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

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2answers
57 views

Derivative with parametric set of equations

I am struggling for hours on a fairly simple problem. I have the following set of parametric equations: $$i = I (q_1^2 + q_1 - q_2^2 - q_2)$$ $$v = V (2(q_1 - 1) + \log q_1 - 2(q_2 - 1) - \log q_2)$$ ...
2
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2answers
26 views

Parametric equations - stating values.

A curve is defined by the parametric equations: $x=\cos2t, y=\sin2t, 0<t<π.$ a) Use parametric equations to find $\frac{dy}{dx}$. Hence find the equation of the tangent when $t=\frac{π}{8}$. ...
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1answer
17 views

General vector curve for the surface of a sphere

I want to prove a theorem about a vector curve ${\bf c}(t): \mathbb{R} \to \mathbb{R}^3$ (for $t \in [a, b]$) which lies on the surface of a sphere in $\mathbb{R}^3$. It is in my understanding that ...
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1answer
71 views

Find a vector function represented by the curve of intersection?

I'm struggling with the following problem: Given $\, z = \sqrt{x^2 + y^2}\,$ and $\, z = y+1\,$ find the vector function represented by the curve of intersection of the surfaces using the ...
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1answer
52 views

solving for initial velocity using the position vector

I am having trouble wrapping my head around this problem. The big picture is that i have to calculate the initial velocity v= needed for a soccer ball to cross a goal line. this is a homework ...
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1answer
26 views

Constructing parametric equation for $x=3z\cos(\ln z)$

I was trying to transform this $$x=3z\cos(\ln z)$$ in parametric form: $$x=x(t)$$ $$z=z(t).$$ To this end I made a substitution $\ln{z}=t$ and I got: $$x=3{e^t}\cos(t)$$ $$z=e^t$$ $$t\in ...
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1answer
48 views

two-dimensional bounded area defined parametrically

How do I define this without using piecewise function? I think it has something to do with Bilinear Surface but not sure how to get started. $x_1=-1, x_2=1, x_3=0, x_4=1$ $y_1=0, y_2=1, y_3=1, ...
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2answers
480 views

Parameterization of a curve for complex integral

I have problem with parameterization of a curve in order to evaluate a complex integral. Most docs that I've tried to read didn't explain the topic very well, especially, in case where the curve in ...
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1answer
84 views

Finding the parametric equation for a longbow curve about a circle

In the figure the circle of radius $a$ is stationary, and for every $\theta$, the point $P$ is the midpoint of the segment $QR$. The curve traced out by $P$ for $0<\theta<\pi$ is called the ...
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1answer
41 views

How to solve parametrized limits?

I am a bit confused at the moment. This exercise in particoular shattered my self confidence: $$\lim\limits_{x\to 0}\frac{\sinh(x) + 1 - (1 + 3x)^{\frac13}}{\ln\left(1 - 2x^α\right) + 2x^3}$$ with ...
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1answer
60 views

What's the equation of this parametric surface?

disclaimer: my math is sketchy at best AND english is not my first language, so... i might have some issues naming things - but i'll try my best to be clear :) given this parametric curve: see it ...
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2answers
45 views

finding t for parametric equation tangent line

Here is the problem I'm trying to solve: Find the tangent line at the point (0,2) $$x=2 \, \cot(t)$$ $$y=2 \, \sin^2(t)$$ $$\frac{dy}{dx} = -2 \, > \sin^3(t) \, \cos(t)$$ The tangent line ...
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1answer
16 views

Proof: Given a $C^1$ parametrization of $f=f(x,y):\mathbb R^2\to \mathbb R$, substitute variables $(x,y) = (s+t,s-t)$

I'm studying Mathematical Analysis and trying to solve example problems as I go. Specifically, this problem comes after an introduction to $C$ parameterizations of $r:I\subseteq\mathbb R \to \mathbb ...
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1answer
55 views

On the implicit function theorem and the gradient.

I was following some MIT notes and came across this proof I had a doubt about the existence of $r(t) = \langle x(t), y(t), z(t) \rangle$ a parametrization of a curve on the level surface. Then I ...
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2answers
39 views

Need help to understand a math task about algebraic and parametric equations

Can anybody please explain this for me?: Find the algebraic and parametric equations of the circle with centre (-2,3) that passes through (1,-1) How do I find the algebraic and parametric ...
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1answer
31 views

Parametric curve parametriced by length

Normally you have a parametric curve with a variable t and you increment t to find the point along the curve. Is it possible to have a curve so that given a value it will give you the point on that ...
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0answers
106 views

Parametric vector form of cartesian equation

Cartestian equation: $$-2x-y+z=6$$ I know to find the parametric vector form we can find any 3 points P, Q and R which satisfy the cartesian equation. $$ \begin{pmatrix} x_1\\ y_1\\ z_1 ...
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1answer
69 views

Identifying self-intersection points in one parametric graph.

My question for you is how to identify self-intersection points in a parametric curve of the form x = f(t), y = g(t). The specific problem asks for the t values of the intersection where $x = ...
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1answer
50 views

Parametric Equation explanation

Explain how the expression $tX + (1-t)Y$, $0\le t\le 1$, produces a segment that connects point $X (x_1, y_1)$ with point $Y (x_2,y_2)$. So I rearranged the problem such that $t(X - Y) + Y$ which I ...
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1answer
201 views

Paramaterization of paraboloid and plane.

Consider the paraboloid $z=x^2+y^2$. The plane $2x-4y+z-6=0$ cuts the paraboloid, its intersection being a curve. Find "the natural" parameterization of this curve. I have set each equation equal ...
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8answers
124 views

How can I parametrize $|x|+|y|=1$

I need parametrize $|x|+|y|=1$ but I don't know how to parametrize. I know that it is a rotated square, I would like understand so if you can explain to me like if I was still, thanks
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1answer
50 views

How do I represent a Mobius Band Triangle Parametrically

I am trying to describe a Mobius band in the shape of a triangle like this: parametrically in terms of its $x$, $y$, and $z$ functions. Is this even possible? I know a basic mobius strip can be ...
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2answers
63 views

Convert between parameteric ellipse equations

I have the parametric equation of an ellipse in this form: $$x(t)= a\cos(t)$$ $$y(t)=b\cos(t+\phi)$$ It's an ellipse centred about the origin, with a tilt angle. So three parameters. How can I ...
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0answers
40 views

Intersection between a parametric equation and a linear equation

2Consider the parametric functions $f_1, f_2$ with $$f_1(x) = 3(60-x)\cdot \sin(3x)$$ and $$ f_2(x) = 3(60-x)\cdot \cos(3x).$$ Suppose you have a linear function: $$f_3 (x) = 1.5 x$$ How does one ...
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2answers
122 views

How can I know whether the airplanes collide by using parametric equations

Recall that a line hes equation y=mx+c. Suppose one airplane moves along the line y=2x+3 while the other airplane moves along the line y=3x-2. By plotting a graph, even though the lines are intersect, ...
2
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1answer
392 views

Converting between explicit function and parametric function

Given an explicit function $y = f(x)$, how to convert it to the respective parametric functions $x = f_1(t)\; y = f_2(t)$? Given parametric functions $x = f_1(t)\; y = f_2(t)$, how to obtain the ...
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1answer
19 views

How to parametrise a parabola with a specific domain

What would be the best method to find the parametric equations for the parabola $y = (x-2)^2$ over a given domain of $(2 ≤ t ≤ 5)$? The figure I've been given has the parabola starting from $(2,0)$ ...
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1answer
48 views

How many solutions depending on the parameter (augmented matrix?)

I have to find how many solutions have got the following equations, depending on p parameter? $ \begin{bmatrix} 5 & p & 5 \\ 1 & 1 & 1 \\ p & p & 2 \end{bmatrix} $ $ ...
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2answers
55 views

Differentiate the parametric function and find $dy/dx$ and $d^2y/dx^2$

Differentiate the parametric function and find $\frac{\mathrm dy}{\mathrm dx}$ and $\frac{\mathrm d^2y}{\mathrm dx^2}$ in terms of "$t$" when: $ x = \frac{1}{t-1}$ and $y = \frac{1}{t+1}$ I have ...
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2answers
82 views

Showing that the image of a curve lies on a surface?

I am looking for an intuitive explanation to a problem in one of my practice tests. I'm given a parameterized curve from $\Bbb R$ to $\Bbb R^3$, called ${\bf r}(t) = (\sin t \cos t, \cos^2 t, \cos ...
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2answers
85 views

Investigating the “Wigglyness” of a 2D-Parametric Curve

I am looking to quantify the (for a lack of a better term) "wigglyness" of a parametric curve. The particular set of curves that I am looking at come from cubic-spline interpolation on a set of points ...
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2answers
42 views

Parametric equations: Finding the ordinary equation in $x$ and $y$ by eliminating the parameter from parametric equations

I am having difficult time solving the following equation: Eliminate the parameter from the parametric equations: $x=\frac{3t}{1+t3}$ and $y = \frac{3t^2}{1+t3}$ where $t \ne−1$ and hence find an ...
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1answer
41 views

Parametrizing motion of string around rod

I'm trying to solve this problem with calculus. A string is wound symmetrically around a circular rod. The string goes exactly 4 times around the rod. The circumference of the rod is 4 cm. and its ...
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0answers
88 views

parsimonious definition of a zig-zag function between two bounds

Suppose I have some strictly increasing function $f:[0,b]\to[0,b]$ with $0<b<1$, $f(x)<x$ and $f'(b)=\frac{1-f(b)}{1-b}$ (i.e. tangent to the secant line to $(1,1)$). Now imagine a 'tunnel' ...
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1answer
36 views

Line integral of a vector field? [closed]

First of all, sorry for the sketch.. I would be glad if you show me another ways to improve this drawn. I'm studying Line integrals, and this question is really boring me. Please, can someone put ...
0
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1answer
31 views

Parabola Question - simultaneous equations?

I'm having trouble with the second part of this question. I can do the first part by finding the normal at P and where it intercepts with U and then for the second part i've substituted each point ...
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1answer
83 views

Locus of solution of an ODE system

I have an ODE system $$ \ddot p = \frac{ p \left( {2p - 4} \right) }{{p - 4}}{{\dot q }^2 } \\ \ddot q = \frac{{3p - 8}}{{p - 4}}\dot q \dot p $$ Short of finding closed-form expressions for ...
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2answers
27 views

Use the discriminant to show that $mx−y + m^2 = 0$ touches the parabola $x^2 =−4y$, for all values of m.

Use the discriminant to show that $mx−y + m^2 = 0$ touches the parabola $x^2 =−4y$, for all values of m. I attempted to solve by letting them both equal each other, but it didn't work. How do I do ...
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2answers
37 views

Find, in terms of $s$,the coordinates of the point where this normal cuts the curve again.

a) Find the equation of the normal at the point $(2s,\frac{2}{s})$ to the curve whose parametric equations are $x=2s,y=\frac{2}{s}$ b) Find, in terms of $s$,the coordinates of the point where ...
3
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2answers
245 views

Gradient vector of parametric curve

I have ellipse $$(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$$ Gradient is $$(\frac{2x}{a^2}, \frac{2y}{b^2})$$ How I can obtain this vector from parametrization of my curve? Let I know only $$(x, y) = (a ...
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2answers
257 views

Radius of a wheel based on parametric equations

I am working on a question and I don't have the slightest idea where to begin. Any nudge in the right direction would be very helpful. Here is the question: A bicycle wheel has radius R. Let P be ...
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4answers
69 views

Eliminate $t$ to give an equation that relates $x$ and $y$

I am having problems understanding how to solve the following parametric equation. I have achieved an answer, but am unsure if my answer is correct or not. Eliminate t to give an equation that ...
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1answer
24 views

If PQis a focal chord, show that the interval RU is parallel to the axis of the parabola.

For part (c) of question thirteen am I only required to find the gradient of RU and prove that is it zero? This is how I have interpreted this question. ANY help on the matter is much appreciated ...
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1answer
40 views

Parametric Curve Tangent Equations

Let a curve be given in the parametrized form by: $r(t) = (2\cos t, 2\sin t), 0 \leq t \leq 2\pi$ Find the equations of the tangents to the curve at each of its points $(X_0, Y_0)$. Having gone ...
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2answers
34 views

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$?

How to parametrise $x^2 + y^2 = z^2; z \in [0, 1]$? I want to parametrise so I can use the divergence theorem to calculate the flux along the surface above. I don't know how to do it and would like ...
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1answer
28 views

Is there a method to parameterise any surface? And how could I parametrise this one given?

I'm having major trouble every time I need to parametrise a surface in order to take a surface integral, I just have no idea where to even start half of the time. Is there some kind of method that can ...
2
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0answers
68 views

Tweaking Reddit's Ranking Algorithm

This image explains how Reddit's Ranking algorithm works. As you know, Reddit is a very high traffic site. Therefore, the post rank decreases quite fast. This algorithm puts emphasis on bringing ...
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3answers
172 views

Find the coordinates of the point where the normal cuts the curve again problem

Find the equation of the normal to the curve $x=2\cos\theta$, $y=3\sin\theta$ at the point where $\theta=\frac{1}{4}\pi$. Find the coordinates of the point where this normal cuts the curve again. ...
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2answers
29 views

To find a extremal point of a function with parameters

I have a function $$f(x) = (x-5m)(x+m)^2$$ I have tried to find the extremal points of this function (and then find if it's local maxima or minima). That means I need to find the x of derivative. The ...
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0answers
29 views

Finding Extrema of a Function Restrained by a Parametrized Surface

Where on the parametrized surface $r(u,v)=⟨u^2,v^3,uv⟩$ is the temperature $T(x,y,z)=12x+y−12z$ minimal? Find all local maxima, local minima or saddle points. I know that one has to insert $r(u,v)$ ...