# Tagged Questions

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

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### Need help understanding what the curve made by two or three intersecting surfaces looks like

I have trouble visualizing what curves are traced out by the intersection of multiple surfaces in $R^3$. for example take the parametric equations $<cos(t),sin(t),sin(t)$ > Clearly this would ...
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### Arc Length parametric curve

I have the following curve: $$x = \cos(t)$$ $$y = t - \sin(t)$$ $$0 \leq t \leq 2\pi$$ I have to draw the graph, point the direction and find its length. The solved the first two questions. The ...
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### When is it possible to eliminate the parameter from a set of parametric equations?

I found this question on-line, I was unable to source it. Is it poorly written? For example, is there a case (C) where you can have a non-parametric form given, and also have a parametric form (...
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### Continuity and differentiability of a function defined parametrically

How do we check continuity and differentiability of a function defined parametrically e.g. $$x=2t-|t-1|$$ and $$y=2t^2+t|t|$$
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### Area of described parametric region

The problem is as follows: A rope is tied to a cow and attached to the side of a circular silo with radius $r$. If the rope has length $\pi r$, what is the area of the land available for grazing ...
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### Parabola that intersects two lines and matching the slope of the two lines?

If I have two lines with equations;$$x=0$$ $$y=0$$ $$z=t$$ and $$x=t$$ $$y=10$$ $$z=t$$ are there any parabolas that cross through the two lines and in which the parabola matches the slope of the ...
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### How to parametrize $\left(4-\sqrt{x^2+y^2}\right)^2 +z^2=1$

How would I parametrize $$\left(4-\sqrt{x^2+y^2}\right)^2 +z^2=1$$ I am really struggling to parametrize this surface. Here is what I observed the surface is $$(4-r)^2+z^2=1$$ so perhaps we can try ...
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### How to parametrise shapes such as petals and cardioids?

Okay for example I want to compute a line integral along the curve described in polar coordinates by $r=\sin(2\theta)$ so I will need to parametrise this curve. (In fact I only need to parametrise one ...
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### Eliminating a parameter from 2 equations

The question given to me was actually of parametric differentiation, and the equations were: $$x = \dfrac{\sin^3 t}{\sqrt{\cos2t}}\ , \ \ \ \ y = \dfrac{\cos^3 t}{\sqrt{\cos2t}}$$ and we had to ...
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### Need help with unit circle trig coordinates.

I'm in over my head and need some help with this question. Sorry if this is too simple for you but I'm really struggling. I can't for the life of me figure out how to write the angles A in terms ...
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### Question about Flux and direction of normal

I am trying to do the following question; calculate the flux Suppose $$F(x,y,z)=(-x)i+(-y)j+(z^3)k$$ over the cone $z=\sqrt{x^2+y^2}$ between $z=1$ and $z=3$ with downward orientation My attempts: ...
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### Can parametric equations graph all kinds of lines?

I saw this question which had a similar viewpoint, but was limited to straight lines and polynomials. Now we know that we can graph some pretty crazy stuff with parametric equations. For example: ...
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### Parametric derivatives

Let $f(x) = \dfrac{2\sqrt{1+x^2}-5\sqrt{1-x^2}}{5\sqrt{1+x^2}+2\sqrt{1-x^2}}$. Hence, find $\frac{dy}{dz}$ when $y=\cot^{-1}(f(x))$ with respect to $z=\cos^{-1}{\sqrt{1-x^4}}$. To get this into a ...
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### Non-Uniform grid

Let say I have $v_0\in [v_1,v_m]$ (say $v_0=0.04\in [0.004,0.24]$) I would like to find a $1$-to-$1$ map that map $[0,1]$ to $[v_1,v_m]$ and more cluster points around $v_0$ from two sides. It seem ...
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### how to choose the suitable parametric form given a boundary?

Find the absolute minimum and maximum values of $g(x, y) = (x^2 + y^2)e^{(−x^2−4y^2)}$ on the set $A = \{(x, y) \in \mathbb R^2 \mid x^2 + 4y^ 2 ≤ 4\}$. here is the solution //see image so my ...
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### Prove that if $x = \sqrt{a^{\sin^{-1} t}}$ and $y = \sqrt{a^{\cos^{-1}t}}$ then $\frac{dy}{dx}$ = $-\frac{y}x$

Prove: If $x = \sqrt{a^{\sin^{-1} t}}$ and $y = \sqrt{a^{\cos^{-1}t}}$ where $\sin^{-1}$ and $\cos^{-1}$ are inverse trig function, show that $\frac{dy}{dx}$ = $-\frac{y}x$ Unfortunately I don'...
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### Solving math problems involving extra variable (p)?

I have a very hard time solving problems in which you have to solve for the additional unknown variable. I would like to know whether there is some method I can learn or approach I can simulate in ...
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### How to determine the period of the following Lissajous figure?

How do I determine the period of the following Lissajous figure? $$x(t) =\cos(2t)-\sin(t)\\ y(t)=\cos(t-\frac{\pi}{3})$$ Highly appreciated, Bowser
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### Write $x^2+y^2=25$ as a vector valued function

How can I write $x^2+y^2=25$ as a vector valued function? At first, I tried letting $x=t$. Then, $y=\pm \sqrt{25-t^2}$. So, $r(t)=t \hat{i}+ \sqrt{25-t^2}\hat{j}$ Would this be correct? What ...
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### Parametric equations and specifications of a logarithmic triskelion (triple spiral)

There is a post in this forum that shows how to create an Archimedean triskelion: Parametric equations and specifications of a triskelion (triple spiral) ...
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### Diffirent ways to write vector equations

Hi I am having some trouble with the following: When I am given some force F and it is in terms of components ie with respect to i, j, k then I have no issue using it to solve line integrals etc, my ...
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### Find the exact area of the region enclosed by the curve of parametric equations.

Find the exact area of the region enclosed by the curve given by $$x=9-t^2$$ $$y=e^t$$ where $-3 \leq t \leq 3$ and the $y$-axis. I tried to take the integral of the $x$ function minus the $y$ ...
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I want to find the surface area $S$ by rotating the curve about the $x$-axis $$\begin{cases} x = t^3 \\ y = t^2 \end{cases} ,t\in [0,1]$$ At some point I find $$S=\... 1answer 26 views ### Graph of parametric vector equation The question is asking me to sketch the graph of the parametric vector function$$\vec r(t) = \vec at+\vec bt^2 $$Where t is a real number, \vec a and \vec b are constant non-parallel non-zero ... 1answer 51 views ### Evaluate the integral \int_C z^2 dz where C is a graph of y=\sin x \int_C z^2 dz C: y=\sin x x=0 to x=3. My attempt: I'm not sure if I have this correct - this is a line integral? If so, a line integral is the same concept as a regular integral in the x-... 0answers 26 views ### Parametric equation for surface of revolution Given a parametric curve that can be rotated about the z-axis:$$P(t) = \langle 0, a \sqrt{t} + b \sin(t), ct \rangle, \quad t \in [0,2 \pi)$$Give a parametric equation and appropriate parameter ... 1answer 18 views ### Finding parameter for quadratic equation Given x^2 - 3ax + a^2 = 0 and$$\frac{x_1^4-x_2^4}{\sqrt{5}x_1x_2} + x_1 + x_2 -20x_1x_2 - 4 = 0$$Find a. The answer is 1 (a = 1) I tried to present x_1^4 - x_2^4 as (x_1+x_2)(... 3answers 36 views ### Converting a Parametric equation into a Cartesian one I was working on converting an parametric equation into a Cartesian one and i cant seem to figure this one out. I was hoping you could help with that for this equation of a cycloid, Thanks x = cos(t)... 0answers 20 views ### The different between Non-parametric statistics and Parametric statistics? I have 1D data. I want to classify the data to N cluster. The two common ways can use Using the mean/average value as a criterion to classify Assumption that the data follows a distribution with ... 0answers 21 views ### Parametric equations for mobius strip and klein bottle How do you fund the parametric equations for a mobius strip and klein bottle? I know I can just look up the equations online but I want to know how you get those equations. 1answer 39 views ### Vector and Parametric equations of Planes I really can't understand what is going on in this example. I understand the vector and parametric equations of a plane in R3, and I understand the examples below and above this one where they give a ... 1answer 37 views ### Problem using Parametric Equation of Semicubical Parabola I've been working my way through an old A'Level maths book and am having a lot of difficulty with a problem given in the Chapter on Loci & Parametric Equations: "Find the equation of the tangent ... 1answer 22 views ### How to find right parameter and calculate the work done Hi I am having trouble calculating the work done in moving a particle from (-1,2,5) to (1,0,1) where F=yi+xj+zk on the curve C, where the curve C is the intersection of z=x^2+y^2 and the plane ... 1answer 31 views ### Find angle of line and time of impact for a line between two parametric circles. I am trying to find the angle of a parametric line so that it will intersect a circular parametric curve when both of their parameters are equal. I also need to have the line's start position be ... 2answers 40 views ### How to predict symmerty of parametric curve Suppose we are given a curve as$$x^{2/3} + y^{2/3} = 1 $$In parametric form it can be written as$$x=\cos^{3}(\theta)$$and$$y=\sin^{3}(\theta) now how can we predict if curve will be symmetric ...
I'm given the equation $4x^2-y^2-24x-6y+23$ and asked to find the foci, vertices and asymptotes. The book showed me how to do it given an equation in the form of $(x^2/a)-(y^2/b)=1$, but didn't show ...
### Find conditions on $C$ and $C^{\prime}$ so that the spirals $r = Ce^{\varphi/a}$ and $r = C^{\prime}e^{\varphi/a}$ are the same
This question is related to one I asked here about the logarithmic spiral. In the linked problem, I had to find and sketch the image of the straight line $z=(1+ia)t+ib$, for \$-\infty < t < +\...