Tangent line Definition and 279 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. J

    Parametric equation of tangent line

    Homework Statement Find parametric equations for the tangent line to the curve with the given parametric equations at a given point. \[x = t^5, y = t^4, z = t^3\] at point (1,1,1) Homework Equations The Attempt at a Solution So we need to have direction vector, and a point. To find...
  2. J

    Finding the a circle's tangent line which intersects a given point

    Homework Statement So, it's my understanding that there must exist a line which is tangent to a given circle and intersects a given point in 2D space. I'm trying to find that line. Any form will do, but I'm currently aiming for the coordinates of the two points: the intersection point, and the...
  3. A

    Find an equation of the tangent line to the graph of the function f

    Homework Statement Find an equation of the tangent line to the graph of the function f defined by the following equation at the indicated point. (x - y - 1)3 = x; (1, -1) The Attempt at a Solution x3-y3=1 3y2(dy/dx)-3x2=0 3y2(dy/dx)=3x22 (dy/dx)=3y2/3x2 (dy/dx)=x2/y2 slope = (dy/dx) = 1...
  4. L

    Multivariable Calculus - Tangent Line

    Homework Statement Find parametric equations for the tangent line at the point (cos (-5*pi/6), sin (-5*pi/6), -5*pi/6) on the curve x(t) = cos t y(t) = sin t z(t) = t (Your line should be parametrized so that it passes through the given point at t=0). Im not really understanding the question...
  5. A

    Which Points on the Curve y = sin(2x) + 2 sin(x) Have a Horizontal Tangent Line?

    Find the x-coordinate of all points on the curve y = sin(2x) + 2 sin(x) at which the tangent line is horizontal. Consider the domain x = [0,2π). f'(x)=2cos2x+2cosx
  6. W

    Finding the parametric form of a tangent line vectors

    Homework Statement Find the parametric form for the tangent line to the graph of y=2x2−5x+3 at x=2 is Homework Equations I have no clue! The Attempt at a Solution I found the tangent line to be y=3x-5 I know that the answer has to be in the form... <x0,y0>+t<x1-x0,y1-y0> I...
  7. J

    Second Derivative: What Does it Represent? - James

    If the first derivative of a function represents the gradient of the tangent line... What does the second derivative represent? Thanks in advance James
  8. Z

    Find Tangent Line to C with Intercept of 150

    Say I have a curve is called C: y=1287*x^-1.5 Find a tangent line to the C, and the tangent line has to have a intercept of 150. This is not a homework, not at all.
  9. S

    Finding the Equation of Tangent line on cos wave

    Finding the Equation of Tangent line on cos wave! Find the equation of the normal line to y = 2cos ( 4x) at x = \pi / 3 I don't even know where to start with this question, i have searched the textbook and internet, help would be appreciated.
  10. K

    Find equation of line that is perpendicular to the tangent line to the curve

    Homework Statement Find the equation of the line that is perpendicular to the tangent line to the curve, y=(3x+1)/(4x-2) at the point (1,2) Homework Equations The Attempt at a Solution I am absolutely confused with this problem. I tried taking a derivative of the equation. And I...
  11. B

    Implicit Differentiation; Tangent Line

    Homework Statement Find the equation of a tangent line at the curve at point (-3√3, 1) x^(1/3) + y^(1/3) = 4Homework Equations Point-slope: y-1=m(x-1) The Attempt at a Solution I took the derivative of that equation and resulted in -y^(2/3)/x^(2/3) When I tried plugging in x and y to...
  12. J

    Triangle and tangent line circle

    A triangle ABC, where ,<A = 60 degrees. Let O be the inscribed circle of triangle ABC, as shown in the figure. Let D, E, and F be the points at which O is tangent to the sides AB, BC and CA. And let G be the point of intersection of the line segment AE and the circle O. (but AE line not cross...
  13. S

    Equation of tangent line (rec. form) to a polar curve

    Equation of tangent line (rec. form) to a polar curve! Homework Statement Quesiton: Find the rectangular form of the equation of the tangent line to the polar curve r=cos^3(theta) at the point corresponding to theta=pi/4 Homework Equations The Attempt at a Solution How to...
  14. M

    Slope of Tangent Line at (0,-10) for y^3+1004=(e^x+1)^2

    Homework Statement A curve is given by the equation: y^3+1004=(e^x+1)^2 Find the slope of the tangent line at the point (0,-10). Homework Equations The Attempt at a Solution I took the derivative of ((e^x+1)^2-1004)^(1/3) and that is (2e^x(1+e^x))/(3((1+e^x)^2-1004)^(2/3)) but...
  15. R

    Slope of the tangent line of an intersection - Directional Derivatives

    Homework Statement Find the slope of the tangent line to the curve of intersection of the vertical plane x - y + 1 =0 and the surface z = x2+y2 at the point (1, 2, 5) Homework Equations Gradients, Cross products The Attempt at a Solution I'm pretty lost here. I think I have to...
  16. S

    How Do You Find a Common Tangent Line to Two Different Functions?

    I was working on a problem, and in my solution I came across a situation which I will try and state in the following question: Given two functions, f(x) and g(x), how would you find a line such that the line is tangent to f(x) at some point x=a, and tangent to g(x) at some point x=b, assuming...
  17. L

    Finding f'(1) with Tangent Line at (1,7)

    Homework Statement If the line tangent to the graph of the function f at the point (1,7) passes through the point (-2,-2), then f'(1) is? Answer: 3 Homework Equations The Attempt at a Solution I got the right answer, but I think it was by pure luck. I found the slope...
  18. L

    Write an equation of each horizontal tangent line to the curve.

    Homework Statement Write an equation of each horizontal tangent line to the curve. Homework Equations y = 2y^3 + 6x^2y - 12x + 6y = 1 y' = (4x - 2xy) / x^2 + y^2 + 1) The Attempt at a Solution Well, horizontal tangent line means the derivative equals zero. Thus, 4x - 2xy = 0...
  19. F

    How can I find the equation of the tangent line at x=4 for y=2x?

    We know that the slope of the tangent line at a point on a curve is found by evaluating the derivative of the curve at that point. Say we have the curve y=2x. Say I wanted to find the tangent line at x=4 dy/dx=2 The first derivative is a constant, which is not surprising since the curve is...
  20. G

    Tangent Line and Coordinates of Trigonometric Function

    Homework Statement There are infinitely many points on the curve y = \frac{sin x}{\sqrt{2}- cos x} at which the tangent line to this curve is horizontal. Find the x- and y-coordinates of one such point. Homework Equations y' = slope of the tangent line Etc., etc. The Attempt at a...
  21. A

    Find which value of x horizontal Tangent Line

    Find which value of x...horizontal Tangent Line Homework Statement What is given is F(x)= -4/(x-3)(x+4) and the problem asks for to find the value of x where f(x) has a horizontal tangent line.Homework Equations I read somewhere else on these forms that using the quotient rule is the key, and...
  22. T

    Finding points on curve where tangent line given

    Homework Statement Find all points on the curve x^2 * y^2 + xy = 2 where the slope of the tangent line is -1. Homework Equations y' = -1 The Attempt at a Solution I got the y' = [ -2xy^2 - y ] / [ 2x^(2)y + x ] to be the gradient which I am sure is right. Then I subbed in y' =...
  23. A

    Find an equation for the tangent line

    suppose the function F is define by F(x)=[integral from 1 to √x] (2t-1)/(t+2)dt for all real numbers x≥0 A.) evaluate F(1) B.) Evaluate F'(1) C.)Find an equation for the tangent line to the graph of F at the point where x=1 D.) on what intervals is the function F increasing? justify your...
  24. 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)...
  25. 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...
  26. 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?
  27. 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.
  28. 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 -...
  29. 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))
  30. 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
  31. 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...
  32. 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...
  33. 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...
  34. 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 ?
  35. 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...
  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. P

    Tangent line of the curve of intersection of two surfaces

    How do I find this?
  40. 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!
  41. M

    Tangent Line Equation for Ellipse: Parametric Equations at (1,2,2)

    Homework Statement The ellipsoid 4x^2 + 2y^2 + z^2 = 16 intersects the plane y = 2 in an ellipse. Find parametric equations for the tangent line to this ellipse at the point (1,2,2)Homework Equations x = x0 + at y = y0 + bt z = z0 + ct The Attempt at a Solution Well i know that x0,y0 and z0...
  42. U

    Implicit Diff: Tangent Line at (-3*31/2,1)

    Homework Statement Implicit Differentiation:Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2/3 + y2/3 = 4 at (-3*31/2,1) Homework Equations ? None? The Attempt at a Solution 2/3 * x-1/3 + 2/3y-1/3*y' = 0 then after a few...
  43. L

    Proof of a focus point on parabola and tangent line equal angles

    Homework Statement Prove that a vertical line and a line going from a point on a parabola to the focus of the parabola form equal angles with the tangent line of the point on the parabola. Homework Equations Focus = 1/4a (maybe relevant) The Attempt at a Solution I know how to...
  44. F

    Determining perpendicular tangent line

    Homework Statement Determine the coordinates of the points on the graph of f(x) = \sqrt {2x+1} where the tangent line is perpendicular to the line 3x+y+4 = 0 Homework Equations f(x) = \sqrt {2x+1} 3x+y+4 = 0 The Attempt at a Solution For this, I was going to find the...
  45. U

    Find parametric equations for the tangent line at the point

    Homework Statement Find parametric equations for the tangent line at the point (\cos(\frac{4 \pi}{6}) ,\sin(\frac{4 \pi}{6}) ,\frac{4 \pi}{6}) ) on the curve x=\cos t,\ y=\sin t, \ z=t Homework Equations The Attempt at a Solution Took the derivative r'(t)=( -sint...
  46. F

    Determining slope of the tangent line.

    Homework Statement I've never seen such a problem before: Find the two straight lines that are perpendicular to y=0.25x and tanget to the curve f(x)=1/xHomework Equations y=0.25x ; f(x)=1/x The Attempt at a Solution Using the power rule, I found the derivatives of: f(x)=-x^{-2} and y=0.25...
  47. S

    Equation for the tangent line

    Homework Statement Find the equation for the tangent line to the curve f(x) = 2x3 - 5x - 3 at x = 2 Homework Equations How do i start this? I'm really confused and have an assignment with a bunch of these questions due tomorrow by midnight... if someone could help me that would be...
  48. A

    Finding Horizontal Tangent Lines on a Parabola

    Homework Statement I have a homework that i couldn't do :( can you explain it to me please ? the problem is : find all values of x=c so that the tangent line to the graph of f(x) af (c , f(c)) will be horizontal http://img13.imageshack.us/img13/4222/scan0002izs.jpg The Attempt...
  49. H

    Finding the Equation of a Tangent Line Perpendicular to a Given Line

    Homework Statement Find the equation of the tangent line on the function, f(x) = X^{2}-4x+1, which is perpendicular to the line, x+2y=10. Homework Equations The Attempt at a Solution x+2y=10 2y = 10-x y=-1/2x+5 Slope of perpendicular is -1/2, so slope of tangent is 2. The...
  50. Ƒ

    Equation of the tangent line at (0, pi/2)

    Find the equation for the line that is tangent to the given formula if y = pi/2 when x = 0 Homework Equations (x+1)dy - [(1/2)secycscy]dx = 0 The Attempt at a Solution I tried to do this, and I got that dy/dx = 1 / (0*0), which is infinity. So... y = \infty?
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