Parametric Definition and 650 Threads

Hey guys, I've got this problem I can't seem to get past. I need to find the tangent line to a parametric curve at t=\frac{\pi}{4} I thought I solved the equation, but my answer doesn't seem to be registered as correct. I'm guessing that means I stuffed up the equation, but I can't see where...

Homework Statement Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singular points at 0, pi/2, 3pi/2, and 2pi. However is there a piecewise smooth curve? Homework Equations $\phi(\theta) =...

Homework Statement [/B] Find the definite integral formula for the length of the curve for $$0 \leq t \leq \frac \pi 2$$ $$x = 2*cos^k(t)$$ $$y = 2*sin^k(t)$$ for general $$k \gt 0$$2. The attempt at a solution I don't understand why this is wrong: $$\int_0^\frac \pi 2\...

Hello. I have graphed a rose curve on Desmos using a parametric point, the equation looking like this: However, I want the graph to be limited so that only values within the circle r=3.45 and r=9 are shown. I have tried using the curly brackets {3.45 <= r <= 9}, however this returns an error...

Homework Statement Find the scalar, vector, and parametric equations of a plane that has a normal vector n→=(3,−4,6) and passes through the point P(9, 2, –5). Homework Equations Ax+By+Cz+D=0 (x,y,z)=(x0,y0,z0)+s(a1,a2,a3)+t(b1,b2,b3) x=x0+sa1+tb1 y=y0+sa2+tb2 z=z0+sa3+tb3 The Attempt at a...

Homework Statement [/B] Suppose the solution set of some system Ax = b , Where A is a 4x3 matrix, is *Bold characters are vectors* x_1= 1 + 3t x_2 = 2 - t x_3 = 3 + 2t where t is a parameter and can be any number. a) How many pivots are in the row echelon form of A? b) Let u, v, w be the...

When I look at certain experiments involving entangled photons generated by spdc, for example quantum eraser experiments, it seems to me that each observed pair of entangled photons is propagated in a horizontal plane. However, aren't the entangled photons generated randomly in all planes...

$\textsf{Find parametric equations .}$ $\textsf{of the line through the point }$ $$P(-3, -4, -2)$$ $\textsf{and perpendicular to the vectors }$ $$u = -6i + 2j + 8k$$and $$v = -7i + 5 j - 2k$$ $\textit{Answer:$\displaystyle x = -44t - 3 , y = -68t - 4, z = -16t - 2 $} $ ok how is this done with...

Homework Statement Only the second part Homework Equations Second derivative: $$\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}$$ The Attempt at a Solution $$dx=(1-2t)\,dt,~~dy=(1-3t^2)\,dt$$ Do i differentiate the differential dt? $$d^2x=(-2)\,dt^2,~~d^2y=(-6)t\,dt^2$$...

Homework Statement $$y=1+t^2,~~x=\frac{t}{1+t^2}$$ What is dy/dx Homework Equations Parametric equation's derivative: $$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$ The Attempt at a Solution $$\frac{dx}{dt}=\frac{1-t^2}{(1+t^2)^2}$$ $$\frac{dy}{dx}=\frac{2t(1+t^2)^2}{1-t^2}$$ I can't translate it back...

Homework Statement If ##x=2-\pi cost## and y = ##2t-\pi sint##, then find the two t's at which the curve crosses itself, where t is on the interval ##[-\pi, \pi)## Homework EquationsThe Attempt at a Solution I really don't know where to start besides just looking at the graph of the parametric...

Ive been working through calculus this year and will be into next year, and as nearly every time I open my calculus book I have found something new and mysterious. This time it's something called parametric representation. It isn't clearly explained what this means or how you go about...

1. The problem statement, all variables and given/k nown data x = (sin(t))^2 y = (cos(t))^2 t goes from 0 to 3 pi Homework Equations ∫ \sqrt{ {(dx/dt)}^2 + {(dy/dt)}^2 } dt The Attempt at a Solution ∫ \sqrt{ {(2sin(t)cost)}^2 + {(-2cos(t)sin(t))}^2 } dt ∫ \sqrt{ 4(sin(t)cos(t))^2 +...

Why ##\mu_1, \mu_2## must be the same as ##\mu_1^*, \mu_2^*## ? What I thought is : If ##\mu_1\mu_2 = \mu_1^*\mu_2^*## and ##\mu_1+\mu_2 = \mu_1^*+\mu_2^*##, then ##\mu_1, \mu_2## are the same as ##\mu_1^*, \mu_2^*## It can be shown by taking the complex conjugate of (27.5) that $$\mu_1\mu_2 =...

Dear all, I have a question regarding the computation of the area of an ellipse. The parametric form of the ellipse with axes a and b is $$x(t) = a\cos{(t)}, \ \ \ y(t) = b\sin{(t)} $$ Using this to evaluate the area of the ellipse, usually one takes one halve or one quarter of the ellipse...

Quick question. I know that if we have a curve defined by ##x=f(t)## and ##y=g(t)##, then the slope of the tangent line is ##\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}##. I am trying to find the second derivative, which would be ##\displaystyle \frac{d}{dx}\frac{dy}{dx} =...

Homework Statement Find the area of the region enclosed by the parametric equation x = t3- 7t y = 8t2 Homework Equations A = ∫ y(t) x'(t) dt The Attempt at a Solution I initially began with A = ∫ y(t) x'(t) dt And got to ∫24t4-56t2dt and then to 24∫t4dt - 56∫t2dt but without a limit/defined...

This isn't exactly homework or coursework, it is a past paper question that I cannot find a solution to (my university doesn't like releasing answers for some reason unknown to me). The question is attached as an image (edit: the image displays while editing but not in the post, so I'll try to...

Homework Statement Find the area between the parametric curve, x(t)=cos(t), y(t)=sin^2(t) and the x-axis Homework Equations - A = ∫ₐᵇ y(t) x'(t) dt The Attempt at a Solution =http://imgur.com/a/UA48d - My work shown in the link provided without the bounds, sorry for not rotating the image and...

Homework Statement Suppose that a particle follows the path r(t) = 2cos(t)i + 2sin(t)j Give an equation (in the form of a formula involving x and y set equal to 0 ) whose whose solutions consist of the path of the particle. Homework Equations None that come to mind The Attempt at a Solution I...

Hello. How can I verify that (u,v) = (x square - x, x+1) is a parametric form of a parabola? Thank you!

see post 3

Homework Statement Given the points P0 = (0,a), P1 = (b,0), P2 = (0,0), write the parametric equation of a circle that intersects the 3 points. Assume that b > a and both are positive. Homework Equations X = h + rcos(t) Y = k + rsin (t) r = √((x-h)2 + (y-k)2 Cos (t) = (x-h)/r Sin (t) =...

Homework Statement parametric equations are x=s+2t y=2s+3t z=3s+4t trying to solve for equation of the plane and then take the coefficients of the equation to get the vector normal to the planeHomework EquationsThe Attempt at a Solution apparently t=2x-y and s=2y-3x and therefore the equation...

Homework Statement Find Parametric equation and vector equation for the portion of parabola y = 1+(x^2) from (1,2) to (2,5) Homework EquationsThe Attempt at a Solution m = (2-1) i +(5-2)j = i +3j x = t , y = 1+t^2 r(t) = ti + (2=3t)j, where 0<t<1 , but the ans given is r(t) = ti + (1+t^2) j...

Homework Statement Find the curvature of the car's path, K(t) Car's Path: r(t) = \Big< 40cos( \frac {2 \pi}{16}t ) , 40sin( \frac {2 \pi}{16}t ), \frac{20}{16}t \Big> Homework Equations K(t) = \frac { |r'(t)\:X \:r''(t)|}{|r'(t)|^3 } The Attempt at a Solution This is part of a massive 6...

Homework Statement x=sec(t), y=tan(t), -π/2 ≤ t ≤ π/2 Try to find y in terms of x Homework EquationsThe Attempt at a Solution 1.[/B] ∂y/∂x = sec(t)/tan(t) y=∫sec(t)/tan(t)∂x =∫x/y∂x =(1/y)*∫x∂x =x2/2y + C 2y2=x2 + C When t=π/4, x=√2, y=1 2(1)2 = (√2)2 + C C=0 So y2 = x2/2 2. y/x = sin(t)...

How did he found x = 1, y = 0, z = 0 and x = 1, y = 1, z = 2?

The derivative for the parametric equations ##x=f(t)## and ##y=g(t)## is given by ##\frac{dy}{dx}=\frac{\Big(\frac{dy}{dt}\Big)}{\Big(\frac{dx}{dt}\Big)}## The proof of the above formula requires that ##y## be a function of ##x##, as seen in...

Hello I have a seminar for my Optic Course in university with this title : Optical Parametric Amplifiers. I searched a lot but I couldn't find why its name "Parametric Amp" I think its about RF parametric amp.. Please if you know why named "parametric" share your knowledge with me. Thanks for...

Homework Statement Consider the parametric curve given by: x=6cos(2t), y=t5/2. Calculate the equation of the tangent to this curve at the point given by t=π/4, in the form y=mx+c. The tangent is given by y= Homework Equations The Attempt at a Solution [/B] the answer that I got was...

Homework Statement Consider the following parametric curve: x=5cos^7(t) y=5sin^7(t) Write it in cartesian form, giving your answer as an equation of the form F(x,y)=c for some function F and some constant c. The Attempt at a Solution [/B] I know that sin^2(t)+cos^2(t) = 1 but I don't...

I have an odd, yet intriguing question. I want to describe a particle that is constrained to move along a straight line in the x direction. The location of the particle can be described with theoretically infinite parametric parameters, d, along the line. For example: R = [d 0 0]T R = [d2 0 0]T...

I am give the following curve r(t) = (t+1,0.5(1-t),0) where t ranges from -1 to 1. I am now trying to derive a new parametric representation of this line segment using the arc length as the parametric variable. I have integrated r'(t) from -1 to 1 and found that the length of the segment ranges...

Hi all! I have recently taught myself parametrics, and I stumbled upon the butterfly curve. So, I was wondering about some cool equations I can plug into a parametric graphing calculator.

Homework Statement Consider the parametric curve x=ln(t) and y= 1+t^2 i need to eliminate the parameter to find the cartesian equation Homework EquationsThe Attempt at a Solution if i solve for t using x I get that t=e^x but if i solve for t using y i get t=(y-1)^(1/2) when i plug in x into...

Homework Statement z = 2x^2 + 5y^2 +2 C is cut by the plane x = 2 Find parametric eqns of the line tangent to C @ P(2, 1, 15) Homework Equations z = 5y^2 + 10 dz/dx = 10y dz/dx (1) = 10 The Attempt at a Solution z = 10y + 15 y = t + 1 if the slope is 10/1 then delta z = 10 and delta y = 1...

Homework Statement Find the parametric and canonical equation of the line L passing through the points A = [1, 0, 2] and B = [3, 1, −2]; check whether the point M = [7, 3, 1] lies on L. Homework Equations Canonical equation of a line in space x-x0 / l = y-y0 / m = z-z0 / n Parametric...

Homework Statement Find parametric equation of the line through the point ##(4,0,-4)## that is parallel to the planes ##x-8y+7z=0## and ##4x+3y-z+4=0##. Homework Equations ## \vec r = \vec r_0 + t\vec v ## (for orthogonal vectors) ## v \bullet w = 0 ##[/B] The Attempt at a Solution So I...

I am reading David Poole's book: Linear Algebra: A Modern Introduction (Third Edition) ... I have a basic (and probably simple) question regarding Poole's introductory discussion of the vector or parametric form of the equation of a plane $$\mathscr{P}$$ (page 38, Section 1.3 Lines and Planes)...

Homework Statement Find parametric equations for the line through P1 and P2 and also for the line segment joining those points. Also find the conditions for the parametric equations. a) P1(0, 2) and P2(-4,-4) b) P1(7,-3,9) and P2(7, -3, 1)Homework Equations Equation of a line (Vector...

Consider the map ##\phi (t,s) \mapsto (f(t,s),g(t,s))##, a point belonging to the envelope of this map satisfy the condition ##J_{\phi}(t,s)=0##. What is the role of the Jacobian in maps like these and why points in the envelope have to satisfy ##J_{\phi}(t,s)=0##?

The site http://tutorial.math.lamar.edu/Classes/CalcII/TangentNormalVectors.aspx talks of "the" unit tangent vector of r→(t) = f(t)*i→(t)+g(t)*j→(t)+h(t)*k→(t) as finding "the" tangent vector r'→(t) = f'(t)*i→(t)+g'(t)*j→(t)+h'(t)*k→(t) and normalizing it, and further with finding "the" tangent...

Hi PF/math ! I've been searching for a program which will draw 3d parametric curves accurately for large variables, eg. f(10^8). Ive tried www.math.uri.edu and Ti-Nspire (the latter may or may not have a setting for the accuracy), but both tend to turn what should've been a smooth curve into an...

Hi PF I've beent rying to model the lunar orbit around the sun (cardioide) as a parametric function, but have run into a problem. f(t) = r(t) : x = a cos(ωt) y = b sin(ωt) z = k t The angular frequency ω as well as the distance from to the center varies around the orbit. Is...

Homework Statement Homework Equations The Attempt at a Solution so to start this off, I choose a random point, by setting u and v = 0 giving me the point (0,3,1) but I have no idea how what to do next. how do I find ua and vb?

http://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx On this page the author makes it very clear that: $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$ provided ##\frac{dx}{dt} \neq 0##. In example 4, ##\frac{dx}{dt} = -2t##, which is zero when ##t## is zero. In simplifying...

How can I find the parametric vector form of a cartesian equation under a specific condition? Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1...

Homework Statement Sketch the trajectory over the time interval 0 ≤ t ≤ 10 of the particle whose parametric equations of motion are given by X= t−3sint . And y = 4 − 3cost find the value of x,y,t Remember that you should be in Radian mode! Answer Do you isolate the t first and plug it in...

Homework Statement [/B] Show that the given parametric curve decribed by the following notations: x=cos(t), y=sin(t), z=2+2cos(t) lies in a single plane ,find the normal vector to this plane Homework Equations --- [/B] r(t)=cos(t) i + sin(t) j + 2+2cos(t) k The Attempt at a Solution My...