Questions tagged [parametric]
For questions about parametric equations, their application, equivalence to other equation types and definition.
0
votes
0answers
19 views
Supply and Demand Functions with Tax
I've been given the below supply and demand functions:
$q^s(p)=50p~~~~~~q^d(p)=100(\frac{12}{p}-1)$
I've answered the first few questions, which include finding the equilibrium etc, and inverting ...
1
vote
2answers
27 views
Area of Parametric Curves
Compute, in terms of $A, B, h$, and $k$, the area enclosed by the curve defined by the parametric equations:
$x(θ)=Acosθ+h$
$y(θ)=Bsinθ+k \quad \quad \quad \quad \quad \quad $ for $0 ≤ \theta ≤ 2π$....
2
votes
3answers
32 views
Area Inside A Loop Formed By Parametric Equations
We are given:
$x=49-t^2$
$y=t^3-16t$
The curve apparently makes a loop which lies along the x-axis. I need help finding total area inside the loop. I don't know where to even start.
If it helps, ...
0
votes
1answer
28 views
Parametric equations derivative's with $0$ in the denominator
Let's consider the following question
Let $x=\frac{t}{1+t^2},$ $y=3t^2-6t$. Find $\frac{dy}{dx}\Big|_{t=1}$.
It is known that $$\frac{dy}{dx}=\frac{dx/dt}{dy/dt}.$$
The problem is that both $\frac{...
1
vote
2answers
40 views
How Can I derive the Parametric equation for Toilet Paper roll
I was going through this wonderful article The length of toilet roll
I have a similar problem where I wanted to find the outer Radius (R) of the Sheet metal roll which can be thought of as Toilet ...
-1
votes
1answer
61 views
Find limit function of $\Phi(\alpha) = \int_\limits{0}^{\alpha}\ln(1+\alpha x)\,\mathrm{d}x$ [on hold]
I have the integral:
$$\Phi(\alpha) = \int_\limits{0}^{\alpha}\ln(1+\alpha x)\,\mathrm{d}x$$
I have to find a limit function but I have no idea where/how to start? I was thinking about L'Hopital ...
1
vote
1answer
25 views
Calculate the derivative of underintegral function and find the integral
I have the integral:
$$ \Phi(\alpha)=\int\limits_{1}^{2\alpha}\frac{\cos(2\alpha x^3)}{x}\,\mathrm{d}x \tag{1}$$
Have to find $\Phi'(\alpha)$ and calculate the integral then.
do I understand ...
0
votes
1answer
9 views
Writing y value of Curtate Trochoid in the function of x?
The parametric equations of a trochoid are
$x = Rt-d\sin(t)$
$y = R-d\cos(t)$
For $d < R$, there should be only one corresponding y value for every $x$ value.
So can we express this equation as ...
1
vote
1answer
48 views
Math competition function with parameter problem
Given a function $$f(x) = \frac{ax + 2}{3x - \frac{1}{a}}$$ find every possible value of the parameter $a$ such that for all real values $x$ for which $f(x)$ is defined it is true that $f(f(x))$ is ...
0
votes
0answers
29 views
Finding the magnitude of displacement with a given parametric velocity function
I am struggling to get the correct answer for the question: what is the magnitude of the displacement of a particle moving in thee xy-plane with the velocity vector given by $v(t) = <e^\left(sint\...
1
vote
1answer
36 views
Parabola tangent to two lines and through two points on those lines
is it possible to calculate parabola that is tangent to two lines exactly on black points? (please see enclosed picture)
And linked question is:
If we assume red line is given by:
$$
f_{1}(x)=S_{1}(...
0
votes
0answers
31 views
Find a parametric equation that draws a quarter circle of radius 9 that crosses the y-axis and is as much in quadrant III as it is in quadrant IV.
In preparation for my precalculus test, I was assigned many challenge problems, including this one. I have been struggling to find a solution to this question, as when I graphed my solution, it seemed ...
0
votes
1answer
17 views
Parametrization of right circular cone
community!
Write the parametric equations of a right circular cone of height $h$
and semi-aperture $α$, lying on the plane $z = 0$, contained in the first
octant, so that the segment between ...
-1
votes
1answer
21 views
why parametrized $(c-2b+6)^2+2(2b-9)^2=2$ is $b=\frac{(8*t^2+5)}{(2*t^2+1)}$and $c=4\frac{(t^2-t+1)}{(2*t^2+1)}$**
parametrized enter image description here(c−2b+6)2+2(2b−9)2=2; is b=(8∗t2+5)(2∗t2+1); and; c=4(t2−t+1)(2∗t2+1);enter image description here
0
votes
1answer
30 views
Simple Displacement of Parametric Equations Dispute
A parametric equation with $\frac{dx}{dt}$ = something, $\frac{dx}{dt}$ = something, has a resultant velocity vector by pythagorean theorem to be $\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}$. ...
2
votes
1answer
29 views
Define a parametrization of the curve $C$
Consider the curve $C\in\mathbb R^3$ defined in Cartesian coordinates, by the equations
$$x=\sqrt{4-y^2-z^2}\qquad y=z+1$$
from $P_0=(\sqrt3,1,0)$ to $P_1=\left(\frac{\sqrt6}2,\frac32,\frac12\...
0
votes
1answer
23 views
Find the tangent line at pont $q=(a,b)$ of the following curves.
I need help with this problem:
Find, where appropriate the equation of the tangent line to $C$ at the point $q=(a,b)$ on $C$. Indicate points on $C$ where no tangent line exists.
the $y$-axis, $x=...
1
vote
1answer
24 views
Cartesian equation of a hyperbolic paraboloid given his parametric equations
I can interpolate a hyperbolic paraboloid given 4 points in space ($\pmb{a_1}, \pmb{a_2}, \pmb{b_1}, \pmb{b_2}$) with the following parametric equation for a ruled surface:
$$ \pmb{r}(u, v) = (1-v)((...
2
votes
1answer
31 views
area under parametric curve
I have difficulty on how to eliminate parameter especially the equation involved trigonometry equation.
The question is asking for the area bounded by the curve , the 2 axes and the line $y=1$.
$x=4 ...
0
votes
1answer
62 views
Solve through Euler integrals [closed]
I know, that it looks like Euler integral should be used here, but no idea, how to simplify
$$\int_0^1\frac{x^{1-a}(1-x)^a}{(x+1)^3}dx$$
0
votes
2answers
31 views
About the slope of a parametric curve at a point. (James Stewart Calculus 8th Edition.)
I am reading "Calculus 8th Edition" by James Stewart.
Suppose $f$ and $g$ are differentiable functions and we want to find the tangent line at a point on the parametric curve $x=f(t),y=g(t)$, ...
0
votes
2answers
27 views
What's the parametric equation of a partial ellipse in 3D space with given major axis and minor axis and start point and end point?
Consider a partial ellipse in 3D space with the following information:
majorAxis (a 3D vector, such as [1, 0, 0])
minorAxis (a 3D vector, such as [0, 1, 0])
startPoint (a 3D vector, such as [1, 0, 0])...
0
votes
2answers
24 views
Parametric equation for y^2 = x+1 [closed]
How do you parametrise
$$y^2 = x+1$$
?
I only know how to parametrise a circle.
0
votes
0answers
15 views
Distance Between Parametric Curve and Plane as a Function of Time
Find the distance from the curve r(t) = cos ti + sin tj + t/(2pi)k to the plane -x-y+z=1 as a function of time.
I suspect that this problem, which presumably asks for the shortest distance between ...
2
votes
3answers
176 views
Is it possible to map from a parameter to a trajectory?
If I have a polynomial trajectory
$$y(t)=at^2+bt$$
The idea I would like to express is that by fixing $a$ and $b$, I will get a unique trajectory $y(t)$. Can I say that there exists a map $M$ such ...
1
vote
4answers
41 views
Is it possible to convert the parametric curve defined by $x = t^3 - 3t$ and $y = t^2 - 4$ to an implicit function?
Is it possible to convert the parametric curve defined by $$x = t^3 - 3t \text{ and } y = t^2 - 4$$
to an implicit function?
0
votes
1answer
25 views
How to differentiate multivariable calculus with parametric equations?
$u=u(x,y)$ and $x=e^s\cos t, y=e^s\sin t$
How do I work with this type of multivariable calculus?
E.g. how do I find $u_{ss}$, $u_{tt}$, $u_{xx}$ etc?
I don't know where to start exactly, I had a ...
0
votes
1answer
53 views
Why $g(t)=(\cos t, \sin t)$ and $h(t) (\cos 2t, \sin 2t)$ have the same image?
I know this might sound silly, but I don't know why $g(t)=(\cos t, \sin t)$ and $h(t)= (\cos 2t, \sin 2t)$ have the same image, that is the unit circle. I understand that if we eliminate the parameter ...
0
votes
1answer
27 views
Parametric equation for the tangent curve
Find parametric equations for the tangent line to the curve with the
parametric equations $x=t, y=t^2, z=t^3$ at the point $(1, 1, 1)$.
For the Solution I know the method of solving it. But have a ...
0
votes
1answer
30 views
Find values of $a$ for which a function is increasing on the interval $(1,\infty)$ [closed]
How to approach this equation?
Find the values of $a$ for which the function $f(x)$ is increasing on the interval $(1,\infty)$, where
$$f(x) =2x^3+3(3a-1)x^2+6ax-a^2.$$
0
votes
0answers
24 views
How to describe the motion of a moving object using parametric equations?
A plane directly above Denver, Colorado, (altitude 1650 meters) flies to Bismark, North Dakota (altitude 550 meters). It travels at 625 km/hour at a constant height of 8500 meters above the line ...
0
votes
0answers
36 views
Paramterizing a walk from $(R, \phi, \theta)=(a,\pi /2, 0)$ to the North Pole, walking North-West
Assuming that the earth is a perfect sphere with radius a, we start our journey in the point $(R, \phi, \theta) = (a, \pi/2, 0)$ (Spherical coordinates) and we travel North-West with constant speed v.
...
0
votes
0answers
28 views
visualising parametric equations
Whenever I'm doing a question on curves, and I am given the equation without the graph I can reasonably visualise in my head what that graph would look like, and for the more complicated equations, I ...
3
votes
0answers
48 views
How do I parametrise the hyperbola $xy = y^2 -1$?
This equation is actually the solution to the intersection of the two surfaces $z = x^2 - y^2$, and $z = x^2 + xy - 1$.
I am to parametrise the solution curve, which is noted in the title. A first ...
0
votes
0answers
22 views
Finding a vector function for the curve of intersection of two surfaces
I wanted to find a vector function $\mathbb{r}(t)$ for the curve of intersection between $z=\sqrt{x^2+y^2}$ and $z = y+5$. I understand that one way would be to let $x=t$ and then we can set $ \sqrt{...
0
votes
1answer
9 views
Surface area of elipsoid created by rotation of parametric curve
I have a parametric curve (elipse) defined as
$x(t)=cos(t)$
$y(t)=2sin(t)$
I need to calculate the surface area of elipsoid made by rotating this curve around x-axis. I know the formula
$S=2\pi \...
0
votes
0answers
25 views
“Slippery Slope” - a Parametric Trajectory problem
I'm designing some equipment that spits out a small ball, which I want to capture when it lands, without it bouncing away. I figure this can be done by having the ball strike a slope tangentially. I ...
0
votes
2answers
30 views
Defining a sphere
Let $c$ be a number, and let $a$ and $b$ be vectors in $R^3$. Let x = $(x, y, z)$. Show that the equation (x$-a)\cdot($x$-b)=c^2$ defines a sphere with centre whose position vector is $1/2(a+b)$ and ...
0
votes
3answers
55 views
How do I convert $(x-1)^2 + (y-\sqrt 3)^2 ≤ 4$ into polar form?
I need to draw the region underneath:
$y ≤ x/\sqrt 3$
and the the circle:
$(x-1)^2 + (y-\sqrt 3)^2 ≤ 4$
My guess would be:
$x = 2cos(\theta) +1$ and $y = 2cos(\theta) +\sqrt 3$.
But the ...
0
votes
0answers
34 views
How can i show a curve is smooth
we have a curve:$$x=f_1(t),y=f_2(t)$$$$t\in I[a,b]$$How can i show that this curve is smooth?$$$$
So far what I read is that the curve must have vertical tangent and perpendicular tangent. But I don’...
1
vote
1answer
32 views
how do i show that a curves have no double point or have only 1 double point. (Double point mean a point in which the curves cut it self)
I have 2 problems in the theme double point that I didn't understand.
1)How can I show that a curve has no double point at all?
2)How can I show that a curve has only 1 double point( or more if needed)...
1
vote
1answer
25 views
Circle and spiral phase portaits
I am having some trouble drawing these phase portraits
[![enter image description here][1]][1]
For 2.a when solving $[\dot{y_1},\dot{y_2}]^T = C * [y_1,y_2]^T$
You get
$\dot{y_1} = -b*y_2$ and $\...
2
votes
2answers
62 views
Draw $r ≤ 3 + 2\sin (\theta)$
Currently I'm stuck at this fairly easy task. All I have to do is sketch the region $r \le 3+2\sin \theta$. My guess would be that the circle has the origin $(0,3)$ with $r = 2$, as I use the formula $...
1
vote
1answer
27 views
Phase portrait of $y_2 = y_{20}*exp(\lambda*t)$ $y_1 = y_{10}*exp(\lambda*t)+ y_{20}*t*exp(\lambda*t)$
$y_2 = y_{20}*exp(\lambda*t)$
$y_1 = y_{10}*exp(\lambda*t)+ y_{20}*t*exp(\lambda*t)$
I am struggling to reproduce the diagram corresponding to the equations above.
I have tried to code it in matlab ...
1
vote
0answers
42 views
Nonlinear optimization with parametric constraint
Is there any way of reformulating the following problem so that it can be solved by means of e.g. Matlab's fmincon?
$\min f(x_1,\dots,x_n)$ subject to $c(x_1,\dots,...
0
votes
0answers
28 views
Question about parametric equations
Let a curve be paramaterized, so x(t)=g(t) and y(t)=h(t) how do I prove that eliminating the parameter yields a relationships which satisfies the points which (g(t),h(t)) traces (and possibly more)?
0
votes
1answer
46 views
Parametric equations intuition
If I paramaterise a function like x= f(t) and y=g(t) like x=t and y=t^2 why why does eliminating the parameter give you the function that the parametric equations? It may be clear for y=x^2 but i'm ...
0
votes
0answers
12 views
Integral representation of time, transforming a graph
I have acquired a solution to a system of differential equations in a parametric form:
\begin{equation}
x= x_{0}u
\end{equation}
\begin{equation}
y=-x_{0}u+\frac{\gamma}{\beta}\ln u - \frac{C_{1}}{\...
2
votes
1answer
24 views
Drawing graphs of parametric equations
the SIR model is an epidemiological model which sorts the whole population into three subclasses. x(t)(susceptibles), y(t)(infected), z(t)(recovered), with initial populations $x(0)= N_{1},y(0) = N_{2}...
0
votes
0answers
9 views
Graphing the SIR model
the SIR model is an epidemiological model which sorts the whole population into three subclasses. x(t)(susceptibles), y(t)(infected), z(t)(recovered), with initial populations $x(0)= N_{1},y(0) = N_{2}...
