Tangent Definition and 1000 Threads

Hi all, I'm quite confused concerning the definition of tangent vectors and tangent spaces as presented in Munkres's Analysis on Manifolds. Here is the book's definition: Given ##\textbf{x} \in \mathbb{R}^n##, we define a tangent vector to ##\mathbb{R}^n## at ##\textbf{x}## to be a pair...

Dear all, in what sense the tangent space is the best approximation of a manifold? The idea is clear to me when we think about a surface in Rn and its tangent plane at a point. But what does this mean when we are referring to very general manifolds? In what sense "approximation" and in what...

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...

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 →...

Homework Statement Need to find the tangent to the curve at: e^(xy) + x^2*y - (y-x)^2 + 3 I just implicitly differentiate the expression to find the gradient and then use the points given to find the equation, right? Or does this involve partial differentiation? Homework Equations...

Homework Statement 1/(u^2+4) Homework Equations The Attempt at a Solution I know that 1/(x^2+1) is the derivative of the inverse tangent function, and that is proved by using tany = x, derivative of both sides with secx=(1+tan^2x) and tan^2x = x^2. I don't know how to use the...

Homework Statement Show that the curve r = (t2,t3-t) Intersects itself at (1,0), and find the slopes of the tangents at this point. Homework Equations The Attempt at a Solution Okay I can show it intersects itself there, but what I am having trouble with is when they say slopes...

Find equations of the tangent lines to the graph of f(x)=\frac{x}{x-1} that pass through the point (-1, 5). Well, first I took the derivative, and afterwards, I made the connection that the derivative was a slope at any instant on the graph. By this, I inferred that f'(x) = m. I knew that the...

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...

When i was at school i used to think that any line that touches a curve at only one point is called as the tangent at that point to the curve! But after reading derivative i think this definition of tangent is not correct at all conditions! For example, x-axis cannot be called as tangent at...

That is, adding up the differential changes in angle between two arbitrarily chosen points on a function, to find the total change in angle between the tangent lines of those two points. How can this be done?

Considering f(x) = tan(x) * 5 / 8 ... how can I find the length of the curve, specifically, between (0, 0) and (1, 1) ? if anyone can help I would be happy. Thanks Keeaga

Homework Statement If the power of the secand is even and positive.. \int sec^{2k} x tan^{n} x dx = \int (sec^2 x)^{k-1} tan ^n x sec^2 x dx The Attempt at a Solution The way I see it, sec^{2k} x = sec^2 x dx * sec^k x dx the next step seems to be to break down sec^k, but on closer...

I've never been able to visualize how the tangent to a curve and the area under a curve are inverses of each other, can anyone give some intuitiveness to this?

Homework Statement See attached file. Homework Equations The Attempt at a Solution I've only been able to do part (a) of this question. I ended up with: tanz= i ({\frac{1-e^{(2iz)}}{1+e^{(2iz)}}}) I'm not sure how to approach the next two parts. If anyone could give me any...

Homework Statement A curve C is defined by the parametric equation x=t^2, y=t^3-3t. a) show that C has two tangents at the point (3,0) and find their equations. b) find the points on C where the tangent is horizontal Homework Equations y-y1=m(x1-x), (dy/dt)/(dx/dt)=m, when dy/dt=0...

How do you determine the equation of all possible tangents to a curve (say, a parabola) that pass through a given point that is not on said curve? This is more of a conceptual question, and it's not homework, so I thought it fit in this forum. I think there might be a question like this on the...

Homework Statement A curve is defined by the parametric equations: x = 2t^3 y = 2t^2 t =/ 0 1)Prove that the equation of the tangent at the point with parameter t is 2x - 3ty + 2t^3 = 0. Proven, and I've no problem with this part. 2.)The tangent at the point t = 2 meets the curve...

Homework Statement Ok so this is a question from last years past paper of my course: X= 1/2 intersects the circle that is centered at origin at two points, one of which is in the lower half plane y<0; what is the equation of the tangent tot the same circle at this point? Homework Equations...

So I'm kinda new to Physics Forums but I've been using threads as guides for about a year now. Basically, I'm hardcore studying for my Calc III exam (the final is in a few weeks) and I came across an interesting lapse in my understanding (well many in fact, but one in particular). First of...

The tangent vector is defined as : T=v/||v|| Where v is some vector. Then how is T the tangent vector to v? It's the unit vector in the direction of v right?

Homework Statement r=2-3cosθ Find the tangent line at any point, and at the point (2,∏) Find the tangent line(s) at the pole Homework Equations Do I have to use x=rcosθ and y=rsinθ to convert it to rectangular to find slopes? The Attempt at a Solution Is the point 2∏ even a...

D is a set of all points (x,y) in R2 distinct from (0,0). I have the funtion f: D --> R which is defined by: f(x,y) = (2xy)/(x2+y2 Im asked to find the equations (in plural) of the tangelnt plane and the normal line to the graph of the function at (1,2) My attempt: I use the tangent plane...

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)...

1. show that there is no line through the point (2,7) that is tangent to the parabola y =x^2 +x 2. y-y1=m(x-x1) 3. y'=f'(x)=2x+1 m1=2x +1 m1=2(2) +1 =5 m2=((x^2 +x)-7)/(x-2) m1=m2? ((x^2 +x)-7)/(x-2)=5 x^2-4x+3=0 2^2-4(2)+3=/=0 -1=0 I'm thinking that i would...

So i was considering minkowski space which is a 4-d manifold, why is that we use the tangent and cotangent space, to construct tensors on the space? The definition of a manifold says that the space is locally homeomorphic to Euclidean space. So is the tangent space and cotangent space...

Homework Statement Find derivative of tan^{-1}(\frac{3sinx}{4+5cosx}) Homework Equations deriviative of tan^{-1}=\frac{U'}{1+U^{2}} The Attempt at a Solution I found U'= \frac{12cosx+15}{(4+5cosx)^{2}} 1+U^{2}=1+\frac{9sin^{2}x}{(4+5cosx)^{2}} I think my components are correct but my...

Homework Statement Find the equations of the lines that pass through (0,0) and are tangent to x^2 - 4x + y^2 + 1 = 0 My confusion I've been given a problem of this sort recently, except now it involves implicit differentiation. I know "how" to get to the correct answer. I just...

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...

Homework Statement Consider the plane curve \overrightarrow{r(t)}=e^tcost(t)\hat{i}+e^tsin(t) \hat{j} Find the following when t= ∏/2 Part A: \hat{T}(t) Part B: \hat{B}(t) Part C: \hat{N}(t) Homework Equations \hat{N}(t)=\frac{\hat{T}(t)}{||\hat{T}(t)||}...

Homework Statement The equation 5x^2 - 6xy + 5y^2 = 16 represents an ellipse. Determine two points on the ellipse at which the tangent is horizontal. Homework Equations The Attempt at a Solution I find the derivative of the equation: (-10x + 6y) / (-6x + 10y) = 0 iff...

Homework Statement Question: "Find the unit tangent, normal and binormal vectors T, N, B, and the curvature of the curve x = 4t, y = -3t^2, z = -4t^3 at t = 1." Answer: T = 0.285714285714286 i - 0.428571428571429 j - 0.857142857142857 k N = -0.75644794981871 i + 0.448265451744421 -...

Homework Statement My problem is one pertaining to my Vector Calculus course. The assignment is asking us to "Find equations for the planes tangent to z = x2 + 6x + y3 that are parallel to the plane 4x − 12y + z = 7." The problem I'm having with the problem is the plural aspect. It states...

Homework Statement Find the equation of the line with slope -1 that is tangent to the curve y=1/x-1 Homework Equations y=1/x-1 The Attempt at a Solution Slope of -1 means y=-1x+k So... -1x+k = 1/x-1 I don't know how to rearrange this into a quadratic equation so that I...

Homework Statement Let c(t) be a path and T the unit tangent vector. What is \int_c \mathbf{T} \cdot d\mathbf{s} Homework Equations The unit tangent vector of c(t) is c'(t) over the magnitude of c'(t) : \mathbf{T} = \frac{c'(t)}{||c'(t)||} The length of c(t) can be represented by ...

Homework Statement attached. Homework Equations The Attempt at a Solution I thought the problem was easy, but my answer is wrong (aparently; I still disagree.) First I defined x = (y^3)(z^3) to be a surface of function F So F(x,y,z) = (y^3)(z^3) - x = 0 Then, the...

Homework Statement lim x^3-2x^2+x/tanx x->0 The Attempt at a Solution All i know is that tan is going to break up into sinx/cosx so the equation will look like this lim x^3-2x^2+x/(sinx/cosx) x->0 I haven't worked with cubic or quadratic functions yet so I don't know...

Homework Statement Find the equations of all the lines through the origin that are tangent to the curve y = (some complicated cubic function) I looked up the question in google and found a much simpler example, y = x^2 passing through (1,-1). However, I don't even get what's going on...

So my problem is this: I need to figure out the center of a circle given two points. At one of the points, I know the tangent angle. So I know (x1, y1, θ1) and (x2, y2) and need to find (xc, yc). I also need to do this on a computer so I need some sort of closed-form solution. The way I...

Homework Statement Find equations of both the tangent lines to the ellipse x2 + 4y2 = 36 that pass through the point (12, 3). Homework Equations The equation of an ellipse is x2/a2 + y2/b2 = 1. I converted the given equation to x2/36 + y2/9 = 1 by dividing each value by 36. The...

Homework Statement Find the unit tangent vector at the indicated point of the vector function r(t) = e(19t)costi + e(19t)sintj + e(19t) kT(pi/2) = <___i+___j+___k>Homework Equations r'(t) / |r'(t)| The Attempt at a SolutionAnswers: 19e(19*∏/2)(cos(∏/2)-sin(∏/2)) /...

Homework Statement How do I solve this equation? lim x>0 (x^3 - 2x^2 + x)/tanx I don't know what to do here, please help. Thank you

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...

I have attached both the question and the solution. I just have questions as to why the solution is the way it is (sorry if they seem stupid but, while I get how to do it mechanically, I don't understand the fundamental reasoning as to why anything is being done): 1) Why are we taking the...

Homework Statement Find the co-ordinates of all points on the curve f(x)= x3 whose tangent lines pass through the point (a,0) Homework Equations f '(x) = nxn-1 The Attempt at a Solution I am really not sure how to attack this question. My initial thoughts are to find f '(x) then...

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...

Homework Statement So I'm a little confused about what a tangent space is. Is it the same as the equation of the tangent plane in lower dimensions? My notes define the tangent space as follows. Let M be a hypersurface of Rd. Let x(s) be a differentiable curve in M such that x(0)=x0 is in...

The curve $\displaystyle y-e^{(xy)} + x=0 $ has a vertical tangent at which point?? I started to differentiate it, then equating dy/dx to 0, then how should i proceed??

Homework Statement Find the direction of the line tangent to the curve x^4+y^4=32 at the point (2, -2) Homework Equations Anything goes, we're in vector calculus now. The Attempt at a Solution So, to find the tangent line, I was thinking of taking the gradient, but I'm not...

Homework Statement f (x) = e^(3x) + sin(2x) + 3x +1 (a) Find a vector V that is tangent to the graph of y = f(x) at the point ( 0, 2). (b) Find a vector N that is perpendicular to the graph of y = f(x) at the point ( 0, 2). 2. The attempt at a solution The first step I took is to...