Questions tagged [parametric]
For questions about parametric equations, their application, equivalence to other equation types and definition.
2,203
questions
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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
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1answer
16 views
Spruce Budworm bifurcation diagram
The ODE that appears in the Spruce Budworm Population problem is the following:
$$\frac{dN}{dt}=r_PN\left(1-\frac{N}{K}\right) -p(N) \quad \quad \mbox{where} \quad \quad p(N)=\frac{BN^2}{A^2+N^2}$$ ...
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How would I find the limit of this recursive sequence?
The Question
$$x_n(c)=\frac{\int_{1}^{c} x_{n-1}(b)y_{n-1}(b)db}{\int_{1}^{c} y_{n-1}(b)x_{n-1}(b)'db}, x_1(c)=\frac{\int_{1}^{c} xf(x)dx}{\int_{1}^{c} f(x)dx} $$
$$y_n(c)=\frac{\int_{1}^{c} y_{n-1}(...
0
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1answer
21 views
Show the given relation related to partial derivatives
Assume $z=f(x,y)$ with $x= \rho \cos(\theta)$ and $y= \rho \sin(\theta)$,then show
$$(\frac{\partial z}{\partial x})^2+(\frac{\partial z}{\partial y})^2=(\frac{\partial z}{\partial \rho})^2+\frac{1}{\...
1
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1answer
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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
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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
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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
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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
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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 ...
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Parameterizing both branches of a hyperbola
Recently I have been studying parametric equations of surfaces and curves, specifically hyperbolic functions. Given by the equations $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \quad\frac{(x-\alpha)^2}{a^2}-\...
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1answer
23 views
Changing the angle / curvature of a mesh created using python and parametric equations
Changing the angle / curvature of a mesh created using python and parametric equations.
I can create a Hyperboloid using Python and parametric equations
Python script formula snippet:
...
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1answer
23 views
Find an equation of the line tangent to the graph of 𝐶 at the point where 𝑡=1
So I was given the following prompt when studying for a test:
"A curve $C$ is defined by the parametric equations $x(t)=3+t^2$ and $y(t)=t^3+5t$. Find an equation of the line tangent to the graph ...
1
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1answer
31 views
How do I prove that $x,y$ and $z$ lie on the same line if they are related to the same parameter?
Related to this question
How exactly does that work? I would be grateful for a proof but barring that just the name of the theorem, so I can look it up myself.
My work/thoughts so far:
If we start ...
0
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2answers
29 views
How to prove that the intersection of two planes is a line (using parameteric equations)?
I'm reading the explanation to why the intersection between two planes is a line in the textbook. This seems reasonable enough, but I don't understand the last part of the proof. This indicates that ...
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1answer
44 views
What is the single equation for a helix?
Is there a way to describe a helix not by its parametric form
$$ x=R\cos(t) ,\ y=R\sin(t) , \ z=ht , $$
but by a single equation like you can for a sphere with $ r^2 = x^2+y^2+z^2 $?
Also the same ...
0
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0answers
13 views
Area under curve, constraints on function for parametric domain
If I have that the area of a curve $y=F(x)$ from $a$ to $b$ is equal to
$$\space A=\int_{a}^{b}y \space dx \tag1 \label{eq1}$$
and the curve is also traced out by parametric equations $x=f(t)$, $y=g(t)...
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0answers
19 views
Irregular statistical model
I am having trouble understanding irregular statistical models.
I know that a parametric model $PP=\{P_\theta:\theta\in\Theta\}$ is said to be regular if each $P_\theta$ has probability density $p(x,\...
2
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3answers
104 views
$x^4-mx^3-2mx^2+2m^2x=0$ roots in arithmetic progression
For which real values of $m$ roots of equation
$$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)(...
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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
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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 ...
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Parametric and Vector-Valued Functions Question
At $t = 0$ a particle is at point $(1, ā2).$ 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
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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
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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 $t$
...
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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
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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 $(\cos(t) + \sin(t), -2\sin(t) + 2\cos(...
0
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1answer
17 views
How to derive the cartesian equation of cycloid expressed for y? [closed]
I found an expression for x in wiki: https://proofwiki.org/wiki/Equation_of_Cycloid_in_Cartesian_Coordinates but I need it expressed for y. How can I do that?
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$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
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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
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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
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1answer
23 views
How do I convert the parametric equation to the corresponding rectangular equation? [closed]
x = t^2 + t ; y = t^2 ā t
I'm getting a couple of answers for this question so I'm confused. I'd love to receive some help, thank you!
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0answers
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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_{...
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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
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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
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1answer
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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!
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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
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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
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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
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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
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2answers
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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
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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
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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 ...
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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
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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
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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
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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$$
has at least one real root and none of its roots are irrational. I made some decent ...
0
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0answers
29 views
Understanding the formula for parametric derivative
I'm trying to make the sense of the identity in this wikipedia
Here is my attempt at proof at this:
$$ \frac{dy(x(t))}{dt} = \frac{dy}{dx} \frac{dx}{dt} \tag{1}$$
If we differentiate this identity ...
