Tangent Definition and 1000 Threads

Homework Statement Find an equation of the tangent plane to the vector valued function at the origin, (0,0,0). Homework Equations The Attempt at a Solution I don't really know how to start. I've been reading and searching around for quite a bit. I know how to do the problem with a regular...

Let M be an n-dimensional manifold, with tangent spaces TpM for each point p in M. Let F(M) be the vector space of smooth functions M --> R, over R, with the usual definitions of addition and scaling. Tangent vectors in TM can be defined as linear functionals on F(M) (Fecko: Differential...

Ok... so I had this down and now I am all confused ;/ I am not posting in the homework sections because its not for homework although I will gie an example of a problem... I just want to understand why/how they use these to find the force/tension... Example: A 20 kg loudspeaker is...

f(x, y, z) = x2 + sin(y) - 2z2 = 0 defines a surface in 3 dimensions. First I need to find the equation of the normal line to the surface at point P0(2pi, 0, 3/2). Then, I need to find the point which is at a distance of 4 from the tangent plane at the point P0 Equation for the normal line at...

I'm completely lost on this question, and it's due tomorrow morning. Help? Homework Statement What point on y=x2+z2 is the tangent plane parallel to the plane x+2y+3z=1?Homework Equations y=x2+z2 x+2y+3z=1The Attempt at a Solution I have no idea what to do... Thanks!

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...

Spivak: Diffrential Geometry, Vol. 1, p. 64: So a tangent vector is a pair (p,v); the tangent bundle T \mathbb{R}^n is the set of all such pairs, in this case, \mathbb{R}^{2n}. But what is \mathbb{R}^n_{\enspace p}? Is it (1) A synonym for T \mathbb{R}^n, the tangent bundle; (2)...

Hello All, I need help in my Calc 3 class and I decided to come here for homework help. What I'm looking for is someone to just check my work for a couple of homework problems. I've already done the problems, I would just like my work checked. Anyone who helps, your kindness is greatly...

I am unable to comprehend the proof for tangent line to conics. Here is the proof as per the book (Multiview Geometry by Hartley and Zisserman). Everything is in homogeneous coordinates. The line l = Cx passes through x, since lT x = xT Cx = 0. If l has one-point contact with the conic...

Homework Statement xy +yz + zx = 3 (1,1,1) Homework Equations equation of tangent plane is z-z0 = fx(x0,y0)(x-x0) +fy(x0,y0)(y-y0) The Attempt at a Solution Right, I've been a few of these exercises, however, this is the first one I've seen that equals a number and not "z". So...

Homework Statement Graph the surface and the tangent plane at the given point. (Choose the domain and viewpoint so that you get a good view of both the surface and the tangent plane.) Then zoom in until the surface and the tangent plane become indistinguishable. Homework Equations...

Hi, I am trying to understand the concept of tangent map and following the ebook of S S Chern. I am a bit confused about the derivation of the tangent map acting on the basis I tried for sometime to type out the equation but it appears I am having problems with the display and not sure what is...

1. This is problem 2.10 from the book "Calculus of Several Variables by C.H Edwards": Let the mapping F: R2->R2 be defined by F(x1 , x2) = (sin(x1 - x2), cos(x1 + x2)). Find the linear equations of the tangent plane in R4 to the graph of F at the point (PI/4, PI/4, 0 , 0 ) The attempt...

What is the solution to: What is the equation of a line tangent to the following surface z=6-(4x^2)-(y^2) at the point (5,3,-103)

Find the equation of the tangent to the level surface of the scalar field theta(x,y,z) =8x^(2) + y^(2) + 3z^(2) At the point (1,3,3) Unsure as where to begin would really like to work through this with someone, thank you

[PLAIN]http://img35.imageshack.us/img35/2033/tangent.jpg I managed to do the first part okay ---- said some stuff about 3x^2 and y^2 term, its not linear etc.. but I am stuck in the part in red. Is it supposed to be something about a normal vector? How do i know what is wrong? and what...

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...

Homework Statement Our teacher was talking about something regarding two tangent lines on a circle who distance between the tangent lines is square root of 3 times the radius of the circle... She wanted us to find the proof of this but I am stumped on where to even look... Does anyone know...

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...

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...

I figured this would be easy, I need to find the normal and tangent vectors to this line: 3x-2y-4 Well simple enough, I got the correct parametric equations for the normal, but the tangent line is being silly. I dumbed it out and got the right answer, but I think it was for the wrong...

Homework Statement Determine the slope of the tangent at x = 0 for the function f(x) = \frac{cosx}{1-x}?Homework Equations Product rule: F'(x) = f'(x)g(x)+f(x)g'(x) Chain rule: f'(x) = nx^n-1·(x)' The Attempt at a Solution So first I rewrite the equation to get rid of the fraction: f(x) =...

The unit tangent vector, T(t) = r'(t) / || r'(t) || always has length 1. Alright, so how do we get a sense of the length of the actual tangent vector itself? Its direction is easy to imagine, but I can't understand how its magnitude changes along the curve (does it have something to do with...

Homework Statement Determine if the surfaces x^3 + 6xy^2 + 2z^3 = 48 and xyz = 4\sqrt{2} have any points of tangency, and if so, find those points. Homework Equations The Attempt at a Solution I'm mostly wondering if anyone can find any problems with my approach. I assume that the geometry...

Homework Statement I need to find the tangent plane of a surface S at a point P without being given the eqn of the surface. I am also given that two curves lie on this surface Homework Equations Point P: (2,1,3) Curve 1: <2+3t, 1 - (t^2), 3 - 4t + (t^2)> Curve 2: <1+ (u^2), 2(u^3) -...

Homework Statement How many tangent lines to the curve y = x / (x + 1) pass through the point (1,2)? At which points do these tangent lines touch the curve? Homework Equations n/a The Attempt at a Solution I'm completely drawing a blank on this one. Help.

Homework Statement Prove that a(s) is a straight line if and only if its tangent lines are all parallel. Homework Equations Frenet serret theorem The Attempt at a Solution I'm confused on the direction "if the tangent lines are parallel then a(s) is a straight line". Assume all the...

Homework Statement find eqn of tangent plane to surface S at point P(2,1,3) the curves r = (2+3t, 1-t2 , 3-4t+t2) r = (1+u2 ,2u3-1 , 2u+1) both lie on S. The Attempt at a Solution i don't really know how to start. am i suppose to find eqn of S first? then use the formula n.(r -...

Homework Statement OK I have a Differential Calculus exam next week and I do not understand about Differential Manifolds. We have been given some questions to practise, but I have no idea how to do them, past a certain point. For example 1. Study if the following system defines a manifold...

Homework Statement Consider the plane z = x + 2y and the cylinder x^2 + y^2 = 1 (a) Find a vector function r(t) describing their intersection. (b) Find the points if any where the tangent to ~r is horizontal (c) Find an equation for the tangent line to ~r at each of these points.Homework...

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...

Hello, I need help for this problem Homework Statement There exist a curve C such that its parametric equation is (x,y,z)=(3−3t,1−t^{2},t+2t^{3}). There is a unique point P on the curve with the property that the tangent line at P passes through the point (−3,−2,2). Find the coordinates of...

Greetings to all. :) This is my first time posting here, so if I do anything wrong, just tell me. :) On to the question. Alright, this question came in 3 parts. I've done the first 2 parts but have no clue on doing the third part. There are two equations (below) and a constant k...

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...

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}...

Homework Statement r(t)=ti+t^2j Find the velocity, speed, acceleration, unit tangent, and unit normal vectors.Homework Equations Velocity=r'(t) Speed=magnitude of r'(t) Acceleration=r''(t) Unit tangent=r'(t)/magnitude of r'(t) Unit normal=d/dt[unit tangent]/magnitude of d/dt[unit tangent]The...

Homework Statement Find the equation of the tangent to the curve y=2cos^3x at x=pi/3 Homework Equations The Attempt at a Solution y=2cos^3x dy/dx=-6sinxcos^2x 0=-6sinxcos^2x set x = pi/3 and solve for the derivative, plug the answer into y=2cos^3x where do I go from here...

Homework Statement Well I would like to prove that any equation that follows the pattern y=rx-r^(-1) is tangent to some sideways parabola (I know this to be true). Problem is that I need help in finding the parabola in question and actually proving my conjecture. I do know, after graphing...

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 -...

Homework Statement Find the slope of the tangent line to the curve: sqrt(4x+2y) + sqrt(1xy) = 9.72 at the point (6,3) Homework Equations Derivative laws The Attempt at a Solution the slope of the tangent line to a curve is the Derivative of the function of the...

Homework Statement Consider the curve x^2+xy+y^2=27 Homework Equations Find all points on the curve where the lines tangent to the curve are vertical The Attempt at a Solution I found dy/dy = (-2x-y)/x+2y) and I think I found the equations of lines visually to be x=6 and x=-6...

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(?)...

hello friends my question is: if we have M a compact manifold, do we have there necessarily TM compact ? thnx .

Homework Statement Let L be the line tangent to the curve \vec{G}(t)=10\text{Cos}(t)\mathbf{i} + 10\text{Sin}(t)\mathbf{j} + 16t \mathbf{k} at the point (\frac{10}{\sqrt{2}}, \frac {10}{\sqrt{2}}, 4 \pi) Find the point at which L intersects the x-y plane.Homework Equations The Attempt at a...

Homework Statement Consider the function f(x,y) = 4-x^2+3y^2 + y. Let S be the surface described by the equation z= f(x,y) where f(x,y) is given above. Find an equation for the plane tangent to S at the point (-1,0,3)The Attempt at a Solution Ok, SO i solved for the gradient of F...

I have a shaky understanding of problems concerning Taylor Series. For example, the question below. Let f(x)=\tan^{-1}\left(\frac{1+x}{1-x}\right) where -\frac{1}{2}\leq x \leq \frac{1}{2}. Find the value of f^{2005}(0) the Taylor Series of \tan^{-1} is...

Homework Statement Three mutually tangent circles with the same radius r enclose a shaded area of 24 square units. Determine the value of r to the nearest unit. Homework Equations do i use the arc length formula to find the answer? [b]3. The Attempt at a Solution A=(central...

Assume you're given a circle with the line AB containing its center O, such that A and B are on the circle (OA=OB=radius). A tangent t is drawn on the point A, and I should calculate the mapping of certain points (a,b,c,d...) of the circle to the points on the tangent (at, bt, ct, dt, ...) such...

Homework Statement Find all the points on the surface x^2+y^2-z^2=-1 where the tangent plane is perpendicular to the vector <1,0,2>. I'm confused! The Attempt at a Solution So, the gradient would be: <2x,2y,-2z>, but what do I do with the gradient vector to find these points...

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...