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

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2
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0answers
645 views

Parametrization of square to calculate Dot-product in line-integrals and area-integrals, electric field from $\frac{dB}{dt}$?

I am calculating 3A of Tfy-0.1064 in Aalto University. I realized here that I am misunderstanding something in vector calculus: the thing market in green particularly. I know $$\nabla\times E= ...
0
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1answer
163 views

equation for the region inside a circle

What equation or group of equations fill the entire or part of a region inside a circle without using inequalities? Update I don't know if this problem is already solved, I'm trying to find the ...
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3answers
303 views

Understanding cubic bezier curve

I do not have experience of Mathematics past a-level, so please excuse the incorrect terminology. I am trying to better understand the fundamentals of how a cubic bezier curve works. I am going by ...
0
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2answers
51 views

what's the algebra (if any) behind converting f(x) for a circle to a parametric equation

I'm sure there has to be some algebra behind it. My problem called to covert $$(x - 2)^2 + (y - 9)^2 = 4$$ if $x = 2 + 2cos(t)$ then $y = ? $ I know the answer is $9 + 2\sin(t)$ but I simply got ...
2
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2answers
286 views

parametric curves, parameter and integration

I just started learning about parametric curves and I find it confusing that we have a 3rd variable but this 3rd variable "t" is some imaginary variable....I dont get what the difference is between ...
4
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3answers
2k views

Writing Polar Equations In Parametric Form

For an example problem, in my textbook, the author wanted to demonstrate how to graph a polar function. Deeming it most convenient, my author took the polar function $r=2\cos 3\theta$, and re-wrote it ...
2
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1answer
116 views

Parametric Equation of a Circle Using a Line

Consider the unit circle $$ x^2+y^2=1. $$ How can I parametrize it using the line $y=m(x+1)$, where $m$ is its slope?
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1answer
382 views

Help me understand a surface integral question?

The question is: Evaluate the surface integral: $$ \iint\limits_S \, x^2yz\ \mathrm{d} S $$ Where S is part of the plane z = 1 + 2x + 3y that lies above the rectangle [0,3] X [0,2] I literally just ...
2
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1answer
229 views

Finding parametric equations

I am trying to understand volume and surface integrals. I do get the idea of the process (find a parametric equation of the volume/surface, integrate afterwards). But I just cannot make up parametric ...
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1answer
116 views

Find the Frenet frame

Consider the following space curve: $$ \gamma(x)=(e^x\cos(x), e^x\sin(x), e^x). $$ My main goal is to find the Frenet Frame T,N,B. So far I have found the arc-length using the following formula: $$ ...
3
votes
2answers
574 views

Converting $x=\frac{1}{2}\cos\theta\;;\;\; y=2\sin\theta $ to Cartesian form

How can we transform these parametric equations to Cartesian form? $x=\frac{1}{2} \cos\theta, \quad y=2\sin\theta \quad\text{ for}\;\;0 \leq \theta \leq \pi$
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2answers
110 views

Converting $x = \sin \frac{t}{2}, y = \cos \frac{t}{2}$ to Cartesian form

How can we transform these parametric equations to Cartesian form? $$x = \sin \frac{t}{2}, \quad y = \cos \frac{t}{2}, \quad -\pi \leq t \leq \pi.$$
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3answers
124 views

Help me to sketch this parametric curves

Is there any defined process to sketch parametric curves? Thanks in advance. $$x = \cos^2 t, \quad y = 1 - \sin t, \quad 0 \leq t \leq 2\pi.$$
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1answer
64 views

Find a function f(x) such that the parametric curve could be obtained by flipping the graph

Find a function f(x) such that the parametric curve could be obtained by flipping the graph of f across the line with slope 1 that goes through the origin. parametric curve with coordinates (t^14, ...
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3answers
306 views

How do we prove that two parametric equations are drawing the same thing?

For example, if I have $$\begin {align} x(t) &= r\sin t\cos t\\ y(t) &= r\sin^2 t\\ \end {align}$$ and $$\begin {align} x(t) &= \frac r 2 \cos t\\ y(t) &= \frac r 2 (\sin t + 1) ...
1
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1answer
226 views

Arc Length Of Parametric Curve

I attached the problem as a file: Where did the trig functions go? I sifted through the different trig identities and formulas, but couldn't find anything that I could use. What should I do?
2
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2answers
2k views

Finding Where A Parametric Curve Intersects Itself

The problem I am working on is to find the where the curve intersects itself, using the parametric equations. These are: $x=t^2-t$ and $y=t^3-3t-1$ For the graph to intersect itself, there must be ...
2
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1answer
2k views

Finding Where A Parametric Curve Crosses Itself

The parametric functions I am dealing with are: $x=2\sin2t$ and $y=3\sin t$ I know for a parametric graph to cross itself, there must be two distinct $t$, $t_1$ and $t_2$, that when placed into the ...
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1answer
1k views

Finding Parametric Equations For A Rectangular Equation

I am trying to find a general way of finding parametric equations for a rectangular equation. The problem I am working on is $y=x^3$, and I have to find two examples of parametric equations. ...
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2answers
245 views

Parametric Equation Problem

The problem is, "to determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain." (a) $x=t;\quad ...
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1answer
142 views

Restriction Of Parametric Functions Domain

The problem I am working on is, "Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the ...
0
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2answers
50 views

Need help with this parametric equation

$s(t)=(\frac{2}{t^2+1},\frac{2t}{t^2+1})$ I need to calculate a line integral along this path. But I have trouble understanding what it is. I did some googling and it looks that it is a parabola, but ...
0
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2answers
49 views

How do I find the 2 slopes at which this parametric function crosses itself?

I have a parametric function. If you graph it, you'll find that it looks like a figure 8. x(t) = 2sin(2t) y(t) = 8sin(t) How do I find the slopes of the function ...
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1answer
2k views

Parametric equation of a cone

I usually use the following parametric equation to find the surface area of a regular cone $z=\sqrt{x^2+y^2}$: $$x=r\cos\theta$$ $$y=r\sin\theta$$ $$z=r$$ And make $0\leq r \leq 2\pi$, $0 \leq \theta ...
5
votes
1answer
77 views

Can (x(t), y(t)) generate a surface? If so, can the surface be continuous?

Intuitively, the parametric equation $z = (x(t), y(t))$ seems to only be able to generate one-dimensional objects, i.e. curves. However... Let $x(t)$ be "the odd-indexed digits of the real number ...
2
votes
2answers
97 views

Parameter values that make function values side lengths of a triangle

I have been trying to solve the following problem for more than a week without any success. Given the function: $$f(x)=\frac{x^2+mx+4}{x^2+x+4}$$ Find all possible values of the parameter $m$ such ...
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1answer
140 views

How can I re-write an equation (or system of equations) in parametric form?

For the equation $y = 3x$ I need to re-write $x$ and $y$ in terms of a variable $t$. How can I find the value of each variable in terms of $t$?
2
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1answer
149 views

How to find a parametric equation?

I want to find an equation for a race track, so I could get the position of a point with respect to time. Let's say I have this track and here are a few points on it: Could it be possible to model ...
0
votes
1answer
184 views

parametrizing quarter of a circle

I am given the circle whose equation is: $(x-\frac{1}{2})^{2}+(y+\frac{1}{2})^{2}=\frac{1}{2}$. So, the coordinates of the origin of the circle are: $(\frac{1}{2},-\frac{1}{2})$ and the radius of the ...
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1answer
330 views

converting a parametric R5 vector into a Cartesian form

How do you solve a problem like this. I'm completely stumped. it seems like there should be an easy solution but I'm obviously over looking it. any help would be greatly appreciated.
2
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2answers
210 views

Constant velocity of a sine function

I am defining the location of an object based on the sine function. The position of the object at s seconds along the x-axis is defined as x=s and its position along the y-axis is defined as y=sin(x). ...
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2answers
373 views

Find the area bounded by the parametric curve…

Find the area bounded by the parametric curve $x = \cos(t)$, $y = e^t, 0 < t < \pi/2$, and the lines $y = 1$ and $x = 0$. I do not even know where to start with this problem. I know that I need ...
2
votes
3answers
2k views

How to find a parametric equation for the tangent line to the curve of intersection of the cylinders?

How can i find a parametric equation for the tangent line to the curve of intersection of the cylinders $x^2 + y^2 = 4$ and and $x^2 + z^2 = 1$ at the point $P_0(1,\sqrt{3}, 0)$?
0
votes
1answer
157 views

How do we find the length of the line (parametric curve)?

A curve in the $xy$-plane is given parametrically by $$x(t) = e^{2t}, \quad y(t) = e^{2t} \sin(2t), \quad t \in [0, \pi/2].$$ What is the length of this curve? Ok, actually I know what to do, ...
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1answer
104 views

Converting parametric equations in a numerical equation

Is it possible convert this parametric equations in a numerical equation? $$ \begin{cases} \displaystyle x(t)=tv_0\cos(\theta)\\ \displaystyle y(t)=tv_0\sin(\theta)-\frac{1}{2}gt^2+h \end{cases} $$ ...
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0answers
55 views

Maximum value for parameter

I am facing the following problem: A number of a adults, b children older than 12, and c children younger than twelve attend an event. The sum of all people a+b+c=100. The prices are \$6 per adult, ...
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1answer
2k views

Equation for making a circle in 3D space

I have a 3D space with axis $(x, y ,z)$ and I can make a circle in the $xy$-plane. To make a circle in the xy-plane I currently use spherical coordinates $(r, \theta, \phi)$ where $r = 1$, $\theta = ...
1
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1answer
432 views

Parameterize a straight line using polar coordinates… without angle.

I had to parameterize a straight line with starting point in $A=(-3,7)\\ $ and endpoint in $B=(4,1)$. My idea was to use the equation for the line that goes through two points. That is: $$ \frac { ...
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1answer
98 views

Parametric Equation of a Particle Movement inside a Vortex in a Rectangular Box

I am trying to simulate the movement of a particle in a vortex in a rectangular box, I am currently using an ellipse but that causes the particle to collide with the walls more that I want. The ...
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2answers
264 views

Finding Tangent line from Parametric

I need to find an equation of the tangent line to the curve $x=5+t^2-t$, $y=t^2+5$ at the point $(5,6)$. Setting $x=5$ and $y = 6$ and solving for $t$ gives me $t=0,1,-1$. I know I have to do ...
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1answer
565 views

Parametric Equation

Let $P_1$ be the plane through the origin containing the vectors $[1,2,-1]$ and $[0,1,1]$. Let $P_2$ be the plane through the point $(1,1,1)$ parallel to the vectors $[-1,2,2]$ and $[3,4,-2]$ I know ...
0
votes
1answer
38 views

Represent sorting position by a parametric form

Given a set of random integers {0,5,100,65,...,0,1,2}, is there a mathematical method existing to construct a parametric form $f$ (the number of parameters $<<$ the number of integers) so that ...
0
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1answer
132 views

Parametric Linear Program: Continuous Solution?

Consider the parametric linear problem $$ x^*(\theta) := \min_{Y , \ Z } \left\| Z \right\|_1 $$ $$ \text{sub. to: } \ \theta A + B Y = \theta C Z.$$ where $Y \in \mathbb{R}^{m \times s} $, $Z \in ...
2
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1answer
181 views

Regular parametrization of a curve

Let $\gamma : \left\{ \begin{array}{ccc} \mathbb{R} & \to & \mathbb{R} \\ t & \mapsto & (t^2,t^3) \end{array} \right.$ and $\Gamma= \gamma(\mathbb{R})$. Because of the singularity at ...
3
votes
1answer
3k views

find length of curve of intersection

I have come to a dead end on a problem and I need someone to tell me either if I did it correctly, or how to fix it if I did not. This is Stewart Calculus 7th edition, problem 13.3.12. Here is the ...
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1answer
433 views

parameterization of helical torus

A Helix is parameterized as $\langle R \cos(t), R \sin(t), \alpha t\rangle$ and one can visualize it as "wrapping" around a cylinder of radius R. I would like to accomplish the same thing but wrapping ...
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1answer
139 views

Parametrization of a solid

Find a parametrization $\sigma : I \subseteq \mathbb{R}^3 \rightarrow \mathbb{R}^3$, with $I$ a parallelepiped, of $\lbrace (x,y,z) \in \mathbb{R}^3 : |z| \leq 4x^2 + 9y^2 \leq 1 \rbrace $.
0
votes
1answer
69 views

Probability distribution for a function of a random variable

I have the distribution of X with respect to parameter t vaying between 0 and 1. However, in nature, parameter t is not uniformly distributed. It has a known probability distribution. What is ...
3
votes
1answer
511 views

Summation of an Arithmetic, Parametric Sequence

I'm trying work on my ability to break complex patterns down, and in this case I'm trying to model the denominators of Lacsap's Fractions: I managed to get the sequence that represents the ...
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vote
3answers
509 views

Write each pair of equations as a single equation in $x$ and $y$.

Write each pair of equations as a single equation in $x$ and $y$. a)$\begin{cases} x=t+1 &\\ y=t^2-t & \\ \end{cases}$ b)$ \begin{cases} x=\sqrt[3]{t}-1 &\\ ...