Questions tagged [parametric]

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

Filter by
Sorted by
Tagged with
0
votes
2answers
29 views

Is this function differentiable at point $A$?

I wonder if this funtion is differentiable at point $A$. I think that it is continuous so therefore I cannot use this to prove that it isn't differentiable. Can someone help me? Edit: My main issue is ...
0
votes
1answer
16 views

Spruce Budworm bifurcation diagram

The ODE that appears in the Spruce Budworm Population problem is the following: $$\frac{dN}{dt}=r_PN\left(1-\frac{N}{K}\right) -p(N) \quad \quad \mbox{where} \quad \quad p(N)=\frac{BN^2}{A^2+N^2}$$ ...
0
votes
0answers
52 views

How would I find the limit of this recursive sequence?

The Question $$x_n(c)=\frac{\int_{1}^{c} x_{n-1}(b)y_{n-1}(b)db}{\int_{1}^{c} y_{n-1}(b)x_{n-1}(b)'db}, x_1(c)=\frac{\int_{1}^{c} xf(x)dx}{\int_{1}^{c} f(x)dx} $$ $$y_n(c)=\frac{\int_{1}^{c} y_{n-1}(...
0
votes
1answer
21 views

Show the given relation related to partial derivatives

Assume $z=f(x,y)$ with $x= \rho \cos(\theta)$ and $y= \rho \sin(\theta)$,then show $$(\frac{\partial z}{\partial x})^2+(\frac{\partial z}{\partial y})^2=(\frac{\partial z}{\partial \rho})^2+\frac{1}{\...
1
vote
1answer
29 views

Rational parametric equation of a circle from a line

I found out that we can define a circle equation as follows: $$\begin{cases}x(t)=\dfrac{t}{t^2+(kt+b)^2},\\y(t)=\dfrac{kt+b}{t^2+(kt+b)^2},\end{cases}$$ where $k, b$ are real numbers. For example, if ...
0
votes
2answers
19 views

Intuitive way to eliminate a parameter

I need to eliminate $\theta$ from the equations $x=\sin\theta+\cos\theta$ and $y=\tan\theta+\cot\theta$. I am actually provide with a hint: consider $x^2y$ , which worked nicely for me. However I am ...
0
votes
1answer
19 views

Parametric Equation for the intersection of a curve

I am trying to find the parametric equation for the cylinder x^2+z^2=4 and the plane through the points (-1,3,0), (1,2,0), and (0,2,2). I obtained 2x + 4y + z = 10. As the equation of the plane ...
0
votes
1answer
20 views

Computing the parametric equation of the line of intersection of two planes

I'm given the following problem: Find the parametric equation of the line of intersection of the two given planes: $y-6z = 7$ and $9x - 8y = 5$ My attempted solution follows: I take the cross ...
0
votes
2answers
32 views

Why is there a point on the unit circle that is not represented by these parametric equations?

$$x=\frac{2t+1}{2t^{2}+2t+1}$$ $$y=\frac{2t^{2}+2t}{2t^{2}+2t+1}$$By squaring $x$ and $y$ and adding them up, I obtained $x^2 + y^2 = 1$ after some algebraic manipulation. But the question asks which ...
1
vote
2answers
41 views

Parameterizing both branches of a hyperbola

Recently I have been studying parametric equations of surfaces and curves, specifically hyperbolic functions. Given by the equations $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \quad\frac{(x-\alpha)^2}{a^2}-\...
0
votes
1answer
23 views

Changing the angle / curvature of a mesh created using python and parametric equations

Changing the angle / curvature of a mesh created using python and parametric equations. I can create a Hyperboloid using Python and parametric equations Python script formula snippet: ...
0
votes
1answer
23 views

Find an equation of the line tangent to the graph of 𝐶 at the point where 𝑡=1

So I was given the following prompt when studying for a test: "A curve $C$ is defined by the parametric equations $x(t)=3+t^2$ and $y(t)=t^3+5t$. Find an equation of the line tangent to the graph ...
1
vote
1answer
31 views

How do I prove that $x,y$ and $z$ lie on the same line if they are related to the same parameter?

Related to this question How exactly does that work? I would be grateful for a proof but barring that just the name of the theorem, so I can look it up myself. My work/thoughts so far: If we start ...
0
votes
2answers
29 views

How to prove that the intersection of two planes is a line (using parameteric equations)?

I'm reading the explanation to why the intersection between two planes is a line in the textbook. This seems reasonable enough, but I don't understand the last part of the proof. This indicates that ...
0
votes
1answer
44 views

What is the single equation for a helix?

Is there a way to describe a helix not by its parametric form $$ x=R\cos(t) ,\ y=R\sin(t) , \ z=ht , $$ but by a single equation like you can for a sphere with $ r^2 = x^2+y^2+z^2 $? Also the same ...
0
votes
0answers
13 views

Area under curve, constraints on function for parametric domain

If I have that the area of a curve $y=F(x)$ from $a$ to $b$ is equal to $$\space A=\int_{a}^{b}y \space dx \tag1 \label{eq1}$$ and the curve is also traced out by parametric equations $x=f(t)$, $y=g(t)...
0
votes
0answers
19 views

Irregular statistical model

I am having trouble understanding irregular statistical models. I know that a parametric model $PP=\{P_\theta:\theta\in\Theta\}$ is said to be regular if each $P_\theta$ has probability density $p(x,\...
2
votes
3answers
104 views

$x^4-mx^3-2mx^2+2m^2x=0$ roots in arithmetic progression

For which real values of $m$ roots of equation $$x^4-mx^3-2mx^2+2m^2x=0$$ are in arithmetic progression? I managed to find a solution using a lot of casework, i.e. factoring the equation into $$x(x-m)(...
-2
votes
1answer
56 views

why am I getting a different answer with $(y_1-y_2) = m(x_1-x_2)$ to when I use $y = mx + c$ ?!?

The question in the book is: 'What is the equation of the tangent of the curve with parametric equations $x = 3 - 2\sin{t}$ and $y = t\cdot \cos{t}$, at the point where $t = \pi$?' The differentiation'...
0
votes
1answer
51 views

Locus of center of tilted rectangle that travels around an ellipse [closed]

In order to determine the stochastic visibility (https://link.springer.com/book/10.1007/978-1-4612-2690-1) in Poisson fields where the center of blockages is randomly distributed according to a ...
0
votes
0answers
5 views

Parametric and Vector-Valued Functions Question

At $t = 0$ a particle is at point $(1, −2).$ Find the position of the particle that moves with velocity vector $v(t) = <t-\pi\cos(\pi t), 2t-\pi\sin(\pi t)>$ when $t=3.$ My Work: We take the ...
1
vote
1answer
40 views

Finding points on $\vec r(t) = \vec At^3 + \vec Bt^2 + \vec Ct + \vec D$ where the tangent is parallel to the line $px + qy + k = 0$

Given a line defined by the equation: $$px + qy + k = 0$$ and a parametric cubic curve defined by: $$\vec r(t) = \vec At^3 + \vec Bt^2 + \vec Ct + \vec D$$ where both curves lie in 2D space, how can I ...
0
votes
1answer
19 views

Determine the closest point(s) from a line to a parametric cubic curve where the distance is less than $L$

Given a parametric cubic curve in 2D space: $$\vec r(t) = \vec At^3 + \vec Bt^2 + \vec Ct + \vec D$$ and a line defined by the equation: $$px + qy + k = 0$$ I can easily determine where these two ...
0
votes
1answer
18 views

Find inflection point of parametric cubic

I'm familiar with turning points and inflection points for "normal" graphs (i.e. those that relate y and x) but how would I get an inflection point for a graph where each dimension has a ...
2
votes
2answers
42 views

Eliminating time from parametric questions

Is there a general technique to eliminate a parameter from two parametric equations? E.g. given the following two parametric equations dictating the motion of a point how can I eliminate parameter $t$ ...
0
votes
3answers
62 views

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2\sin\theta}{2\sin\theta-1}$

Parametric equation of curve is $ x=2\sin\theta +\cos(2\theta) $ , $y=1+\cos(2\theta)$ , for $0\leq\theta\leq\pi/2$ Show that $$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2\sin\theta}{2\sin\theta-1}$$ I ...
0
votes
1answer
38 views

Parametrization of a circle and an ellipse [closed]

Can someone explain to me why $(\sin(t) - \cos(t), \cos(t) + \sin(t))$ is a parametrization of a circle and it "travels" in the clockwise direction but $(\cos(t) + \sin(t), -2\sin(t) + 2\cos(...
0
votes
1answer
17 views

How to derive the cartesian equation of cycloid expressed for y? [closed]

I found an expression for x in wiki: https://proofwiki.org/wiki/Equation_of_Cycloid_in_Cartesian_Coordinates but I need it expressed for y. How can I do that?
0
votes
0answers
25 views

$C$ border of surface $x^{2} + y^{4} + z^{4} = 1$ and $v= (3zx^{2}, 12\sin y, x\cos z^{3} )$. Calculate $\int_{C}^{}v dt$

Let $C$ be the border of the surface $S$ with equation $x^{2} + y^{4} + z^{4} = 1$ and $v= (3zx^{2}, 12\sin y, x\cos z^{3} )$. Then calculate $\int_{C}^{}v dt$. I know that this is a line integral. I ...
1
vote
1answer
33 views

Find the length of C which has parametric equations $ x=9 \cos (t)-2, \quad y=9 \sin (t)+1, \quad-\frac{\pi}{6} \leqslant t \leqslant \frac{\pi}{2} $

I solved it by first finding the Cartesian equation of $C$ and its domain & range. $$ (x+2)^{2}+(y-1)^{2}=81,\;\;-2 \leq x \leq 7,\;\;-\frac{7}{2} \leq y \leq 10 $$ After sketching the curve we ...
0
votes
3answers
46 views

How to find inverse function of function with parameter?

I'm trying to find the inverse function of this function: $$ f_a(x) = \frac{x^a}{x^a + (1-x)^a} $$ $a$ - parameter $x$ - variable Is it possible to find if there is a parameter here? Hope for your ...
0
votes
1answer
23 views

How do I convert the parametric equation to the corresponding rectangular equation? [closed]

x = t^2 + t ; y = t^2 − t I'm getting a couple of answers for this question so I'm confused. I'd love to receive some help, thank you!
1
vote
0answers
24 views

Limits for the parameters of the parametric equation of a surface of revolution

According to Lipschutz's Differential Geometry book, A surface of revolution $S$ is obtained by revolving a plane curve $C$ (the profile curve) about a line $L$ (the axis of $S$) in its plane. If $x_{...
0
votes
0answers
14 views

Determine the smallest 3D cuboid which entirely contains a domain-restricted parametric curve

If I have a parametric cubic curve in 3 dimensions defined by: $$\vec r(t) = \vec At^3 + \vec Bt^2 + \vec Ct + \vec D$$ where $t\in [0; 1]$, how can I determine the co-ordinates of the smallest 3D ...
2
votes
3answers
81 views

Finding the self intersection point of two parametric equations.

Find the values of t where the graph of the parametric equations crosses itself: $x = t^3-4t^2+t+7$ and $y = t^2-t$. I am asking this problem on behalf of me and my Calculus 3 class. We understand the ...
0
votes
1answer
16 views

Parametric equation gives dy/dt = sin(t) +3 and x(t)= 6t^2+ln(t) and asks for when the graph of the position will have a horizontal tangent

To my understanding if $\dfrac{dy}{dt}=0$ then the tangent is horizontal, but $\sin(t)+3$ will never equal $0$, therefore there is no horizontal tangent. Thanks!
0
votes
1answer
39 views

Finding a such that a parametric equation has solution

I've been trying to solve this parametric equation: Find all $a$ for which there exist $x$ and $y$ such that $a(3x+2y) = y + \sqrt{2a^2(4x^2+y^2)} + \sqrt{2a^2x^2+2(a-1)^2y^2}$. I noticed x and y can ...
0
votes
0answers
27 views

Find point on curve given tangent vector

Given a parametric curve $\vec r(t)$ and a vector $\vec v$ (not necessarily a vector that was calculated using the derivatives of $\vec r(t)$ , and not necessarily one that is normalized) what is the ...
0
votes
1answer
39 views

Parametric equation of an ellipse in the 3D space

I have found here that an ellipse in the 3D space can be expressed parametrically by $$\mathbf x (t)=\mathbf c+(\cos t)\mathbf u+(\sin t)\mathbf v$$ with $\mathbf c = (c_1,c_2,c_3)$ being the center ...
0
votes
2answers
39 views

Finding a parametrization of a curve from cartesian equations

I'm in need of some help to address this problem. Let $C$ be a curve given by two equations: $x^2+y^2-z^2-1=0$, $x^2-y^2-z^2-1=0$ Express the curve by means of parametric equations. Any ideas on how ...
2
votes
2answers
55 views

How to be sure about any parameterization?

This is the problem: Find intersection of this two surfaces and then calculate its length. $$x^2+4y^2=4$$ and $$y\sqrt3 +z=1$$ Method 1 : Common way is to take: $$x=2\cos t ,\: y=\sin t ,\: z=1-\...
2
votes
4answers
70 views

Why does the intersection of two parametric curves have different t results for each curve?

I'm slightly confused after reading the problem posed here: Points of intersection of two parametric curves Why is it that the "t" result at the intersection point of the two curves is not ...
0
votes
1answer
41 views

Parametric equation of ellipse

Find the curvature and the radius of curvature for (f) $x = 2 \cos t$ and $y = 3 \sin t$, $0 < t < 2\pi$ at point $(2, 0)$ and $(0, 3)$, where the parametric equation given is a ellipse $\frac{x^...
0
votes
0answers
12 views

Relationship of cartesian and parametric form of a line in R3

When one converts the parametric representation of a line in R3 to its cartesian form doesn't that create some ambiguity? If I look at the plane expressed by that cartesian equation and the previously ...
1
vote
0answers
24 views

Can you derive a parametric function which describes a crystallographic screw axis?

A $2_{1}$ screw axis is defined as a 180-degree rotation followed by a translation of $\frac{1}{2}$ along a particular unit cell vector. In matrix form: \begin{equation} \begin{bmatrix} -1 & ...
0
votes
2answers
43 views

Vector form of a plane

Express the following plane in vector form: $\mathcal P_1\subseteq \Bbb R^3$ with equation $4x-z=0$. The answer is $t(1,0,4)+s(0,1,0)$. I don't understand how they got $(0,1,0)$ for the direction ...
0
votes
1answer
16 views

How do I find the value of k based on the fact that x+k is a tangent to a parametric equation?

I have been given the question of: A curve has the parametric equations x=2$t^2$ and y=4t. Find the value(s) of k if y=x+k is a tangent to the curve. Being the first question I've gotten of this ...
0
votes
0answers
16 views

Converting this parametric curve to a level curve

I want to convert the parametrized curve $\gamma(t) = (\cos^{3}(t), \sin^{3}(t)), \ t \in \mathbb{R}$ to a level curve. Let $C = \{(x,y) \in \mathbb{R^{2}}: x^{2/3} + y^{2/3} = 1\}$. I claim that $x^{...
6
votes
2answers
101 views

Solving parametric in rationals

I want to find all values of a such that the equation $$\sqrt{a - x} = a - x^2$$ has at least one real root and none of its roots are irrational. I made some decent ...
0
votes
0answers
29 views

Understanding the formula for parametric derivative

I'm trying to make the sense of the identity in this wikipedia Here is my attempt at proof at this: $$ \frac{dy(x(t))}{dt} = \frac{dy}{dx} \frac{dx}{dt} \tag{1}$$ If we differentiate this identity ...

1
2 3 4 5
45