Tangent Definition and 1000 Threads

Let me first confess this a copy/paste of a question I asked on another forum; I trust it's not against the rules. Let M be a C^{\infty} manifold, and, for some neighbourhood U\ni p \subsetneq M let there be local coordinates x^i such that p=(x^1,\,x^2,...,x^n) Suppose that T_pM is a...

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 +...

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 =...

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...

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...

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...

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...

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...

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...

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'=...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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 -...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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

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...

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...

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...

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?

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...

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.

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...

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).

[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 =...

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

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) -...

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 ?

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

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...