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. Greg Bernhardt

    Insights What is a Tangent Line? A 5 Minute Introduction

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  2. PsychonautQQ

    Is There a Point of Tangency Between f(x) = x^(1/2) and y = (x/4) + 1?

    Homework Statement Q: Does the graph of f(x) = x^(1/2) have a point of tangency with the line y = (x/4) + 1? Homework Equations lim x->a (f(x) - f(a)) / (x-a) The Attempt at a Solution If the limit exists of the relevant equation than there is a point of tangency. So I'm...
  3. R

    Finding the Tangent Line at a Point on a Curve

    I need to find the tangent line to the curve xe^Y+ye^x=1 at the point (0,1). I took the derivative and found to be: dy/dx=-(ye^x-e^y)/(xe^y+ye^x) I set that equal to 0 so: 0=-ye^x-e^y I have tried using a natural log to get y on one side and x on the other, but so far no good. How can...
  4. M

    Point of intersection of tangent line with another line

    Homework Statement Find the pt. at which the tangent line to the curve x=3t^2 - t, y=2t+t^3 at t=1 intersects the line y=2-x. Homework Equations Possibly <6t-1, 2+3t^2> if the tangent is not already present The Attempt at a Solution I am confused about how to go about solving...
  5. karush

    MHB How Do You Find a Common Tangent Line to Two Parabolas?

    https://www.physicsforums.com/attachments/2227 The region $R$, is bounded by the graphs of f(x)=x^2 -3 and g(x)=(x-3)^2, and the line T, as is shown in the figure above. T is tangent to the graph of f at point (a,a^2-3) and tangent to the graph of g at point (b,(b-3)^2) a. Show that...
  6. MarkFL

    MHB Find Tangent Line to Curve (x+2y)^2+2x-y-3=0: Answer at Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  7. C

    Tangent line, to a surface, through a point, parallel to a plane

    Hey y'all, this is my first post. I am currently stuck on a multivariable question. Please let me know if you can help. Homework Statement The point, P = (1, 2, 2) lies on the surface z = x^2 + y^2 -3x. Find parametric equations for the tangent line to the surface through the point P parallel...
  8. H

    Is the tangent line really a point when the slope is zero?

    Homework Statement I was given parametric equations. x(t) = a(t) y(t) = b(t) z(t) = c(t) where a, b, and c are functions that depend on t. I was supposed to find equation of the tangent line at t = f given: x(f)= m y(f) = n z (f) = o where m,n,o are some constant numbers...
  9. J

    Finding the slope of the tangent line using the f(a+h)-f(a)/h form?

    Find the slope of the tangent line to the curve f(x)=x-x^3 at the point (1,0) So, I went through and plugged (a+h) in everywhere I saw "x" (a+h)-(a+h)^3 Factoring: (a+h-a^3-3a^2h-3ah^2-h^3) I'm really stuck on what to do next...I don't see anything you can cancel/pull out? I know...
  10. H

    Finding the tangent line at pi/3 for the following equation

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  11. Lebombo

    Find an equation of the tangent line at the given point on the curve

    Given the parametric equations, find an equation of the tangent line at the given point on the curve. Homework Statement Find an equation of the tangent line at each given point on the curve: x = 2cotΘ and y=2sin^{2}θ at point (\frac{-2}{\sqrt{3}},\frac{3}{2}) Homework...
  12. S

    MHB Find the equation of the tangent line pt 3 (Need someone to check my work)

    find the equation of the tangent line.y = e^{3x + cos x} @ x = 0 y1 = e^{3(0) + cos(0)} = e^1 = e y1 = e y = e^{3x+cos x} y' = e^{3x + cos x} * (3 - sinx) m = 3e y - e = 3e(x - 0)
  13. S

    MHB Find the equation of the tangent line (Can someone check my work?)

    Find the equation of the tangent line to g(x) = arctan(x) + ln(x) @ x = 1 y - \frac{\pi}{4} = \frac{3}{2}(x - 1)
  14. S

    Help with basic Calculus (Equation of a Tangent Line) Please?

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  15. P

    Find Tangent Line: f(2) & f'(2) at y=4x-5

    1. If an equation of the tangent line to the curve y=f(x) at the point a=2 where is y=4x-5, find f(2) and f'(2). Homework Equations m=\frac{f(x)-f(a)}{x-a} The Attempt at a Solution To be honest, I really don't know where to start. Here's what I have so far...
  16. MarkFL

    MHB Solving Point of Tangency for AP Calculus BC Problem

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

    MHB Katie's Question: Find f(2) & f'(2) of Tangent Line y=4x-5

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

    MHB How Do You Find the Equation of a Tangent Line at a Given Point?

    The line y = ax + b is tangent to y=x3 at the point P = (-3,-27). Find a and b. I'm pretty lost on this one. Here's my initial thoughts. Find (a) first. (a) is equal to the slope which is the derivative of x3 So (a) would be equal to 3(-3)2 ? Not quite sure where to start on this one.
  19. PsychonautQQ

    Polar Coordinates Tangent line

    Homework Statement I don't know how to make theta so ∅ = theta. find the slope of the tangent line at r = sin(6∅) when ∅ = pi/12 Homework Equations y=rsin(6∅) x=rcos(6∅) r=sin(6∅) tangent line equation y-y' = m(x-x') m = dy/dx The Attempt at a Solution when ∅ = pi/12 then...
  20. PsychonautQQ

    Finding the equation for a tangent line (Parametric Equation)

    Homework Statement Find an equation of the tangent to the curve given by x=tan(∅) y=sec(∅) at the point (1,sqrt(2)) The answer should be in the form y=f(x) without ∅ The Attempt at a Solution Tangent line equation... y-y*=m(x-x*) m = dy/dx m = sec(∅)tan(∅) / (sec^2(∅) m = tan∅...
  21. C

    MHB Finding the equation of a tangent line at a given point (tan and e^x questions)

    I don't need an answer, I'm just stumped as to how to properly turn these into point slope form to find the tangent, can anyone guide me through this? Help would be greatly appreciated. I believe I understand the formulas that are used to solve problems such as these. It starts by finding the...
  22. 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.
  23. J

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

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  24. 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...
  25. 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.
  26. 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...
  27. 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...
  28. 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.
  29. 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...
  30. 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...
  31. 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...
  32. 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...
  33. 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.
  34. 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...
  35. 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...
  36. 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∏
  37. A

    Equation of Tangent Line to Curve at Point

    Homework Statement Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Homework Equations x = t \\ y = e^{-4t} \\ z = 5t - t^5 \\ P = (0, 1, 0) The Attempt at a Solution \vec{r}(t) = < t, e^{-4t}, 5t - t^5 > At the point...
  38. T

    Find the equation of y^2=x(x-3)^2 of tangent line at (3,0)

    Homework Statement Find the equation of y^2=x(x-3)^2 of tangent line at (3,0) Homework Equations Given above. I think implicit differentiation is involved or no since there is no xy's on the same side? The Attempt at a Solution Anyways... My attempt: 2ydy/dx = x*2(x-3)*1...
  39. C

    Finding Slope of a Tangent Line to a Parabola

    Homework Statement I've got the equation of a parabola y=2x^2-4x+1 with point (-1,7) and a tangent line running through it the point. I'm supposed to find the equation of the line. Simultaneously solve this equation with that of the parabola, place the results in form ax^2+bx+c, and find the...
  40. L

    Find the y-intercept of the tangent line to: -.4/ sqrt(3 + x) at

    1.Find the y-intercept of the tangent line to: y= -.4/√(3 + x) at [2.5, -.170560573084488] 2. So I thought the first step would be to find the slope of the tangent line. I think we find the slope of the tangent line by taking the derivative. So I am going to use the Quotient Rule to take...
  41. S

    How to find equation of tangent line?

    Find the equation of the line tangent to f(x)= 3x^3 + 2 at x = 1. a) y = 9x-4 b) y = 9x+5 c) y = 3x 2 d) y = 3x+1 e) Not enough information given. Im confused on this one, but I am thinking about E, because it doesn't specify if the equation should be parallel or perpendicular to the...
  42. M

    MHB Approximation Problems (Finding an equation of a Tangent Line)

    I am asking for simple guidance on this problem. f(x) = 3x^2-1, (2,11)I do believe I need to obtain an equation for tan line so first step I think is to use point slope or slope intercept (a friendly reminder to the name of formula would be very nice :)) y - ysub1 = m(x-xsub1) = y -...
  43. R

    Finding the tangent line using implicit differentiation.

    Homework Statement The equations ##2x^3y+yx^2+t^2=0##, ##x+6+t-1=0## implicitly define a curve $$f(t) = \begin{pmatrix} x(t)\\y(t) \end{pmatrix}$$ that satisfies ##f(1)=\begin{pmatrix} -1\\1 \end{pmatrix}.## Find the tangent line to the curve when ##t=1##. Homework Equations The...
  44. T

    Find equation of tangent line of tan(xy^2)=(2xy)/pi (implicit diff.)

    Homework Statement Find the equation of the tangent line of tan(xy2)=(2xy)/\pi at (-\pi,1/2) Homework Equations The Attempt at a Solution I managed to get the equation into its dy/dx form and for the slope to be (1-.5pi)/(2pi-2pi2) This seems far to complicated to be correct though.. can...
  45. T

    Find the points on a graph at which the tangent line is parallel

    Homework Statement Find the points on the graph y=x^3/2 - x^1/2 at which the tangent line is parallel to y-x=3. Homework Equations The Attempt at a Solution First I found that the derivative of y=x^3/2 - x^1/2 is 1x. I then rewrote the other line as y = 3+x and found the...
  46. B

    Determine the equation of the tangent line

    Homework Statement \frac{3x+6}{2-x} at x=3 Homework Equations y - y_{o} = m(x-x_{o}) The Attempt at a Solution f(3) = -\frac{15}{4} m = \frac{3}{0} DNE I have to write the equation in the form of the point-slope formula. I can get x_{o} and y_{o}, but I am...
  47. B

    Vector valued functions: finding tangent line

    Homework Statement Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the space curve at point P r(t)= <2sin(t), 2cos(t), 4sin2(t)>, P(1, √3, 1) The Attempt at a Solution I found T(t) using the formula T(t)= r'(t)/||r'(t)|| r'(t)= <2cos(t)...
  48. J

    Tangent Line to f(x) Without Specified Point

    Homework Statement Hello again. The question asks me to find an equation of the tangent to the graph: f(x)= - sin^2 x + 1/2, ~x~\epsilon~[0, \frac{\pi}{2}] which makes an angle of 135° with the x-axis (measure anti-clockwise from the positive x-axis). Assume that the scales along...
  49. J

    Tangent Line to f(x) Without Specified Point

    Homework Statement Hello again. The question asks me to find an equation of the tangent to the graph: f(x)= - sin^2 x + 1/2, x \epsilon [0, \frac{\pi}{2} which makes an angle of 135° with the x-axis (measure anti-clockwise from the positive x-axis). Assume that the scales along the...
  50. G

    Find the tangent line that passes through the origin

    Homework Statement This problem is 2.95 of University Physics, 11th edition. Catching the Bus: A student is running at her top speed of 5.0 m/s to catch a bus, which is stopped at the bus stop. When the student is still 40.0 m from the bus, it starts to pull away, moving with a constant...
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