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

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0answers
23 views

Find all real numbers a such that equation 3^(x2+2ax+4a−3)−2=|(a−2)/(x+2)| Has exactly two different roots x1,x2 those belong to [−4;0]

Find all real numbers $a$ such that equation $${3^{(x^2+2ax+4a-3)}}-2=|{a-2 \over x+2}|$$ Has exactly two different roots $x_1,x_2 $ those belong to $[-4;0]$ Tried plenty different things to solve: ...
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2answers
23 views

How do you find a parametric representation for a specific surface?

I trying to find a parametric representation of the plane which goes through the origin and contains the vectors $i-j$ and $j-k$. I found the cross product for these vectors and found that the formula ...
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2answers
39 views

How do you parameterize a circle?

I need some help understanding how to parameterize a circle. Suppose the line integral problem requires you to parameterize the circle, $x^2+y^2=1$. My question is, if I parameterize it, would it ...
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3answers
27 views

Express angular position of the Earth as a function of time

Say I have for example the Earth orbiting the Sun (in circular orbit) and I want to express angular position (in radians) as a function of time. Would I be correct in thinking that $2\pi/t$ does the ...
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1answer
16 views

Where Am I Going Wrong on this Curvature Problem?

Massively deviating from the given answer on this one, have no idea what's going on. I'm sure that I'm using the correct formulas, but the answer provided is very different from what I come up with. ...
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0answers
18 views

Shannon entropy on the simplex

The state of a trit $$\{p_1,p_2,p_3=1-p_1-p_2\}$$ can be represented as a triangular simplex. The centre of the simplex is the maximally mixed state $$m=\{\frac{1}{3},\frac{1}{3},\frac{1}{3}\}$$. And ...
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19 views

Parameterized Curvature Problem

Massively deviating from the given answer on this one, have no idea what's going on. Find the curvature of the following function: x = 5cos(t) ; y = 4sin(t) ; t = $pi$/4 Formula for curvature is k ...
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0answers
14 views

Horizontal and Vertical Tangents of Limicons

I feel like I am over thinking this problem, and am probably just confusing myself... So I need to find the values of $t$ where the equation $r=a+b\cos(t)$ has horizontal and vertical tangents, for ...
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2answers
22 views

Can't Seem to Solve This Parametric Arclength Problem

I'm working on an arclength question right, now, and I'm consistently landing very close to the answer but not on it. I'm trying to find the arclength of the following: $x = 6e^tcos(t) ; y = ...
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0answers
21 views

volume under the surface with parametric equations

with parametric equations: What is the volume under the surface $z=x+y$ on the plane $xy$ and interior to the surface $$(x^2+y^2)^2=2xy\space, \space \space x>0, y>0 $$
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2answers
46 views

Can the volume of a partial sphere be re-written as $V= \frac{4\pi}{3}r^{2}h$?

I am doing a related rate problem where I have a spherical surface given by the spherical equation: $$V_{sphere} = \frac{4}{3}\pi r^3,$$ $$ρ = 4(1+cos(ϕ)),$$ bounded by the $xy$ plane I have a rate ...
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0answers
25 views

Is this the correct parametrization for the intersection between a sphere and a plane?

I'm studying Stokes' Theorem. I want to compute $\oint_C (-y^3,x^3,-z^3)$ where $C$ is the intersection between the sphere $x^2+y^2+z^2=4$ and the plane $x+y=0$. Applying the Theorem, I wanted to ...
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1answer
63 views

Verify Stokes' Theorem

I need to verify Stokes' Theorem for the vector field $F(x,y,z)=(x^2,xy,z^2)$, for the surface, $S$, given by the part of the plane $x+y+z=1$ that is inside the cylinder $x^2+y^2=x$. For this I ...
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1answer
32 views

Show in parametric form plane in $R^3$

I have this question in my book. Show in parametric form the plane of $R^3$ that determined by these points : $$(1,0,0)$$ $$(0,1,0)$$ $$(0,0,1)$$ Does $(0,0,0)$ found on this plane? My answer The ...
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0answers
26 views

Challenging Path Integral

Let $c$ be the curve of intersection of the plane defined by $x+y+z = a$ and the cylinder $x^{2} + y^{2} = a^{2}$ ($a > 0$). Evaluate the path integral: $\displaystyle\oint_{c} \sqrt{a^2 + xy} ...
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0answers
47 views

Rational dynamical system with nonnegative paramaters and with nonnegative initial conditions

let $A$ be a rational system of the form :\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...
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1answer
30 views

Is it possible to parametrise $x^{\frac{1}{x-1}}=y^{\frac{1}{y-1}}$?

I don't know if there is a process for parametrising $y^\frac{1}{y}=x^\frac{1}{x}$ and suspect it is not possible to do so. But if it is possible, is it also possible for the similar ...
2
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1answer
38 views

Surface integrals: Find the area of the portion of the cone $x^2+y^2=z^2$ above the $xy$ plane and inside the cylinder $x^2+y^2=ax$

I need to find the area of the portion of the cone $x^2+y^2=z^2$ above the $xy$ plane and inside the cylinder $x^2+y^2=ax$ . For this, I used cylindrical coordinates to parametrize the region: ...
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1answer
63 views

Let $S$ be the surface generated by the circles of radius $b$, find a parametric expression for $S$

Let $C$ be the curve associated to a regular, simple path $\theta:[0,l]\rightarrow \Bbb R^2 $; also assume that $((x'(s))^2+((y'(s))^2=b^2$ and let $S$ be the surface generated by the circles of ...
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0answers
32 views

Perpendicular line to plane

Consider the plane $\large2x+y-5z=7$ and the line with parametric equation $\large r=r_0+tu$ a) Give a value of u which makes the line perpendicular to the plane. So I'm confused on what to do. I ...
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0answers
11 views

Parametric Surfaces: rank of linear transformation is the plane spanned by $T_u$ and $T_v$

Let $\phi$ be a parametrized surface which is both simple and smooth in a point $(u_0,v_0)$. I want to prove that: i) The rank of the linear transformation $D\phi (u_0,v_0)$ is the plane spanned by ...
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1answer
13 views

How can I parameterize these angles

My angle starts at 90, goes down to 0/360, then down to 270. This is a 180 degree range of motion. How can I express these angles from 0 - 180 instead of 90 - 270, where 90 gives 0 and 270 gives 180. ...
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1answer
48 views

Hysteresis Model with two real parameters

I would like to ask the following: I am trying to make a throughout analysis of a Hysteresis model in one dimension, with two real parameters: $\frac{dx}{dt}=f(x,ν,μ)=νx-x^3+μ$, where ...
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0answers
19 views

Existence of Lebesgue measure in parametric function space

I am thinking about this question but I can not solve. The question is: Can I define Lebesgue measure in the space of parametric functions and if the answer is yes what is that Lebesgue measure? Could ...
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0answers
20 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
17 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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13 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
18 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
15 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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2answers
42 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
47 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
23 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 ...
2
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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
26 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
28 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
18 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
11 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
24 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
45 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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0answers
113 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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1answer
42 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
28 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
24 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
66 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
32 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 ...
12
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1answer
222 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 ...
0
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1answer
47 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?