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

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1answer
32 views

Help with arc length

I have a curve defined by parametric equations $$y=a\sin^{5}t \\ x=a\cos^{5}t, $$ where $t\in(0;2\pi)$. I solve it by well-known formula: $L=\int\limits_{0}^{2\pi}\sqrt{(y')^2+(x')^2}dt$. $$x'=-5a\...
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1answer
15 views

Eliminate the parameter

Given the parametric equations: $x = sin(\frac{1}{2} \theta)$ $y = cos(\frac{1}{2} \theta)$ Eliminate the parameter. I am completely lost. Please help.
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1answer
35 views

Proof that Epicycloids are Algebraic Curves?

Epicycloids are most commonly described by the parametric equations, $x(t) = (R + a)\cos(t) – a \cos \left(\frac{R + a}{a} t \right),$ $y(t) = (R + a)\sin(t) – a \sin \left(\frac{R + a}{a} t \right)...
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1answer
15 views

Last Step of a Parametric to Cartesian Conversion

I need to figure out how to combine the (4) line to make the t=y-z/R+x and I just don't have any ideas. I'm sorry if these seem basic but I'm 16 and struggling through a topic I've never done before. ...
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0answers
31 views

Find an equation to the tangent line to the curve at the given point

\begin{align} x &= \cos t + \cos 2t, & y &= \sin t + \sin 2t, & \left(−1, 1\right) \end{align} Using the above information I found that $\;\frac{dy}{dx}\;$ is: \begin{align} \...
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2answers
30 views

Find all values of parameter a, when sum of solutions of following equation is 100

Find all values of parameter $a$, when sum of solutions of following equation is $100$. $$ \sin(\sqrt{ax-x^2})=0 $$ I tried to get rid of that $sin$ and there was quadratic equation with two ...
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0answers
10 views

I need input and help understanding how the formula for x arises in a cycloid that is parameterized with theta with the cusp at the origin

Disclaimer: I attempted to answer some of it by using my own deductions. I would feedback on that. The book gives the formulas for how x arises but my problem is understanding how the formulas arose. ...
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0answers
27 views

Eliminate the parameter to find a cartesian equation for the curves

For the first part I am just unsure as to how the book has a different answer than mine. The book has the answer $y = \frac{3}{4} x - \frac{1}{4}$ but given the functions $x(t) = 3 - 4t$ and $y(t) = ...
2
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2answers
54 views

Find curve parametrization

I am asked to find the work of $f(x, y, z) = (x, z, 2y)$ through the curve given by the intersection of two surfaces. I have been doing a series of exercises on this and my question has simply to do ...
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1answer
47 views

Find $\frac{dy}{dx}$ when $t=0$ for $\begin{cases}x = t^2 + 2t \\ y = 2t^3 - 6t\end{cases}$

A dot is moving on a grid following this rule: $$\begin{cases}x = t^2 + 2t \\ y = 2t^3 - 6t\end{cases}$$ I need to find $\frac{dy}{dx}$ when $t =0$. It seems like I should use implicit ...
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1answer
39 views

Parameterise linear combination of cosines

How do I parameterise the following implicit surface? $$ \cos x + \cos y + \cos z = 0 $$ Motivation for this problem comes from attempting to find stable motion for an object balanced on one point. ...
0
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1answer
22 views

Sampling a curve (parametric)

I am working with parametric curves, I need to find the maximum curvature of these curves. I know the starting point, ending point and length of a curve. I want to use sampling method to know the ...
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2answers
31 views

Intersection between roses with given polar equations

$ r_1 \ = \ 4 \sin(3 \theta) \ $ and $ \ r_2 \ = \ 3 \cos(3\theta) \ $ a) find the solutions to the system using polar coordinates I was able to solve this by setting $ \ r_1 \ $ and $ \ r_2 \ $ ...
4
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1answer
67 views

Which way should you run from the lions?

This is a fun problem that I saw somewhere on the internet a long time ago: Suppose you are at the center of an equilateral triangle with side length $s$. At each of its vertices, there is a lion ...
1
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1answer
22 views

Maximizing this parametric expression with a certain range of integer inputs

Let $a,b$ be integers with $1 \le b < a \le n$ and $s,t$ be integers with $0 \le s < t \le m$ I would like to maximize the expression: $b^s (a^{t-s} - b^{t-s})$ My intuition says this should ...
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0answers
13 views

Maximum of this parametric expression

Let $a,b$ be integers with $1 \le b < a \le n$ and $s,t$ be integers with $0 \le s < t \le m$ I would like to maximize the following expression: $b^s~(a^{t-s}-b^{t-s})$ My intuition says this ...
5
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1answer
61 views

Solving a Diophantine equation in three variables as a parametric equation in one variable

Let’s say that $a$, $b$, and $c$ are integers such that $$(b^2+2)^2=(a^2+2c^2)(bc-a). \tag{$\star$}$$ By brute force search, I think I’ve discovered that $$(a,b,c)=(5d+1,3d+1,d+2), \qquad d=\dots,-2,-...
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2answers
31 views

parametric polar equation of a circle

I discovered that Mac's Grapher has a parametric polar mode, i.e. where $r$ and $\theta$ can be specified in terms of a parameter, usually $t$. I am attempting to convert the generic equation for a ...
2
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1answer
23 views

Why are the parameterizations of the circle in cartesian coordinates defined in open intervals?

The parameterization of the circle in rectangular coordinates is given by the the following functions $$ y = g_1(x) = \sqrt{(1-x^2)} \\ y = g_2(x) = -\sqrt{(1-x^2)} \\ x = g_3(y) = \sqrt{(1-y^2)} \\ ...
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4answers
300 views

How did the answer key get $h=40-2r$?

A cone has radius of $20\ \rm cm$ and a height of $40\ \rm cm$. A cylinder fits inside the cone, as shown below. What must the radius of the cylinder be to give the cylinder the maximum ...
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0answers
31 views

Parametric Equation of Elliptical Cycloidal Sine Curve

I am trying to find the parametric equations of a cycloidal curve, which, instead of using a circle, uses an ellipse to oscillate around a base circle. Below are equations of the standard, circular ...
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2answers
23 views

Maximum Height is giving me negative

Hey guys for this parametric equations its giving me negative Question is: A dart is thrown from a point 5 feet above the ground with an inital velocity of 58 ft/sec and angle of elvation of 41∘. ...
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1answer
24 views

Parametric Problem: Throwing a Dart <Test Review>

Yup it's me ... Parametrics, who would have thought xD! Anyways, again ... I am doing review and I really need this grade to get an A in math class; that's why I am asking questions here. And you guys ...
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2answers
42 views

Parametric Problem: Kicking a football (Getting Ready For Test)

As I told you on title I'm getting ready for a test. I have this Test-Review Problem.... A football is kicked from the ground with an initial velocity of $28$ ft/sec at an angle of $28^\circ$. How ...
0
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1answer
36 views

Parametric Problem

i have a question on parametric.. The question states A vector equation $(x,y) = (2,-1) + t(3,2)$. Write as a parametric equation. Show a table with x,y values. Sketch a picture of vector ...
0
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1answer
29 views

Parametric Equations Problem

Im back! Um, i have a simple question im trying to get ready for test after 5 days.. I slacked of sadly :( on math, so i have to pick up my skills.. On my test review i have this question: The ...
1
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0answers
17 views

Heuristics for putting $f(x_0,x_1,\ldots,x_n)=0$ into parametric form?

Suppose I have an implicit equation: \begin{equation} f(x_0,x_1,\ldots,x_n)=0 \end{equation} Which might be 'paramaterizable'; i.e. put it into the form: \begin{align} x_0 &= g_0(t_0,t_1,\ldots,...
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2answers
46 views

Conic Sections: Hyperbola (Finding the Locus)

This is a multipart question so bear with me until I get to the part where I am stuck on. $H$: $xy=c^2$ is a hyperbola (i) Show that $H$ can be represented by the parametric ...
0
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0answers
28 views

Rotate an Ellipse

$x = h + a \cos(φ) \cos(θ) + b \sin(φ) \sin(θ)$ $y = k + b \cos(φ) \sin(θ) - a \sin(φ) \cos(θ)$ Hi, I have basic question of parametric equation for ellipse. I'm trying to rotate horizontal ellipse ...
1
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1answer
42 views

Find the point where the curve $\big(x(t),y(t)\big)=(t^2-1,t^3-t)$ crosses itself

Consider the curve defined by $x(t)=t^2-1$ and $y(t)=t^3-t.$ Find the point where the curve crosses itself. I know that the curve will cross itself if there are two distinct values, say $t_1$ and $...
1
vote
1answer
24 views

Finding algebraic curve satisfying given parameterization

Is there an easy way to find an algebraic curve that satisfies a given parameterization? Specifically, I am talking about the following parameterization: $$ x=z(1-z),\hspace{10pt} y=\sum_{n=1}^r \...
2
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0answers
43 views

Given any parametric curve, finding its general form?

I'll illustrate the problem I'm trying to solve with an example. Let's consider the equations $$ x = \cos (t) $$ $$ y = \sin (t) $$ We know that these are a parametric form of the unit circle. In ...
0
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1answer
31 views

How to fit a set of 3D points to a helical curve?

suppose I have a set of points in $\mathbb{R}^3$, and I want to find an arbitrary helix which best approximates these points. An arbitrary helix in $\mathbb{R}^3$ can be parametrized as $$\vec{r}(t)...
2
votes
4answers
109 views

How does one parameterize $x^2 + xy + y^2 = \frac{1}{2}$?

Parameterize the curve $C$ that intersects the surface $x^2+y^2+z^2=1$ and the plane $x+y+z=0$. I have this replacing equations: $$ x^2+y^2+(-x-y)^2=1$$ and clearing have the following: $$ x^2+...
0
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0answers
21 views

How to find the limits of integration for parametric

In this question: Find the area bounded by: $x=\ln(t)$, $y=\frac{t-3}{t-1}$, $3\leq x \leq 5$, and by the $x$-axis (it is above the $x$ axis). I solved the integration parametric curve, $3\ln(t) -...
0
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1answer
66 views

Integral of logarithmic fuctions with parameter

Hello I am solving an integral with a natural logarithm that has a parameter. Let say $I(a)=\int_0^\pi\ln(1-2a\cos(x)+a^2)dx$ Then for differentiation under integral sign and that yields $I'(a)= \...
3
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1answer
26 views

Finding minimum plus maximum of $g(a)=\int_{0}^{\pi/2} |\sin 2x-a\cos x|dx$

Let $$g(a)=\int_{0}^{\pi/2} |\sin 2x-a\cos x|dx,\quad a\in[0,1].$$ If $L$ and $M$ are the minimum and maximum values of $g(a)$ for all $a\in [0,1]$. Find the value of $L+M$. The first thing ...
0
votes
1answer
40 views

How do parametric equations work?

I was given a graph like this in my exam. Its defined para-metrically by x=c^2 and y =c^3. It won't help me now but could someone explain this to me why I have two seemingly different lines I know it ...
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1answer
26 views

Paramteric Curves and the exponents of $\cos$/$\sin$/$\tan$

Lets say we have the curve $\frac x7=\cos^7t$, $\frac y7=\sin^7t$ Now I know that $\sin^2x+\cos^2x=1$. So $\cos^2=(\frac x7)^{\text{some exponent}}$. What is that exponent? How do you work it out?
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1answer
40 views

Parametric curve, Write it in cartesian form, giving $y$ explicitly in terms of $x$ [closed]

$x=4\ln(t)$ $y=2t$ Write it in cartesian form, giving $y$ explicitly in terms of $x$.
0
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1answer
22 views

Parametric Equation of Cycloidal Sine Curve

I am trying to find the parametric equation of a sine curve, which oscillates around a circle as it's $x$-axis. I have done preliminary approximations using Epicycloid parametric equations for the top ...
0
votes
2answers
38 views

An equation with a parameter

Given the equation $(|x+1|+|x-a|)^2-2(|x+1|+|x-a|)+4a-4a^2=0$ find all possible $a$ such that this equation has only one solution. I wanted to solve it like this: $(|x+1|+|x-a|)^2-2(|x+1|+|x-a|)+4a-...
0
votes
1answer
27 views

2D parametric equation for an arc between two points with a start angle

What's a parametric equation (eg. $(x,y)=(\cos(t \cdot 2\pi),\sin(t \cdot 2\pi)$ plots a circle where $t$ is the 'time' along the circle) that draws an arc between the two points $(x_0,y_0)$ and $(x_1,...
0
votes
1answer
23 views

How to find parametric equation between two points in line integral?

[In this example how can we find parametric equations of x and y.] [1] [question]: http://i.stack.imgur.com/lTOnW.png [1] [Solution]: http://i.stack.imgur.com/l8ao7.jpg
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1answer
18 views

Convert line parametrization into two equations

Consider the following parametrization on $\mathbb{R}^3$ $$g(t) = (t^2,t\cos(t),t\sin(t))$$ This is a line, and as such can be characterized by two equations. I already found the first one to be $$...
0
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0answers
13 views

Derive the parametric form of the locus of point where difference between distance to two points is constant

Given two points $P_1=(x_1,y_1)$ and $P_2 = (x_2,y_2)$, the locus of the point whose (signed) difference between the distance to the two points is a constant $\Delta$ is one branch of a hyperbola ...
3
votes
2answers
34 views

Parametric version of a simple equation

I have a simple relation that I need to plot in a plane. I could do it, but I believe that I don't get the best way. A plane curve is defined implicitely by the following equation : \begin{equation}\...
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votes
1answer
23 views

Parametric Equations: Having trouble with finding two tangents (Calculus) [duplicate]

curve C defined by these parametric equations $x = t^3 - 3t^2$ $y = t^3 - 3t$ I need to find the equations of two tangents at the point $(-4,2)$ I have attempted to solve this problem myself but I ...
0
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1answer
21 views

Deriving parametric equations for a cubic equation

I've been looking at elementary cubic equations for curves and seem to understand them well enough. Going the other way and driving parametric equations has been mystifying. For example: given a ...
2
votes
0answers
26 views

Graphing/visualizing a complex parametric plot without using mathematica

I am trying to visualize the parametric plot in $\mathbb{C}$ of the curve $\gamma$ defined for $t\in[-\infty,\infty]$ as $$\gamma(t)=\exp\left(-t^{2}+\frac{t}{\sqrt{1+t^2}}i\right).$$ I think I find ...