Tangent Definition and 1000 Threads

If the tangent bundle is trivial, then the cotangent bundle is trivial. To see this, consider (X_i) a global frame for TM. Then define a global frame (\alpha^i) for T*M by setting \alpha^i(X_j)=\delta_{ij} and extend by linearity. Does trivial cotangent bundle implies trivial tangent bundle? A...

Homework Statement Use implicit differentiation to find an equation of the tangent line to the cardioid at the point (0, 0.5). x2 + y2 = (2x2 + 2y2 - x)2Homework Equations Derivative rules point slope formula The Attempt at a Solution I got y' = [16x3-4x2+16xy2-4y2-4y2-8x2+2x] / [2y -...

Homework Statement Question: Determine the equations of both lines that are tangent to the graph of f(x) = x2 and pass through point (1,-3). Homework Equations Some of the equations that I could use for this problem are: y-y1=m(x-x1) (Point-slope Equation) the derivative of the...

Homework Statement Find the polar form of 2i − 1 Finding polar form is easy r(cosx + isinx) call the real part a and imaginary part b r = sqrt(a+b) theta = arctan (-2) = - 63.43 This is the wrong angle for theta as it's 116.57 (which is 180 - 64.43), and I guess this if...

Homework Statement Hi guys, need some help finding the common tangent of two curves. Two curves y = 3X3+6X2+6X+3 and y = 3X3+6X+3 touch each other. Find the common tangent. Homework Equations y = 3X3+6X2+6X+3 and y = 3X3+6X+3 The Attempt at a Solution well i made the two equations equal...

Homework Statement Find points on curve y=sinhx where the tangent line has slope 2 Homework Equations The Attempt at a Solution y' = 1/(sqrt(1-x^2)) = 2 4 = 1/(1-x^2) 4-4x^2=1 x=sqrt(3/4) the actual answer should be x = +or- ln(2+sqrt(3))

Find the equation of the tangent line to the curve y=(lnx)^cosx at the point (pi/2, 1)? The Attempt at a Solution lny = cosx(ln(ln(x))) d/dx = -sinx(ln(ln(x))) + cosx/(ln(x)(x)) y' = y(-sinx(ln(ln(x))) + cosx/(ln(x)(x))) this is the part where I get stuck

Homework Statement Given a surface parameterized by the function f(x) and a point p on that surface, assume that P is a point on the tangent space of f at p. Find the normal vector to the hyperplane at P . The Attempt at a Solution The tangent hyperplane to f at p is given by the...

Homework Statement Find the equation of the circle that passes through the point (3,-2) and tangent to the line y=3x+5 at (-1,2). Answer in standard form. Homework Equations d= |mx0+b-y0| ____________ sqrt(1+m2 is needed to find the radius of the circle (x-h)2+(y-k)2=r2...

Homework Statement Find the parametric equation for the line tangent to the curve: x=t^3-1, y=t^4+1, z=t at the point (26, 82, 3). Use the variable t for your parameter. Homework Equations The Attempt at a Solution dx/dt=3t^2, dy/dt=4t^3, dz/dt=1 I got then that...

Homework Statement Give the equation of the two lines through the point (-1, 3) that are tangent to the parabola y= x^2 Homework Equations The Attempt at a Solution

Homework Statement Show that the tangent to the curve y=(x^2+x-2)^2+3 at the point where x =1 is also tangent to the curve at another point. Homework Equations y=(x^2+x-2)^2+3 The Attempt at a Solution y'=2(x^2+x-2)(2x+1) y'(1)=0 0=2(2x+1)(x+2)(x-1) x= -1/2, -2, 1 Would this be correct?

Homework Statement Find the vectors T, N, and B at the given point. r(t) = (sin(t), cos(t), ln(cos(t))), P = (0,1,0) Homework Equations T(t) = r'(t) / | r'(t) | N(t) = T'(t) / | T'(t) | B(t) = T(t) x N(t) The Attempt at a Solution I am stuck on how to solve for t. I am not...

Homework Statement Find the equation f(x) who's integral is \int x^{3} and has a tangent x+y=0 Homework Equations The Attempt at a Solution I know that f(x) is 1/4x4+c because of the integral. The tangent is the derivative of f(x) at some point i have the equations y=x3+c y=-x but...

Homework Statement At x=3 does the function y=x^3 + (x-3)^(1/3) have a vertical tangent, cusp, corner or none? The Attempt at a Solution I took the derivative y'=3x^2 + 1 / 3x^(2/3) then i replaced 3 in which gave 27 + 1 / 0 I don't understand how to come to the correct...

Where is the use of the "tangents at every point on the curve" in the Riemann sum? Riemann sum allows us to find the area of under the curve, and this involves only the height of each rectangle (i.e. the function f(x) at each x), and the width (i.e. the x), and the two are multiplied together...

Is it possible to find a recurrence relation of tan(nx) where n is a positive integer and x is a real variable? My friend said that it is possible. I don't see how to do it. Does anyone have a way to do it?

To understand differentials better, I'm trying to use differentials dy and dx in the equation of the tangent line to the curve x^2 at point 3. Here is the equation of the tangent line to the curve x^2 at point 3: y=f'(3)(x-3)+f(3)=2(3)(x-3)+9=6(x-3)+9 But since we are dealing with the...

Homework Statement Find two straight lines that are perpendicular to y=0.25x and tangent to the curve f(x) = 1/x. Homework Equations y=0.25x f(x) = 1/x. The Attempt at a Solution What I did was equate y and f(x) and determined when they equal which is 2 and -2. The points are (2, 1/2) and...

Using a graphing software, I'm trying to graph three things: 1. The function x^2. 2. It's derivative 2x. 3. The tangent to the curve at point 3. Now I know that that if I want to find the slope of the curve at point 3, I should substitute 3 into the derivative 2x, which will give 6. What...

Homework Statement A curve is defined parametrically by x=sin3t, y=cos3t, 0≤ t ≤ 2pi. Find the equation of the line tangent to the curve at the point defined by t=2pi/9. Homework Equations The Attempt at a Solution ?

Homework Statement Find two points on curve y=x4-2x2-x that have a common tangent line. Homework Equations *the one stated above dy/dx = 4x3-4x-1 The Attempt at a Solution equation of a tangent line: y=mx+b (4x3-4x-1) = m at two different points? So there are two points for which...

I am trying to write an Excel template which will chart two circles and their external tangent lines - similar to a belt and two pulley system. I have the formulas to calculate and chart the circles. I am looking for a formula to calculate the tangent points between the circles, given the...

hello everyone, :confused:i'm having can't seem to solve these two questions... 1) Find the equation of the tangent of the parabola y^2= 4px, perpendicular to the lines 4Y- X + 3=0, and find the point of contact? 2) Find the equation of the tangent to the parabola Y^2= 10x, at the...

Homework Statement Find the slope of the tangent line using a specific formula g(x)=3t-t2 at (0,0) Homework Equations Im told to use this equation by the book f(c+deltax) - f(c) Deltax The Attempt at a Solution Everytime i plug it in by way of the books style i...

Homework Statement Find the equation of the tangent line to f(t)=<cot(t),csc(t)> at the point (1/sq.rt of 3, 2/sq.rt of 3) Homework Equations n/a The Attempt at a Solution I started by finding the slope, y'/x', so I got csc(t)cot(t)/csc^2(t). I then used the equation of a line...

Hello - I've been stuck on this for a while now and I really need some help. Homework Statement Find the equations for all lines that are tangent to the circle x^2+y^2=2y and pass through the point (0,4).Homework Equations y=mx+b ax^2+bx+c=0The Attempt at a Solution From the given equation of...

I have that P is the tangent plane to the surface xyz=a^{3} at the point (r,s,t). I need to show that the volume of the tetrahedron, T, formed by the coordinate planes and the tangent plane to P is indepedent of the point (r,s,t). I have found that P is; \frac{x}{r} + \frac{y}{s} +...

Homework Statement # 23 http://img26.imageshack.us/img26/1008/questionsh.jpg Homework Equations m1 * m2 = -1 The Attempt at a Solution i get -4 not sure what's wrong: http://img4.imageshack.us/img4/6092/15608741.jpg

Homework Statement Find the tangent vector at the point (1, 1, 2) to the curve of intersection of the surfaces z = x2 + y2 and z = x + y. Homework Equations The Attempt at a Solution I haven't started the problem, because I'm not sure what the first thing to do is. Do I have to parametrize...

How do I find this?

Homework Statement Hi all. I have to solve the differential equation \frac{dv}{dt} = g(1 - \frac{\rho}{g}v^2). The Attempt at a Solution Apparently the solution should involve the inverse hyperbolic tangent function - with the equation in this form it should just be separable, correct...

Hello, Say you have a function f on the domain R^n, and an integral transform P which integrates f over all possible straight lines in R^n. I am lead to believe that the range of this is R^(2n), or a tangent bundle, which I am having MASSIVE problems visualising! Am I right in saying the...

Homework Statement Find the equation of the tangent line to y=sqrt(25-x^2) at the point (4,3) Homework Equations The Attempt at a Solution (25-x^2)^1/2 1/2(25-x^2)^-1/2 (2x)1/2(25-x^2)^-1/2 x(25-x^2)^-1/2 I have no idea what to do or where to go after this... Please Help!

Homework Statement Find the points on the graph of z=(x^2)(e^y) at which the tangent plane is parallel to 5x-2y-.5z=0 Homework Equations An equation of the tangent plane to z=f(x,y) at (a,b) is: z = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b) The Attempt at a Solution Partial derivatives...

I saw this identity somewhere and have been looking for a derivation but I can't seem to find one. It would be of great help if someone can show me where this comes from. tan\frac{\theta}{2} = csc\theta - cot\theta

Homework Statement Find the angle of inclination of the tangent and plane to the surface at the given point. (I we are comparing the tangent plane to the XY-plane) Homework Equations 2xy-z3=0 Point: (2,2,2) The Attempt at a Solution This one was thrown into the textbook just to piss...

I have this graph The blue line is a tangent to the black The blue line starts at (0,0) I need to show that the blue line y=Hx is a tangent to the black line y= x(x-L)(1-x), (0<L<1) By deducing that H = (1-L)2/4 I took the example L=0.25 in the pic, where H = 9/64 where it's a...

Homework Statement Find the x values of the points on the curve f(x) = (x^3) + (3x^2) + 3x + 6 where the tangent has a slope equal to m = 6 Homework Equations The Attempt at a Solution This question was on an online quiz for my into calculus course... can't seem to wrap my head around...

OK, this is really confusing me. Mostly because i suck at spatial stuff. If the gradient vector at a given point points in the direction in which a function is increasing, then how can it be perpendicular to the tangent plane at that point? If it's perpendicular to the tangent plane...

Homework Statement Find the equation of the tangent and normals of y=x+\frac{1}{\sqrt{x}} at x = 4. Homework Equations \lim_{h \rightarrow 0}\frac{f(x_{0} + h) - f(x_{0})}{h} = m Also: slope of normal = \frac{-1}{slope of the tangent}The Attempt at a Solution m=\lim_{h \rightarrow...

Homework Statement Does the surface level of a curve influence the tangent plane of that curve? If so, how do I find the tangent plane specific to that level?

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 x = x0 + at y = y0 + bt z = z0 + ct The Attempt at a Solution Well i know that x0,y0 and z0...

Homework Statement Implicit Differentiation:Use implicit differentiation to find an equation of the tangent line to the curve at the given point. x2/3 + y2/3 = 4 at (-3*31/2,1) Homework Equations ? None? The Attempt at a Solution 2/3 * x-1/3 + 2/3y-1/3*y' = 0 then after a few...

Homework Statement Prove that a vertical line and a line going from a point on a parabola to the focus of the parabola form equal angles with the tangent line of the point on the parabola. Homework Equations Focus = 1/4a (maybe relevant) The Attempt at a Solution I know how to...

How would find the equation of a line that has to be tangent to a curve and parallel to another line (i know slope has to be equal) (you are given the equation of the line and the curve) x^3 3x-y-6

How would find the equation of a line that has to be tangent to a curve and parallel to another line (i know slope has to be equal) (you are given the equation of the line and the curve) x^3 3x-y-6

Homework Statement a.)\sqrt{x^2+y^2} Find the equation of the tangent plane at the point given by: x = 1, y = 1 Draw the 3d-graph of the surface and the tangent plane. \stackrel{\rightarrow}{n} = the normal vector to the tangent plane. b.) If the surface is intersected with the plane y =...

Homework Statement Find the equations of both the tangent lines to the ellipse x2 + 9y2 = 81 that pass through the point (27, 3). One is horizontal the other is not. Homework Equations The Attempt at a Solution horizontal, easy: y = 3 x^2+9y^2=81 derivative: 2x +...

Homework Statement Find equations of the following. x2-2y2+z2+yz=7, (5,3,-3) (a) the tangent plane (b) the normal line to the given surface at the point Homework Equations I know it involves fx, fy, fz The Attempt at a Solution I got 10x-15y-3z=7. Is this correct? Because its not true at...