For questions about parametric equations, their application, equivalence to other equation types and definition.

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17 views

Parametric derivative of $x^2+y^2+sin(4x)+sin(4y)=4$.

I am trying to parametrize $x^2+y^2+sin(4x)+sin(4y)=4$. I need to find a way of taking the intersections between $x^2+y^2+\sin(4x)+\sin(4y)=4$, and $\tan(nx)$, as n increases from $0\le{n}\le{2\pi}$. ...
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1answer
14 views

Equation of the Tangent Line and Area of Parametric Equation

I need to find the equation of the tangent line to the point (1,0) for the equation: $x=e^{-0.1t}cos(t) \\ y=e^{-0.1t}sin(t)$ I also need to calculate the area in the first quadrant bounded on the ...
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3answers
14 views

Find the slope of a parametric equation [on hold]

Find the slope of a line whose parametric equations are $$x=2-t \\y=1+2t$$
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0answers
12 views

Parametric tangent lines?

$x=t^4+t$ and $y=t^5+2$ at the point (-1,1) $x'=4t^3+1$, $y'=5t^4$ $\frac{dy}{dx} = \frac{5t^4}{4t^3+1}$ Plug in the value of -1, and I get $y'(-1)= -5/3$ What do I do from here? I use ...
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0answers
14 views

Parametric curve arc length! Am I doing it right?

$x=3t^2+2$ and $y=2t^3-1$ on $[1,3]$ The formula for parametric arc length is $\int\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt$ $x'(t) = 6t,y'(t)=6t^2$ Under the radical, I ...
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1answer
13 views

Finding the points of a paramertized curve where a tangent line has slope 3?

I have a curve at $c(t) = (-5t^2-3t+4,t^3-9t+5)$ and given a slope for the tangent line of $3$. I would like to find the point $(x,y)$ where this occurs. What I did is took the derivatives of $x(t)$ ...
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0answers
10 views

parametric representations 3d object

I'm trying to model a 3 dimensional body that is sort of ellipsoidal and am looking for parametric representation of 3D objects similar to the quadratic surface representation of a sphere or ...
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2answers
40 views

How do I convert this parametric expression to an implicit one

I have: $$x=5+8 \cos \theta$$ $$y=4+8 \sin \theta$$ With $ -\frac {3\pi}4 \le \theta \le 0$ If I wanted to write that implicitly, how would I do it? I get that it's a circle, and I can easily write ...
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2answers
40 views

Ellipse with center in origin

The purpose is to fit data to a ellipse which center is the origin $(x_0=0,y_0=0)$. I found the general quadratic curve: $$ax^2+2bxy+cy^2+2dx+2fy+g=0$$ Reference: ...
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0answers
20 views

Non-linear integral equation

Show that the function $$x(t) = \frac{1}{{\sqrt {k \cdot m} }} \cdot \int_0^t {F(\tau ) \cdot \sin \left( {\sqrt {\frac{k}{m}} \cdot (t - \tau )} \right)\,d\tau } $$ satisfies the initial conditions ...
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1answer
23 views

Computing derivative of parametric equation

This is probably a silly question but I am just not sure if I understand what to do. So I have the parametric equations: $x=6\cos (t)-2\\ y=5\sin (t)+3$ I am asked to compute $\dfrac{dy}{dx}$ at ...
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1answer
25 views

When is $x=sin(at), y=sin(bt)$ symmetric to x and y axes?

Take the simple system of parametric equations, $$x=\sin(at)$$ $$y=\sin(bt)$$ where $a,b \in \Bbb{N}$. When is this curve symmetric with respect to both the $x$ and $y$ axes? In other words, what ...
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0answers
22 views

Famous parametric curves that are solutions to differential equations

I know that the cycloid satisfies the differential equation $ \left( \frac{dy}{dx} \right)^2 - \frac{2r}{y} + 1 = 0. $ Are there other famous plane curves that are also solutions to a differential ...
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0answers
12 views

Parameterization of a closed curve on a sphere

I'm looking for a parameterization of a closed curve C on a sphere. assume the projections of C on y-z, x-z, x-y plane are f(x), g(y), h(z), respectively, and ${\oint}f(x)dx={\oint}g(y)dy=0$, and ...
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1answer
9 views

Determine if a parametric equation's trajectory lies on a circle?

I have a question as follows: Determine whether the following trajectory lies on a circle. If so, find the radius of the circle and show that the position vector and velocity vector are everywhere ...
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0answers
18 views

Make equation on nonparametric form

I have the following Points $(-1,-2,-6)$ and $(-1,-2,-12)$ if I write the line on parametric form I get $$x = -1 + (0*t)\\ y = -2 + (0*t)\\ z = -6 + 6t $$ I know how to solve it if I have more ...
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0answers
12 views

Find a complete sufficient statistic

Here is my problem : Suppose theta is a nonrandom parameter satisfying theta > 1. Suppose further that, given theta, Y1 , Y2, ... , Yn are i.i.d. observations with each density f_\theta(y) = (\theta - ...
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1answer
37 views

Parameterizations of Lines.

Which of the following equations give alternate parameterizations of the line L parameterized by: r(t)=(1+2t)i +(2+2t)j -(1+4t)k? a. -(1+t)i-t*j+(3+2t)k b, (3-2t)i+(2-2t)j+(3-4t)k c. ...
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104 views

Length of parametric curve $\phi(t)=(f(t)\cos(t),f(t)\sin(t))$

Define the curve $\phi$ by $\phi(t):=(f(t)\cos(t),f(t)\sin(t))$, where $f$ be a strictly increasing infinitly many differentiable function . Find an explicit formula for the length of $\phi$ ...
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0answers
8 views

Parametric equation ( Planes)

The three planes α, β and γ have the equations: α: x-2y+z=3 β:2x+y-3z=1 γ: x+y+az=1 where a is a real number 1) Given that α intersect the xy-plane (with equation z=0) in a straight line. Find a ...
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1answer
35 views

Parameterizing an ellipse

Given the ellipse $(x-1)^2 + \frac{y^2}{4}= 1$, parametrize the curve in polar coordinates. I've forgotten something very basic here. Can someone help get me started?
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2answers
32 views

Where does the line x = 2 − t, y = 3t, z = −1 + 2t intersect the plane 2y + 3z = 6?

form this 2y + 3z = 6 equation i take the x = 0. therefor 2 - t = 0 and ...
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0answers
25 views

An example of a space curve with given normal and osculating planes

I am student currently taking calculus 3 and I recently was given a quiz with a very difficult question. The question relates to the chapters in my book which talk about "Arc Length and Curvature" and ...
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1answer
21 views

Horizontal and vertical tangents to a parametric curve

I'm not sure what my procedure should be when solving this problem: find all points with a horizontal tangent find all points with a vertical tangent find all inflection points $$x(t) = ...
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2answers
20 views

Paramertized ellipses

I have $a^2 = (4x-8)^2 + 4y^2$ and $b^2=(4x+8)^2 + 4y^2$ which I switch between every $t=\frac{n\pi}2$ How do I draw this touching the origin, and moving outwards, noting that ...
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5answers
62 views

Evaluating a contour integral where C is a square

I've been working problems all day so maybe I'm just confusing myself but in oder to do this, I have to the take the integral along each contour $C_1-C_4$ My issue is how to convert to parametric ...
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1answer
30 views

How can I get a smooth distortion on a circle with a function g(x,y)

Let's say, $$f(x,y)=x^2+y^2=1$$ gives the unit circle. Now I would like to get a smooth distortion on the circle with a function $g(x,y)$. my guess is to consider the perimeter as one dimension, so ...
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1answer
199 views

Connected unbounded sets $S\subset \Bbb{R}^n$ such that $x\mapsto ||x||^t$ is uniformly continuous on $S$?

Spending the night perusing my old answers, and this question left me wondering about the following. Let's equip $\Bbb{R}^n$ with the usual Euclidean metric, and let us consider the map ...
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1answer
43 views

How to parametrize a curve using polar coordinates

How do I parametrize a curve $(x^2+y^2)^2=40^2(x^2-y^2)$ using polar coordinates? then what period is $\theta$ in?
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1answer
18 views

Equation for simple parametric curve

My math skills are rusty. I want to find the parametric equation for the 5 vertices curve below. It consists of an ellipse with a rotating axes. I get stuck after this: $$x = a \cos(t) \cos(\theta) - ...
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0answers
12 views

How to design own the parametric vector?

I try to design the parametric vector that looks like a roller coaster I know that my equation will like $r(t) = A\sin(t)i + Btj+ C\cos(t)k$, but i want ...
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1answer
15 views

curve with vanishing tangent vector assumption

I am just reviewing some assumptions in Parametric representations The book says we assume 3-d curve has non-vanishing tangent vector. Why do we need to assume this Simply if we take $R^3$ then ...
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0answers
25 views

Parameterize the equation

Find a way of parameterizing the following curve: $y^2=\sin x $. I have already tried $x(t) = (\sqrt t, \sin^{-1} t) $ but this only gives part of the curve because of the nature of the sqrt function ...
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2answers
22 views

Show that if $v_1$ and $v_2$ are any two vectors in this plane, then for any two scalars, $c_1v_1 + c_2v_2$ is also a vector in the plane

Let $a,\,b$ and $c$ be constants (not all zero) and consider the equation $ax + by + cz = 0$, which has a graph that is a plane that passes through the origin in $\mathbb{R}^3$. Show that if $v_1$ and ...
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1answer
22 views

Is this parametric equation correct?

Am I able to just put the planes point into the equation and leave it at that? or am I wrong here? My parametric equation: (x,y,z) = (3,-2,1)+ t(2,1,-3) + s(1,-2,4)
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0answers
15 views

Evaluate the line integral with Euler.

Need some help evaluating this line Intergral. $\int$$_c$ xy${e^y}$$^z$ dy Where C: x = 4t ; y = 3t$^2$ ; z = 3t$^3$ ; 0$\le$t$\le$1 Any help would be great. Thanks.
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2answers
33 views

Parametric / vector question.

Question 10 [10 points] Let L be the line with parametric equations $$ x = −6−3t $$ $$ y = 6+3t $$ $$ z = −8+2t $$ Find the vector equation for a line that passes through the point P=(−1, 2, 3) and ...
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2answers
24 views

How to define distance between two functions in a non-linear space (example of non-linear space: shape space)?

Suppose I have two parametric circle $f_1=(acost,asint)$ and $f_2=(bcos t,bsint)$, $t\in(0,2\pi),a>0,b>0$, which lies in some non-linear space. Are there any way, how to define the ...
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2answers
23 views

Creating a parametric Equation when given the points of a collinear line?

$(-70, 3)$, $(88, 81)$, and $(246, 159)$ are three collinear points. Write parametric equations for $x$ and $y$. (In other words, write equations that produce points when $t$-values are assigned.) ...
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1answer
32 views

Parametrizing shapes, curves, lines in $\mathbb{C}$ plane

I've been struggling with parametrizing things in the complex plane. For example, the circle $|z-1| = 1$ can be parametrized as $z = 1 + e^{i\theta}$. I'm not sure how this was done. I understand how ...
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1answer
24 views

Orthogonal parameterization

Consider the function $$f(a,b,c,d):=\frac{\left(a^*\right)^2b^2-\left(b^*\right)^2a^2+\left(c^*\right)^2d^2-\left(d^*\right)^2c^2}{a^*a+c^*c}$$ With complex parameters $a,b,c$ and $d$ Now find any ...
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1answer
34 views

Ray-sphere intersection: t-value of the intersection points

You have a sphere centered at [1,2,3] with radius 3, and a ray from [10,10,10] in the direction [-1,-1,-1]. Write the implicit equation for the sphere, the parametric equation for the ray, and compute ...
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0answers
67 views

Parametrization for intersection between sphere and elliptic cylinder

Given the sphere: $$x^2 + y^2 + z^2 = 12$$ and the ellyptical cylinder: $$(x-1)^2 / (7/3) + (y-2)^2 / 7 = 1$$ Give a parametrization for the intersection curve. I'm confused on how to do it. The ...
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1answer
8 views

How do I successfully combine these two paramaterized equations?

I'm working on a set of equations that would tell a hypothetical robot soccer player whether or not to pass a ball to a teammate. After a lot of algebra, I arrived at these equations for the partial ...
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1answer
23 views

Parametrization of the implicit curve $F(x,y)=a_1\cos x+a_2\sin x+a_3\cos y+a_4\sin y-b=0$

I am trying to find a parametrization for the curve defined implicitly by $$ F(x,y)=a_1\cos x+a_2\sin x+a_3\cos y+a_4\sin y-b=0, $$ where $a_1$, $a_2$, $a_3$, $a_4$ and $b$ are constants satisfying ...
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2answers
53 views

How to show that the curve $ (x,y,z) = \langle \cos t, \sin t, c\sin t\rangle $ is an ellipse?

Show that the curve $$(x,y,z) = \langle \cos t, \sin t, c\sin t\rangle $$ is an ellipse in the plane it lies on. $$x^2 + y^2 = (\sin t)^2 + (\cos t)^2 = 1$$ $$x^2 + (z/c)^2 = (\sin t)^2 + (\cos ...
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4answers
46 views

Using parametric differentiation for $\frac{\operatorname d \! y}{\operatorname d \!x}$?

Hi so I'm in my calculus class and the teacher gave us a problem to do. I'm not quite sure how to attack this question. He's given us a couple of steps but I don't understand. If someone can further ...
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1answer
23 views

Determine whether the points (5, -6, 10) and (3, 3, 8) are on the line x = 2 + t, y = 3 - 3t, z = 4 + 2t

I've gone about trying to solve this by assuming the x co-ordinate lies on the line, and then determining whether the other points lie on that line according to that ie. If x = 5, then 5 = 2 + t, ...
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2answers
38 views

Parametric form of square

What is the appropriate parametric equation of the boundary of a square? For example, the unit circle has a parametric equation $x(t)=\cos(t)$ and $y(t)=\sin(t)$.
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1answer
29 views

Parametrization of a Complex Path/Contour Integration

How would I parametrize the path which is a straight line from 1 to a complex point z? Does $\delta (t) = z^t$ make any sense?