What is Tangent line: Definition and 279 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. 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...
  3. S

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

    Homework Statement Given that near (1,1,1) the curve of intersection of the surfaces x^4 + y^2 + z^6 - 3xyz = 0 and xy + yz + zx - 3z^8 = 0 has the parametric equations x = f(t), y = g(t), z = t with f, g differentiable. (a) What are the values of the derivatives f'(1), g'(1)? (b)...
  4. 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...
  5. 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...
  6. 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...
  7. ArcanaNoir

    Find direction of tangent line to equation

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  8. B

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

    Homework Statement Find an equation of the tangent line to the curve at the given point. y = tan x at point (pi/4,1) Homework Equations The Attempt at a Solution step 1. find the derivative of tan x, which sec^2 x step 2. find the slope. this is where I mess up. I assume...
  9. C

    Tangent line that passes through origin

    Homework Statement Find a > 0 such that the tangent line to the graph of f(x) = x^{2}e^{-x} at x = a passes through the origin. Homework Equations The Attempt at a Solution First I found the derivative to be: -e^{-x}(x-2)x , which is the slope of the function. I know the tangent line...
  10. K

    Tangent line and points it intersects a curve

    This is another question from last year's calculus exam that I'm a bit stuck on. " Let f be the real function given by f(x) = −x4+2x2+3x. (a) Determine the tangent line to the curve y = f(x) at x = 1. (b) Show that the tangent line to the curve y = f(x) at x = 1 is also the tangent line to...
  11. G

    How Accurately Can We Estimate Volume Error Using Tangent Line Approximation?

    Homework Statement You measure a cubic container and find it to be about 10 cm on each side. From this, you conclude that it holds 1000 cc. However, your measurement is accurate only to within ±0.1 cm. (So you can be sure that the side is between 9.9 and 10.1 cm.) What are x, f(x) , and a in...
  12. F

    Tangent line approximation with sinx/x + siny/y = C

    Homework Statement Suppose the point (pi/3, pi/4) is on the curve sinx/x + siny/y = C, where C is a constant. Use the tangent line approximation to find the y-coordinate of the point on the curve with x-coordinate pi/3 + pi/180. Homework Equations TLA: f(a) + f'(a)(x-a) Where a is...
  13. H

    Equation of tangent line to curve

    Homework Statement Find all x coordinates of points (x, y) on the curve y = (x − 2)5 /(x − 4)3 where the tangent line is horizontal Homework Equations Quotient Rule - Differentiation The Attempt at a Solution So I could guess, that if tangent line is horizontal, the equation of...
  14. V

    Find EQ of tangent line for y=sin x at x = 6*pi

    Homework Statement Find the tangent line equation for y = sin x at x = 6\,\pi. Homework Equations m_{tangent}\,=\,\lim_{h \to 0}\,\frac{f(c+h)\,-\,F(c)}{h} y\,-\,y_1\,=\,m\,(x\,-\,x_1) The Attempt at a Solution m_{tangent}\,=\,\lim_{h \to 0}\,\frac{f(c+h)\,-\,F(c)}{h}...
  15. L

    Finding the tangent line of the curve of intersection

    Given the paraboloid z = 6 - x - x2 -2y2 and the plane x = 1, find curve of intersection and the parametric equations of the tangent line to this curve at point (1,2,-4).So I plugged x=1 into the paraboloid equation and got z = 4-2y2. Then I take the derivative of the curve of intersection...
  16. V

    Finding points on a tangent line

    Homework Statement Find the point(s) on the graph of the function at which the tangent line has the indicated slope. (If an answer does not exist, enter DNE.) g(x) = (1/3)x^3 - (1/2)x^2 - 4x +8 mtan=-4 Homework Equations [b]3. The Attempt at a Solution firstly i derived...
  17. C

    Tangent Line Problem: Find Point P & Compute Slope m_L

    I'm on mobile so I can't use latex. Let C: y=8x^5+5x+1 and suppose L is a line through the origin tangent to C at a point P=(a,f(a)) on C. -Find the coordinates of P -Compute the slope m sub L of L Where should I begin? I'm guessing I would need the derivative of the equation f(x)...
  18. C

    Equation of the tangent line at the indicated point

    Homework Statement Find an equation of the tangent line at the indicated point on the graph of the function. y=f(x)=x^3/4 , (x,y)=(6,54) Homework Equations The Attempt at a Solution I did the derivative which I get 3x^2/4 and then I plugged in the 6 and get 162. Is that the...
  19. L

    Differentiation and tangent line

    Homework Statement Function f(x) = KxL K= 1.78 L= -1.39 Problem 1: Find f'(x). ____________________ v= 0.89 w= 0.5 "v" and "w" are two points located on the x-axis. Problem 2: Calculate f'(v). ____________________ Problem 3: Find the equation of the tangent line of f(x) over the point "w". The...
  20. P

    Tangent line through arbitrary point?

    Homework Statement Consider the function f(x)=x2+5x+5 and the point A(4,5). Find a formula for the slope of a line passing through point A, and an arbitrary point on the function. Your answer should be a formula in terms of x. slope=? Homework Equations f′(x)=2x+5 The Attempt at...
  21. D

    If every tangent line of a curve passes through a point, it is a line

    If every tangent line of some curve B(s) which has the unit speed parametrization passes through a fixed point P, then the curve B(s) must be a line. As a hint, my book says p = B(s) + r(s)B'(s) where r(s) is some function. so i have that any tangent line L(t) = B(s) + t B'(s) and for...
  22. T

    Tangent line to the curve f(x)= x + 1/x

    Homework Statement So I have to find the tangent line when x=5. I use the difference quotient, but I'm stuck at the algebra? I can't seem to figure out what steps I should take. Here's what I have: [(x+h)+1/(x+h)] - (x+1/x) I've tried numerous steps on how to solve this, but can't do...
  23. Z

    Equation of a Tangent Line to a Polar Curve

    I need to find the equation of the tangent line to the curve x=t4+1, y=t3+t; t=-1 I have already found that the slope of the line is -1 by finding (dy/dt)/(dx/dt) I just need to figure out how to solve for y1 and x1 Thanks in advance
  24. A

    How to Find the Tangent Line to a Curve in R3 Using Vector Functions?

    So the curve is defined as x-y^2 = 0, z= x . I'm supposed to find the tangent line to the curve. How do I find a slope in R3?
  25. 1

    Equation for Tangent Line at (2,2) on f(x)=xy+y^3=12

    Homework Statement Find the equation for the tangent line at (2,2) f(x)=xy+y^3=12 Homework Equations The Attempt at a Solution I really feel like I know what I'm doing here, but the key disagrees. My equation for the tangent slope comes out to be: (-y)/(x+3y^2) when...
  26. H

    Tangent Line MultiVar Calc Problem

    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=x^2+ y^2 at the point (1, 2, 5) . Homework Equations The Attempt at a Solution Normal Vector 1 is i-j Normal Vector 2 is 2i+4j-k Cross them to get...
  27. D

    Finding the points on an ellipse where the slope of the tangent line equals 1

    If there is an an ellipse x^2/9 + y^2/16 = 1, and the slope of the tangent is dx/dy = -16x/9y, how do you find what points at which the slope of the tangent is 1? I have no idea how to answer this and I've been trying for like an hour. Can anyone help me?
  28. nukeman

    How Do You Find the Equation of a Tangent Line to a Circle?

    Homework Statement Center of circle is: (3,2) Tangent point: (8,4) Question: What is the equation of the tangent line? Homework Equations The Attempt at a Solution I am just not getting it. So, would the radius be 5? Now, would i go: (x-3)^2 + (y-2)^2 = 5 ...
  29. D

    What Is the Equation for a Tangent Line Parallel to a Given Line?

    Homework Statement What is the equation of the tangent line to the curve y = Sqrt( 2x - 1) where the tangent line is parallel to the line x - 3y = 16 The Attempt at a Solution My teacher briefly mentioned to find the derivative of the given, and then plug it into the slope-y intercept...
  30. H

    Solving for the Tangent Line at (1,e^-2)

    Does anyone know how to find the equation of a tangent line to y=e^(-2x) at the point (1,e^-2) I honestly have no idea how to even start this problem when I tried it I came up with y=1(x+.27)+ln2
  31. J

    Tangent line parallel to a plane

    Hi guys, I'm stuck with a problem here: Let a curve be given by the following parametric equations: x=t, y=t^2, z=t^3. At which points is the tangent line (of the curve) parallel to the plane x + 2y + z = 0? What is the underlying principle behind this? My thoughts: The tangent line...
  32. T

    Determining perpendicular tangent line

    Homework Statement Determine the coordinates of the points on the graph of f(x) = _/'2x+1 where the tangent line is perpendicular to the line 3x+y+4 = 0 _/' -means square root Homework Equations f(x) = _/'2x+1 3x+y+4 = 0 The Attempt at a Solution I made it equal to y like y= -3x-4...do I...
  33. A

    Find parametric equations for the tangent line to the curve

    Homework Statement Homework Equations r = <x,y,z> r' = <x',y',z'> The Attempt at a Solution I started by finding the derivatives of each part of the vector and got: x= 2/sqrt(t) y= 3t^2+1 z= 3t^2-1 Then I plugged the point (5,2,0) into that and got (2/sqrt(5), 13, -1). This should be...
  34. S

    Help Solving Derivatives using Tables and Equation on Tangent Line

    Help! Solving Derivatives using "Tables" and Equation on Tangent Line 1. I got a worksheet that was given to me on Derivatives to finish for homework but hardly understand it since my teacher assigned it for the same night and she had taught it I was wondering if an explanation on how to do...
  35. P

    Tangent line to curve, derivative.

    Homework Statement See the attachment. I feel like my answer is right, but I keep getting told I am wrong. This is the first time I have taken any math in 5 years so I am figuring I must not be simplifying enough? But I can't see what I need to simplify. Perhaps I have made a dumb error and...
  36. K

    Equation of a Tangent Line - PreCalc

    Find the equation of the line with slope -1 that is tangent to the curve y = 1/(x-1). The equation of a line with slope -1 is y = -x + k The curve is y = 1/(x-1) Set the y-values to each other: 1/(x-1) = -x + k Rearrange and set equal to 0: 1 = -x² + x + k x² - x + 1 – k = 0...
  37. M

    Finding the Tangent Line of x^4+2x^2 at x=1

    Homework Statement Okay, so I was helping someone study for the AP calculus exam, and I don't really know why we got this question wrong. It says to find the tangent line to the equation f(x)=x^4+2x^2 at x=1. Homework Equations Taylor series, we'll just use first order. f(x)|_{x=x_0}...
  38. C

    Find the slope of the tangent line (ii)

    Homework Statement Find the slope of the tangent line to the curve: 2(x^2 + y^2)^2 = 25(x^2 - y^2) at the point (-3, -1) Homework Equations Implicit differentiation The Attempt at a Solution 2(x^2 + y^2)^2 = 25(x^2 - y^2) 1. 4(x^2 + y^2)(2x + 2y(dy/dx)) = 25(2x -...
  39. C

    Find the slope of the tangent line

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  40. K

    Tangent Line and Normal Line

    Homework Statement Find the lines that are tangent and normal to the curve x^2 + xy - y^2=1 at (2,3) Homework Equations Umm.. first derivative and second derivative? The Attempt at a Solution For the tangent line i found the first derivative and used the implicit differentiation(?)...
  41. L

    FInding Points of Tangent Line w/ Vectors

    Homework Statement Consider the ellipsoid 4x^2+2y^2+z^2 = 19. Find all the points where the tangent plane to this ellipsoid is parallel to the plane 2y−8x+z = 0. Homework Equations The Attempt at a Solution So I found the normal vector to the tangent, <8x,4y,2z>. I also...
  42. L

    How to Put Gradient Vector into Implicit Form?

    Homework Statement Let f(x,y) = 5y^(2)-(2x^(2)+xy) Then an implicit equation for the tangent plane to the graph of f at the point (0,-2) is Homework Equations The Attempt at a Solution I understand that I should take the derivative to find the gradient vector. For the...
  43. L

    What is the parametric form for the tangent line to y = 2x^(2)+2x-1 at x = -1?

    Homework Statement The parametric form for the tangent line to the graph of y = 2x^(2)+2x-1 at x = -1 is Homework Equations The Attempt at a Solution I am confused about where to begin this problem. Any thoughts? Thanks!
  44. B

    Minimum area between f(x) and a tangent line

    How would you write a proof that proves that the minimum area between a function and its tangent line is the tangent line evaluated at point p, where p is the midpoint on a given interval? i.e. The minimum area between x^2, and its tangent line on the interval [0,1]...
  45. D

    Parametric equations for Tangent line of an ellipse

    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 sin(t)^2 + cos(t)^2 = 1 The Attempt at a Solution After plugging 2 in for y, I get...
  46. H

    Finding the Tangent Line to a Parametric Curve

    Homework Statement This is a very basic problem, though it did confuse me a little: Find the tangent equations to the curve x=3t^2+1 \ , \ y = 2t^3+2 which intercepts the point (4,3). Homework Equations --- The Attempt at a Solution I took \frac{dy}{dx} = t =...
  47. R

    Calculus - Tangent Line Question

    Homework Statement Hello, this is a problem from the practice test for the GRE subject test. For what value of b is the line y=10x tangent to the curve y=e^{bx} at some point in the xy-plane? A) \frac{10}{e} B)10 C)10e D)e^{10} E)eHomework Equations The Attempt at a Solution For the line to be...
  48. C

    Tangent and Normal Lines at (1,0) on Curve y = pi*sin(pi*x-y)

    Homework Statement Verify that (1,0) is on the following curve and find the tangent line and normal line to the curve at the point. y=pisin(pix-y)The Attempt at a Solution i think i got it is y ' {-1/pi*cos(pi*x-y)} + pi
  49. E

    How Do You Find Tangent Points on a Unit Circle from an External Point?

    Homework Statement We are given the unit circle and the point (5,2). There are two lines that are tangent to the unit circle and they both intersect at the point (5,2). What are the points where these lines are tangent with the unit circle. Homework Equations Tangent line of a circle at...
  50. C

    Help with Derivative and Tangent Line Problem

    First Problem Homework Statement Find the derivative of x^6+y^6=18xy Homework Equations Find derivative The Attempt at a Solution 6x^5+6y^6=18*(dy/dx) Second Problem Homework Statement Verify that (1,0) is on the following curve and find the tangent line and normal...
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