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. W

    Unit Tangent Vector at a Point

    Homework Statement r(t) = costi + 2 sint j Find the tangent vector r'(t) and the corresponding unit tangent vector u(t) at point P:(.5, 3.5,0) Homework Equations r'(t) = r(t)dt u(t) = r'(t) / |r'(t)| The Attempt at a Solution r'(t) = -sinti + 2costj |r'(t)| = [sin2t +...
  2. J

    Equation of tangent plane at (2, -1, ln 7): z = ln 7 + (4/7)(x-2) - (6/7)(y+1)

    Just when I thought I got the hang of tangent planes and surfaces there comes a question I haven't quite seen before z = ln (x^{2}+3y^{2}) Find a normal vector n and the equation of the tangent plane to the surface at the point (2, -1, ln 7) So keeping the cartesian equation in mind: z =...
  3. A

    Parametric Tangent Problem driving me insane

    Homework Statement x = e^{t} , y = (t-1)^{2} , (1,1) Find an equation of the tangent to the curve at a given point by two methods. Without eliminating the parameter and by first eliminating the parameter. The answer in the book says y = -2x + 3 and I cannot see how you get it. So...
  4. J

    Finding Tangent Planes and Normal Vectors to Surfaces

    Suppose that F(x,y) = x^{2}+y^{2}. By using vector geometry, find the Cartesian equation of the tangent plan to the surface z = F(x,y) at the point where (x,y,z) = (1,2,5). Find also a vector n that is normal to the surface at this point...
  5. P

    How Do You Determine the Equation and Tangency Point of a Sphere on the X-Axis?

    Homework Statement Find the equation of a sphere of radius 9 which is tangent to the plane x-2y+2z=4 and whose center lies on the x axis. b. Determine the point where the sphere is tangent to the plane. Homework Equations The Attempt at a Solution My TA told me it might be...
  6. I

    Trouble with a tangent series expansion

    Okay, so I am supposed to expand ln(cos(x)) basing my calculations on tan(x) = x + (1/3)x^3 + (2/15)x^5 + (17/315)x^7 + ... and expanding out to that fourth term is sufficient. I am having a major brain meltdown, I can't seem to find any equivalencies present, and I'm sure there are. Any help...
  7. M

    Vector function, position and tangent vectors

    Homework Statement If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t) show that the curve lies on a sphere with center at the origin Homework Equations The Attempt at a Solution I have no idea how to even approach this...
  8. M

    How do I find the slope of the tangent line using Definition 1 and Equation 2?

    Homework Statement Find the slope of the tangent line to the curve y=x3 at point (-1,-1) (i) Using Definition 1 m=limx-->a\frac{f(x)-f(a)}{x-a} (i) Using Equation 2 m=limh-->0\frac{f(a+h)-f(a)}{h} Homework Equations The Attempt at a Solution For the first one, I got as far...
  9. B

    Solving Tangent Line Problem: Adding & Subtracting 5 Explained

    Homework Statement Ok, how do i solve this question I've attached the picture.. The answer is A but i don't understand how to get that... The Attempt at a Solution The solution is also given , but i don't understand the adding and subtracting 5. the Derivative of y=ln (e^x + 5) y'=...
  10. M

    Area between a function and its tangent

    Homework Statement Find the area of the region bounded by the graph of f(x) = 4x^2, the tangent line to this graph at P(2, f(2)), and the x-axis Homework Equations Integral of [f(x)-g(x) dx] The Attempt at a Solution I first tried to find the equation for the tangent line The...
  11. S

    Tangent Lines to f(x) at Point (2,7)

    Homework Statement f(x) = 4x-x2 Question: Find the equations of the lines that pass through P(2,7) and are tangent to the graph of f(x). (P is not on f(x).) Thats all the problem states. Homework Equations f(x) = 4x-x2 Point (2,7) The Attempt at a Solution Ive tried...
  12. K

    Is There a Simpler Way to Find a Normal Using the Given T-Value?

    Write a in the form a=a_{Y}T+a_{N}N without finding T and N at the given t-value. r=t\widehat{i}+3t^{2}\widehat{j}+t^{2}/2\widehat{k} , t=2 The first thing I did was to find the first derivative. Which turned out to be i+6tj+tk Then I found the length \sqrt{1+37t^{2}} Then I took the...
  13. S

    Equation of a line that is tangent to f(x)

    In order to find the equation of a line that is tangent to f(x) and goes through point P on f, you got to find the derivative of f(x) at P, but how would you go about solving a problem where you have to find the equation of a line tangent to f(x) that goes through point P, but P is NOT on the...
  14. F

    Calculting the slope of the tangent to find Instantaneous velocity

    EDIT: Ugh... I meant "calculating", of course. This is an assignment that was due on the 16th. It's already been handed in, I just need some help with a question that I obviously was not able to answer in due time. I am coming back from a long, long hiatus from my academic pursuits and so some...
  15. I

    Where Does a Ball Depart on a Tangent from a Circle?

    Homework Statement A ball that is circling with x=cos(2t), y=sin(2t) flies off on a tangent at t=Pi/8. find its departure point and its position vector at a later time t (linear motion; compute its constant velocity v). Homework Equations v=dR/dt T=v/|v|= (dR/dt)/(ds/dt)=dR/ds...
  16. U

    How Do I Simplify a Fraction Within a Fraction?

    Find the slope of the functions graph at the given point. F(x) = x / x-2 point (3,3) f(x+h) - f(x) / h is what we have to use to find the answer. so I've plugged it all in and have came to this.. ((3+h) / (3+h-2)) - 3 / h I need some help with my simplification...
  17. R

    Find Tangent Line to Path: x(t)=4costi-3sintj+5tk, t=\pi/3

    Find an equation for the line tangent to the given path at the indicated value for the parameter. x(t)=4costi-3sintj+5tk, t=\pi/3 So what I did here was take x'(t) and then plugged in \pi/3 after that to get an equation containing i, j, and k. x'(t)= -4sinti-3costj+5k...
  18. 2

    Unit Vectors for Tangent Line at (\pi/6, 1)

    Homework Statement Find the unit vectors that are parallel to the tangent line to the curve y=2sinx at the point (\pi/6, 1). Thereafter, find the unit vectors that are perpendicular to the tangent line. Homework Equations The Attempt at a Solution I took the derivative of y=2sinx and...
  19. C

    Finding the Tangent of a point on a curve (problem driving me crazy )

    Homework Statement I've done this problem about 5 times, several different ways following many different examples but I can't seem to find the right answer or even figure out how they got one of the answers (y answer is given) The Attempt at a Solution r(t) = (1+4*sqrt(t), t5-t...
  20. JasonJo

    Tangent bundle of a differentiable manifold M even if M isn't orientable

    This is a problem many of the grad students have probably encountered, it's in Chapter 0 of Riemannian Geometry by Do Carmo. Do Carmo proved that the tangent bundle of a differentiable manifold is itself a differentiable manifold by constructing a differentiable structure on TM, where M is a...
  21. K

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

    Homework Statement http://img214.imageshack.us/img214/4673/mathproblemnw5.png Homework Equations lim x->a\frac{f(a+h)-f(a)}{h} The Attempt at a Solution Ive tried so many times to figure this out. I first substituted the equation into the formula above and multiplied by the...
  22. F

    Easy tangent line problem but I don't know how to approach it

    Let f be the funtion given by f(x) = 4x^2 - x^3 , let L be the line y = 18 - 3x, where L is tangent to the graph of f. Show that L is the tangent to the graph of y= f(x) at the point x = 3. I'm equaling both graphs like this: 4x^2 - x^3 = 18 - 3x and then I isolate everything...
  23. A

    Calculating acceleration from 2 tangent lines

    I have a position vs. time graph which is slightly curved. I found the slope of 2 tangent lines which I know are the velocity. My question is how do I get the acceleration using these 2 slopes. One slope is 295cm/s and the other is 575cm/s. I know that avg acceleration is final velocity -...
  24. kreil

    Taylor series expansion of tangent

    Homework Statement find the first four nonzero terms in the power series expansion of tan(x) about a=0 Homework Equations \Sigma_{n=0}^{\infty} \frac{f^n (a)}{n!}(x-a)^n The Attempt at a Solution Well the series has a zero term at each even n (0,2,4 etc) for n=1 I got x, for...
  25. H

    What Are the Equations of Tangent Lines to a Circle Without Using Calculus?

    Homework Statement You have a circle with the equation x^{2} + (y + 1)^{2} = 1. You can draw to two tangent lines to that circle that intersect the point (0,1) What are the equations of these lines? And you can't use any calculus, derivatives and the like. Homework Equations y=mx+b...
  26. A

    Finding a Tangent Line to a Curve: Exploring Slopes

    A tangent to a curve is a line that touches the curve but at what specific point where there is a slope to the curve. Equations like y = X^2 have many slopes because the curve is shaped differently at different points. We can choose 2 points on the graph and find the slope but it will...
  27. CalleighMay

    Symmetric equations of tangent lines to curves

    Hello, my name is Calleigh and i am new to the forum! I am in Calculus II and have a few questions on some problems. I am using the textbook Calculus 8th edition by Larson, Hostetler and Edwards. Could someone please help me? The problem is on pg 950 in chapter 13.7 in the text, number 46. It...
  28. B

    Equation of a tangent to a complicated function

    The problem: The y-intercept of the line tangent to y=(x4-2x2-8)ecosx2 at x=1 is (a) -15.499 (b) -25.999 (c) -41.448 (d) 15.449 (e) 25.99 To find the y-intercept I have to find the whole equation of the line y=f(1) m=f'(1) x=1 b=?? My problem is that I don't think I got the the...
  29. T

    Normal line to a tangent plane

    Homework Statement I have two problems that don't seem to match. The task is to find the parametric equation for the normal line to the tangent plane. I'm seeing that the vector value of "z" is (-1) with one formula, but not in the other. These two seem to be the same, but the first was solved...
  30. C

    Answer Check - Find an equation of the tangent line

    Homework Statement Find an equation of the tangent line to the curve y = 5x3 at the point (-3,-135). Enter your equation in y = form. Homework Equations The Attempt at a Solution Taking the derivative of the equation I get y = 15x2, following that I plug in the x value of...
  31. E

    Equations of circles tangent to graph

    Two circles of radius 3√2 are tangent to the the graph y^2 =4x at (1,2). Find the equation of each circle. I have found the derivative of the graph, which is 1/√x. I know that the equation of the circle is X^2 + Y^2=r^2 where r is the radius so the equation of each circle is X^2 + Y^2=18...
  32. C

    Solving the Limit of Tangent Function at h=0

    Solve: lim x->0 (tan 3(x+h)-tan(3x))/h i hv no clue where to start =(
  33. H

    Help finding equation of the tangent line to the graph

    Homework Statement Find the equation of the tangent line to the graph of x^2 - xy + y^2 = 19 (where y=y(x)) at (3,-2). Homework Equations The Attempt at a Solution So, this is what I did: d/dx(x^2-xy+y^2) = (19)d/dx 2x*dx/dx-dy/dx+dy/dx+2y(dy/dx)=0 2x+2y(dy/dx)=0 And I...
  34. T

    Find the equation of tangent line

    Homework Statement Find equation of the tangent line at t=0 x=2sint y=tant The Attempt at a Solution Really don't know where to start, can someone give me the hint, don't solve it, i want to try it, just need a hint
  35. R

    Tangent Help: Expert Solutions for Your Queries

  36. T

    Length of Tangents from (10,0) to (x-3)² + (y-4)² = 5² Circle?

    Homework Statement I have the circle of equation (x-3)² + (y-4)² = 5² Find the exact length of the tangents from the point (10,0) to the circle Homework Equations The Attempt at a Solution Honestly I can't acutally visualise on a co-oridinate graph what it's asking. A tangent...
  37. C

    Quick Help with Tangent Sum Formula: Solving for Tan 15 and Tan 30

    Hi, I'm studying for my pre-calc test tomorrow and I've run into a snag. I just can't seem to get the correct answer for a Tangent sum problem, and I'm hoping someone could help me out with it. It goes like this: Sum Formula for Tangant: Tan (a+b) = Tan a + Tan b/1-Tan a * Tan b Problem...
  38. J

    How can I find the equation of tangent lines from a circle to point A?

    Circle: (x-2)² + (y+1)² = 25 Point A: (11,8) Therefore Center of Circle = (2,-1) and Radius = 5 Looks somewhat like: O> Basically there are two tangents coming from the edge of a circle with the above equation to a point (pt. A). I have figured out to make two right triangles by...
  39. B

    Finding Tangent Lines at a Given Point on a Graph

    Verify that (1,1) is a point on the graph of y + ln xy = 1 and find the equation of the tangent line at (1,1) to this graph how do you go about answering this?
  40. T

    Finding a value of theta for which a tangent line is horizontal

    Homework Statement Find the smallest positive value for `theta` for which the tangent line to the curve `r = 6 e^(0.4 theta)` is horizontal The Attempt at a Solution I tried to use the (r'sin(theta)+rcos(theta))/(r'cos(theta) - rsin(theta)) formula to get the tangent line, which I got...
  41. B

    Proof: Everywhere Tangent to Curve?

    Proof: Everywhere Tangent to Curve?? If the function v depends on x and y, v(x,y) and we know there exists some function psi(x,y) such that vx = partial w.r.t (y) of psi vy= -(partial w.r.t (x) of psi) show that the curves psi(x,y) = constant, are everywhere tangent to v.
  42. A

    Answer Equations of Lines Tangent to xy+y=2 and xy+x=2

    i have 2 simmilar questions 1) Find the equation of the line tangent to the curve at the point xy + y = 2 ; (1, 1) and i did this xy + y = 2 y + xy' + y' = 0 xy' + y' = - y y'(x + 1) = -y y' = -y/(x + 1) y' = -1/2 y- y_1 = m(x - x_1) and i hope so this is OK but how can i...
  43. D

    Find the Tangent Lines of y=x/(x+1) Through (1,2)

    Can someone help me with this problem? Consider the curve defined by y=x/(x+1). (1,2) is a point NOT on the curve. Find the equations of the two tangent lines to the curve passing through the point (1,2).
  44. K

    What Other Point(s) Have the Same Tangent Plane?

    [SOLVED] Tangent plane to surface Homework Statement Find an equation for the tangent plane to the surface z^2 = x^2 + 2y^2 at the point P = (1,2,3). Which other points on the surface have the same tangent plane? 2. The attempt at a solution I find the derivatives: fx =...
  45. Q

    Trigonometric formulas for tangent?

    using those formulas, how would you solve tan 11 pi ----- 8 and cot 195 also, how would you prove tan (x +y ) = tan x and tan ( pi/4 -x) = 1 - tan x --------- 1 + tan x thank you so much for any help
  46. T

    Finding the Unit Tangent Vector for a Given Curve

    Homework Statement Let bar r(t) = < -1t^(2)+2, -3e^(5t), -5sin(-4t) > Find the unit tangent vector `bar T(t)` at the point `t=0` The Attempt at a Solution Attempt: r(t) = -1t^2 + 2, -3e^5t, -5sin(-4t) v(t) = -2t, -3e^5t, -5cos(-4t)*-4 T(t) = (-2t - 3e^(5t) -...
  47. R

    Tangent and normal acceleration, curvature radius

    What exactly are the tangent and the normal accelerations of a projectile motion and how are they expressed mathematically? What is curvature radius? What is its expression? How is it derived ?
  48. D

    Tangent Planes: Proof of Tangential Surfaces at (1,2,3) with Differentiation

    Two surfaces are said to be tangential at a point P if they have the same tangent plane at P . Show that the surfaces z = √(2x²+2y²-1) and z = (1/3)√(x²+y²+4) are tangential at the point (1, 2, 3). differentiate first then evaluate both at 1,2,3
  49. N

    How Do You Find Points with Horizontal Tangents on f(x) = 2sinx + (sinx)^2?

    Homework Statement Find all points on the graph of the function at which the tangent line is horizontal. f(x) = 2sinx + (sinx)^2 Homework Equations Hi guys I have an issue. I do not know how to approach this problem. I know the chain rule but I still do not know how to solve the...
  50. M

    Equation of a tangent, with a slope of 'e'.

    Help my guys. I'm so hopelessly lost on this question... Problem: Determine the equation of the tangent to the graph of y=ln x that has slope e. The answer in the book is y=ex-2. Please help. :smile:
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