equations The Attempt at a Solution This question was on an online quiz for my into calculus course... can't seem to wrap my head around...
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