Tangent Definition and 1000 Threads

Homework Statement Consider the two space curves r1(t) = <cos(t − 1), t^2 − 1, 2t^4> r2(s) = <1 + ln s, s^2 − 2s + 1, 2s^2>, where t and s are two independent real parameters. Find the cosine of the angle between the tangent vectors of the two curves at the intersection point (1, 0, 2)...

Homework Statement Find a parametric equation of the line that satisfies the condition: The line that is tangent to the parabola y=x^2 at the point (-2,4) The Attempt at a Solution My answer came out to <x,y> = <-2,4> + t<1,2>

Homework Statement So I have to find the tangent line when x=5. I use the difference quotient, but I'm stuck at the algebra? I can't seem to figure out what steps I should take. Here's what I have: [(x+h)+1/(x+h)] - (x+1/x) I've tried numerous steps on how to solve this, but can't do...

Homework Statement Find all points for which the curves x^2+y^2+z^2=3 and x^3+y^3+z^3=3 share the same tangent line. Homework Equations Sharing the same tangent line amounts to having the same derivative. The constraint then is that 3x^2+3y^2+3z^2=2x+2y+2z. The points must obviously also...

Hi there:) so i have gotten this question in my homework, and its not that i don't understand how to find a tangent to a curve, i just don't understand this question!:(:( so here it is: "The line x+y-k=0 is a tangent to the circle x^2+y^2-2x+4y-72=0. Find the value(s) of k. Does this mean i am...

Hello people! Well, I am doing some excercises for fun. Picked some Precalculus stuff, and found this excercise: "Construct a function that has the same slope at x = 1 and x = 2. Then find two points where y = x^4 - 2x^2 has the same tangent line (draw the graph)." I have found a solution, but...

Homework Statement In triangle ABC, the altitude from B is tangent to the circumcircle of ABC. Prove that the largest angle of the triangle is between 90◦ and 135◦. If the altitudes from both B and from C are tangent to the circumcircle, then what are the angles of the triangle...

defining a tangent vector v as the equivalence class of of curves: v = [\sigma] = \left. \frac{df(\sigma)}{dt} \right|_{t=0}, i want to show that this definition is independent of the member of the equivalence class that i choose. where \sigma represents a function from the reals to the...

Homework Statement Find equation of the plane The plane though P(6,3,2) and is perpendicular to vector <-2,1,5> Why would it be -2(x - 6) + (y - 3) + 5(z - 2) = 0? If it is perpendicular to <-2,1,5>, shouldn't be the cross product of some other vector with this? Using <-2,1,5>...

Homework Statement A point of mass m is placed on a frictionless plane that is tangent to the Earth’s surface. Determine Hamilton’s equations taking: (a) the distance x (b) the angle q as the generalized coordinate. Homework Equations The Attempt at a Solution Take the...

Homework Statement Draw a diagram to show that there are two lines tangent to both of the parabolas y = - x^{2} (1) and y = 4 + x^{2} (2) Find the coordinates of the four points at which these tangents touch the parabolas. Homework Equations y - y_{o} = m(x - x_{o}) The Attempt at a...

What is the difference? According to my text... Tangent Plane: z-z0=fx(x0,y0)(x-x0) + fy(x0,y0)(y-y0) Linearization: L(x,y) = f(a,b) + fx(a,b)(x-a) + fy(a,b)(y-b)

I need to find the equation of the tangent line to the curve x=t4+1, y=t3+t; t=-1 I have already found that the slope of the line is -1 by finding (dy/dt)/(dx/dt) I just need to figure out how to solve for y1 and x1 Thanks in advance

Homework Statement The graph shows the vector-valued function r(t). It is given that r(1) = <-1,1> and r'(1) = <1,2>. Find the unit tangent vector T(1) and unit normal vector N(1). (the graph given is that of y = x^2 shifted 2 units left.) Homework Equations (1) T(t) =...

Consider the hyperbola y^2-x^2=1 (y>0) a.) Find a parameterization for the curve and write it in vector form, R(t) (b) Calculate the unit tangent vector as a function of the parameter. (c) Calculate the unit normal vector and the curvature vector as a function of the parameter.

How do you find the equation of the (sometimes 2 possible) tangent lines between two (or more) circles? like the 2 tangents that cross in the picture on this page: http://mathworld.wolfram.com/Circle-CircleTangents.html. The application for this is for a program that would draw this tangent...

Homework Statement Take the infinite limit of that sequence. (click to expand) Homework Equations The Attempt at a Solution I have no idea where to start from, hints and ideas would be greatly appreciated. Thanks.

So the curve is defined as x-y^2 = 0, z= x . I'm supposed to find the tangent line to the curve. How do I find a slope in R3?

Homework Statement Find the equation for the tangent line at (2,2) f(x)=xy+y^3=12 Homework Equations The Attempt at a Solution I really feel like I know what I'm doing here, but the key disagrees. My equation for the tangent slope comes out to be: (-y)/(x+3y^2) when...

Homework Statement Find the equations for two spheres that are tangent to the plane x+y-z=3 and x+y+z=9 and the line x=t , y=2t, z=3t passes through its center. Preface: This problem was on a test I took yesterday. My professor handed it back today. The relevant equations and work are what...

I am currently going through some online notes on differential geometry and general relativity. So far I have been following pretty well until I got to 3.4 cont'd letter (e) of http://people.hofstra.edu/Stefan_Waner/diff_geom/Sec3.html" document and the definition directly after. My first...

Demontration of: cos(x)² = 1 \overline{1+tan(x)²} Anyone know? If you don't understand: cos(x)² = 1/1+tan(x)²

The problem is: Tangent Lines: Determine equations of the lines tangent to the graph of Y = x√(5-x²) at the points (1,2) and (-2,-2). Graph the function and the tangent lines. I have no IDEA where to go with this. I am taking calculus over the summer and we are in week 2 and I'm...

Homework Statement Find the slope of the tangent line to the curve of intersection of the vertical plane x-y+1= 0 and the surface z=x^2+ y^2 at the point (1, 2, 5) . Homework Equations The Attempt at a Solution Normal Vector 1 is i-j Normal Vector 2 is 2i+4j-k Cross them to get...

Homework Statement I know that x+y+z=0 is a tangent plate to my sphere at (0,0,0). I want to find the tangent plane for my sphere at (1,-2,3).Homework Equations Suppose that his is my sphere (x-a)^2+(y-b)^2+(z-c)^2=R^2 (a,b,c) is the center of my sphere while R is the radius of it. The Attempt...

please help me about this get Equation of the tangent plane on the surface(x^2-2y^2+2z=4) on p0(x0,y0,z0) and then find a point that tangent plane on the curve on this point is Horizontal ? i get the Equation of the tangent on the surface(x^2-2y^2+2z=4) on p0(x0,y0,z0)...

Homework Statement So I'm studying for my test. doing even and odd problems from the book. I wanted to see if this answer is right. Q: find an equation for the line in the xy-plane that is tangent to the curve at the point corresponding to the given value of t.Also, find the second derivative...

Homework Statement If there is an an ellipse x^2/9 + y^2/16 = 1, and the slope of the tangent is dx/dy = -16x/9y, how do you find what points at which the slope of the tangent is 1? I have no idea how to answer this and I've been trying for like an hour. Can anyone help me? Homework...

If there is an an ellipse x^2/9 + y^2/16 = 1, and the slope of the tangent is dx/dy = -16x/9y, how do you find what points at which the slope of the tangent is 1? I have no idea how to answer this and I've been trying for like an hour. Can anyone help me?

Homework Statement Find the slopes of the two tangent lines of x^3-y^2+x^2=0 at 0,0. Homework Equations Differentiating implicitly we get (dy(x))/(dx) = (x (2+3 x))/(2 y). The Attempt at a Solution I'm not sure how to deal with the derivative being undefined at 0,0 when there are...

Homework Statement Find equations of the tangent plant and the normal line to x-z=4arctan(yz) at (1+∏, 1, 1) Homework Equations The Attempt at a Solution I took the partials and got fx=0 fy=0 fz=(-4y)/((yz)2+1) so for the plane i got -2z-2=0 and for the normal line I got .5z-2=0...

Homework Statement Center of circle is: (3,2) Tangent point: (8,4) Question: What is the equation of the tangent line? Homework Equations The Attempt at a Solution I am just not getting it. So, would the radius be 5? Now, would i go: (x-3)^2 + (y-2)^2 = 5 ...

I sometimes see that the basis vectors of the tangent space of a manifold sometimes denoted as ∂/∂x_i which is the ith basis vector. what i am a little confused about is why is the basis vectors in the tangent space given that notation? is there a specific reason for it? for example, i know...

Homework Statement What is the equation of the tangent line to the curve y = Sqrt( 2x - 1) where the tangent line is parallel to the line x - 3y = 16 The Attempt at a Solution My teacher briefly mentioned to find the derivative of the given, and then plug it into the slope-y intercept...

Given: let f (x,y,z) = x2 + y2 + z2 = 2y - 3x and g(x,y,z) = 3x + y2 - z2 A. Find an equation for the tangent plane for the surface g(x,y,z) = 9 at the point (3, -1, 1) B. Find the line tangent to the intersection of the surfaces f(x,y,z) = 0 and g(x,y,z) =9 at the point (3,-1,1) A. I...

Homework Statement Let r(t) = <\cos(e^{-t}),\sin(e^{-t}),3e^{-t}>, find the equation of the line tangent to r(t) at the point \left ( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, \frac{3\pi}{4} \right)Homework Equations Okay, in just normal Cartesian Coord, we have y - y_0 = f'(x)(x - x_0) So I...

Homework Statement Suppose you need to know an equation of the tangent plane to a surface S at the point P(2,1,3). You don't have an equation for S but you know that the curves: r_1(t) = <2 + 3t, 1 - t^2, 3 - 4t + t^2> r_2(u) = <1 + u^2, 2u^3 - 1, 2u + 1> both lie on S. Find an...

Homework Statement Given the circle (x+1)^2 + (y-3)^2 = 25, determine the equations of the tangents to the circle with the slope -3/4.Homework Equations y = mx + bThe Attempt at a Solution I thought that if I could find the equation of the line that passed through the center of the circle and...

Does anyone know how to find the equation of a tangent line to y=e^(-2x) at the point (1,e^-2) I honestly have no idea how to even start this problem when I tried it I came up with y=1(x+.27)+ln2

1. Homework Statement : Find the points on the hyperboloid of two sheets with equation x²-2y²-4z²=16 at which, the tangent plane is parallel to the plane 4x-2y+4z=5. Homework Equations The hyperboloid with two sheets: x²-2y²-4z²=16 The given plane: 4x-2y+4z=5 Equation of tangent plane...

Find the equation of the tangent plane to the level surface of the scalar field \xi(x,y,z) = x2+y2+z2 at the point (1,1,2) Looking the work through this question with someone, not to sure where to start.

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 the origin. Homework Equations -1/r'(t)= slope of position vector x^{2}+y^{2}=1 The Attempt at a...

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...