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Questions tagged [parametric]

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

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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 ...
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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 $0 ≤ \theta ≤ 2π$....
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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, ...
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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{...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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1answer
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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}(...
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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 ...
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1answer
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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 ...
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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
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1answer
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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 $\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}$. ...
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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\...
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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=...
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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)((...
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1answer
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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 ...
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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 $$\int_0^1\frac{x^{1-a}(1-x)^a}{(x+1)^3}dx$$
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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 $x=f(t),y=g(t)$, ...
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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])...
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Parametric equation for y^2 = x+1 [closed]

How do you parametrise $$y^2 = x+1$$ ? I only know how to parametrise a circle.
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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 ...
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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 ...
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4answers
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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?
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1answer
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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 ...
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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 ...
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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 ...
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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 $$f(x) =2x^3+3(3a-1)x^2+6ax-a^2.$$
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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 ...
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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. ...
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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 ...
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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 ...
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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{...
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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 \...
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“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 ...
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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 ...
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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 ...
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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’...
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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)...
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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 $\...
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2answers
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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 $...
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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 ...
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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,...
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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)?
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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 ...
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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}}{\...
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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}...
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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}...