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

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1answer
85 views

Simplify a parametric equation with hyperbolic trigonometric functions

I've the following parametric equations for a curve: $$\begin{cases}x(t)=a\cdot \operatorname{sech} (t) \\ y(t)=a\cdot(t-\tanh(t))\end{cases}$$ Now let $\theta(t)=-\arctan(\sinh(t))$ how does the ...
0
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1answer
70 views

Parametric equations for the tractrix

The cartesian equation of the tractrix is:$$y=\pm\left(a\cdot \operatorname{arcsech}^{-1}\left(\frac{x}{a}\right)-\sqrt{a^2-x^2}\right)$$ where $a>0$ is a real parameter and $x$ varies from $0$ to ...
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1answer
129 views

Jacobian of parametrized ellipsoid with respect to parametrized sphere

I'm not even sure how best to phrase this question, but here goes. Given $\theta$ (elevation) and $\phi$ (azimuth), the unit sphere can be parametrized as $ x = \cos(\theta)\sin(\phi) \\ y = ...
2
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1answer
31 views

length of intersection of parabolic cylinder and a surface

Let $C$ be the curve of intersection of the parabolic cylinder $x^2 = 2y$ and the surface $3z = xy$. Find the length of the part of $C$ from $(0, 0, 0)$ to $(6, 18, 36)$. (Hint: It may be useful to ...
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1answer
34 views

Finding the acceleration

So I am given a problem stated as: a point moves in the plane at speed 1 along the curve $y = x^2$. Find the acceleration at the point (x,y). I know that the velocity is y' = 2x, and that at a ...
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1answer
130 views

Loxodrome parametric equations

I have been trying to understand HOW one arrives at the equations $x=cos(t)cos(c)$ $y=sin(t)cos(c)$ $z=−sin(c)$ of the loxodrome. I can see that if the transformation to spherical coordinates is ...
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2answers
24 views

Eliminating $\theta$ between the two expressions

How do we find the equation of this parametric curve $$2x=\cos {\theta}\left(\sqrt {\dfrac{3}{5}}\sin {\theta}+\cos {\theta}\right)$$ $$2y=\sin {\theta}\left(\sqrt {\dfrac{3}{5}}\sin {\theta}+\cos ...
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2answers
41 views

Create paramatric shape wihtout 'dents'

I am plotting a shape with the following equation $$\left\{ \begin{array}{c} x=r_{in} \cos(4 t)+r_{out} \cos(t)\\ y=r_{in} \sin(4 t)+r_{out} \sin( t) \end{array} \right. $$ Given various parameters ...
2
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1answer
61 views

Mathematical Description for Steam Rising from a Cup

I was staring at a cup of coffee I have on the desk just now. The light shines through the water vapor as they rise from the cup. The shape of the steam is not completely random, as it drift from ...
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1answer
27 views

Curvature of curve

$r(t) = (-3sint)i + (-3sint)j + (cost)k$ I got as far as:$$||r'(u)|| = sqrt{(18cos^2u + sin^2u)}$$ But I cannot evaluate $\int_0^t||r'(u)||dt$
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1answer
19 views

Finding the normals of an equation based on their parametric representation

A curve is defined parametrically by the equations $$ x = t^3 - 6t + 4, y = t - 3 + \frac{2}{t} $$ The first question, which I've partially solved, was to find the equations of the normals to the ...
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3answers
143 views

Finding cartesian equation from parametric trigonometric equations

I'm trying to find the cartesian equation of the curve which is defined parametrically by: $$ x = 2sin\theta, y = cos^2\theta $$ Both approaches I take result in the same answer: $$ y = 1 - ...
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1answer
42 views

Line Integral with Arclength Parametrization

Suppose we have an arclength parametrization of a curve in the $xy$-plane given by $x(s)$, $y(s)$ where $0 \leq s \leq L$. We want to integrate a scalar function $f(x,y)$ along this line. Since we are ...
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2answers
38 views

How do you eliminate the parameter to find a cartesian equation of the curve?

$$x=1/2cosθ$$ $$y=2sinθ$$ $$0 \le θ \le π $$ So I know the parameter that must be eliminated is θ. How should I do this? Are there trig identities that I can use?
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1answer
33 views

Find the area of the surface obtained by rotating the curve about the x-axis?

Given this curve: $$y=\frac{x^3}{6}+\frac{1}{2x} 1/2 \le x \le 1 $$ This is what I get for my (dy/dx)^2: $$\frac{x^4+x^{-4}+2}{4}$$ I'm unsure about this. Can anyone confirm that I did it ...
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2answers
110 views

Find the exact area of the surface obtained by rotating the curve about the x-axis?

So given this curve: $$y=\sqrt{9x-18},\ \ 2 \le x \le 6$$ And using this lovely formula: $$\int2πy\sqrt{1+(\frac{dy}{dx})^2} dx$$ This is what I get for a set up: $$\int_2^62π ...
5
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2answers
259 views

Finding the length of a curve?

With the information given: $$x=\frac{y^4}{8}+\frac{1}{4y^2}\,,\ \ 1 \le y \le 2$$ I must find the exact length of the curve. I use this formula to find it: ...
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2answers
212 views

Find the cartesian equation from the given parametric equations

I'm tasked with converting these parametric equations into one cartesian equation. $$ x = a*sin(t) $$ $$ y = b*cos(t) $$ So I begin with my reasoning, which is potentially 100% wrong. I want to ...
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2answers
47 views

Parametrised curves.

I've been working through the following question: Q1= What points on the parameterised curve $x(\theta)=\cos^2{\theta}, y(\theta)=\sin{\theta}\cos{\theta}$ correspond to the parameter values ...
1
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2answers
153 views

Can there be variations on the Witch of Agnesi?

The function known as "The Witch of Agnesi" can be constructed using a circle of radius $a$, and is written in Cartesian coordinates as $$f(x)=\frac{8a^3}{x^2+4a^2}$$ The family of functions that ...
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2answers
43 views

Finding cartesian equation for trigonometric parametric forms

I'm trying to find the cartesian equation for these parameteric forms: $$ x = sin\theta + 2 cos \theta \\ y = 2 sin\theta + cos\theta $$ I tried: $$\begin{align} x^2 & = sin^2\theta + ...
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1answer
43 views

Sinewave riding on sinewave help

Consider this image Top one is Cos(t), I know that. What is the equation for the second one? (sinewave within sinewave) And how would I get to the third one and then on n-amount of recursion? Mind ...
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0answers
30 views

Smoothness depends on parametrization

Does the smoothness (meaning infinitely differentiable) of a function depend on its parametrization? Suppose we have the function $f(t) = [t^2,t^{\frac{1}{3}}]^T$ on [0,1]. Then $\nabla f(t) = ...
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0answers
41 views

What's the next “recursion” here?

Plotting a single 3d helix is x = cos(t); y = sin(t); z = t; From this equation: x = [R + a cos(\omega t)] cos t y = [R + a cos(\omega t)] sin t z = h t + a sin(omega t) Comes the awesome ...
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1answer
136 views

Multivariable optimization - how to parametrize a boundary?

A metal plate has the shape of the region $x^2 + y^2 \leq 1$. The plate is heated so that the temperature at any point $(x,y)$ on it is indicated by $T(x,y) = 2x^2 + y^2 - y + 3$. Find the ...
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0answers
519 views

Helix around helix parametric equation?

I know the parametric equation for a $3D$ helix is: $x = R \cos t$ $y = R \sin t$ $z = h t$ Can somebody explain to me this parametric equation (image and equation from Wolfram) for a "Helix ...
0
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2answers
64 views

Find a normal vector onto the line

How can I find normal vector on the given line. For example if I have a line $3x - 5y = 1$, what would be the normal vector of this line? I am not sure whether it's useful or not, but we have one more ...
1
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1answer
112 views

Vector Parametrization of a Hyperbolic Paraboloid and a Plane

So I need to find the intersection between a hyperboloid ($z=\frac {y^2}{b^2}-\frac{x^2}{a^2}$) and some related plane ($bx+ay-z=0$). I have tried solving for $z$ and equating the two: ...
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2answers
50 views

Sketch curve $r(t) = cos(t)i+sin(t)j+tk$

This is from "Multivariable Calculus, Concepts and Contexts" by Stewart. He says "The parametric equations for this curve are: $x = cos(t)$, $y = sin(t)$, $z = t$ This makes sense. However, he then ...
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1answer
29 views

Where do I go wrong? (cartesian form of parametric eq.)

Basic stuff but I'm not sure where I'm going wrong. Much appreciated if someone could check through my working to see where I misstep! $$x(t)=t+\frac{1}{t}, y(t)=1-\frac{1}{t}$$ So: ...
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3answers
136 views

Using trig identities to change from parametric to Cartesian equation

$$x=\sin t\\ y= 3\cos (3t)$$ Find $y$ in terms of $x$. I have graphed the function and it appears to follow $y(x)=-4x^2 +2$ from $-1\le x\le 1$ and $-2\le y\le 2$ . Thanks
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3answers
318 views

Circle rotating within a circle (roulette)

This was something in a course of mine I'm a bit too thick to see. If one takes a circle of radius $3$ and a circle of radius $1$, and rolls the smaller circle smoothly inside the larger one until the ...
3
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1answer
51 views

Inverse mapping for a simple $\mathbb{R}^3$ surface given by $(\sin u, \sin 2u, v)$.

For a domain $U=\{\, (u,v) \in \mathbb{R}^2 \mid -\pi<u<\pi,\ 0<v<1 \,\}$ we have a mapping $X \colon U \to \mathbb{R}^3$ defined by $X(u,v) = (\sin u, \sin 2u, v)$. The resulting surface ...
0
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1answer
56 views

Cartesian Equation and Parametric Equation Help

I just need some help with a maths question that I am trying to get done for a maths tutorial homework sheet. The question is... Let L be the line through D = (6,5,4) and E = (1,0,6), and let P be ...
0
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1answer
38 views

Find the parametric equation of the following parabola?

It doesn't give me $2$ equations this time just $1$ and I have no clue what to do; $y^2 = 4x$ ANSWER IN BOOK: $x = t^2, y = 2t$
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0answers
22 views

How to know which variables to parametize in a large matrix?

(dont want anyone to solve the problem, just don't understand one thing) So I have a homework problem where I got a 3x6 matrix, and I have to parametrize the equations and solve for each variable in ...
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2answers
39 views

How to apply chain rule to:$\frac{d}{dx} \Big( \frac{dy}{dx} \Big)$?

How do I apply chain rule to the following: $$\frac{d}{dx} \Big( \frac{dy}{dx} \Big)$$ Where $$\Big( \frac{dy}{dx} \Big) = \Bigg(\frac{\frac{dy}{dt}}{\frac{dx}{dt}} \Bigg)$$ I don't see the ...
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1answer
101 views

Intersection of graphs of parametric equations with trig functions

I've had this problem for a while, but I have not been able to solve it. Any help would be really appreciated, thanks in advance! Problem: Find the sum of all possible values of the constant $k$ such ...
0
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1answer
49 views

Sketching a curve and finding where the parameter increases

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. $$x = ...
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0answers
94 views

Finding the equation for ellipse velocity

I am trying to figure out how to do a homework problem for my math class. The homework for the night is, given a set of parametric equations, has three parts, the first of which is to find the speed ...
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0answers
21 views

Comparison of Parametric and Polar Equations

Having been introduced to parametric equations, I cannot help but question the similarities between parametrized functions and polar functions. A parametric circle is defined by the following: ...
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1answer
531 views

Eliminate the parameter to find a Cartesian equation of the curve

I've done every problem on this subject except I can't get this one. (every other problem had x = something and y = something. $$y = (t+1)^{1/2},\quad y = (t-1)^{1/2}$$
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1answer
64 views

Evaluation of $\int_{-\pi}^{\pi} \cos(ax) \sin^n(bx) dx$

As it is a kinda famous integral I thought I would find something on MSE but I didn't so here I am. If there is, link it in the comments and I will delete the question. How do I evaluate ...
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2answers
106 views

Not sure how to differentiate implicitly using parametric equations…

I am not sure (not taught before explicitly) how to apply implicit differentiation on parametric equations when I am solving the question posted below. Question Two positive numbers $x$ and $y$ ...
0
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2answers
38 views

Determine max/min speeds of trochoid

Find the minimum and maximum speeds of the point of a trochoid and the locations where each occurs. I know a trochoid has equations $ (x)t = at - b \sin{t} $ ; $ y(t) = a- b \cos{t} $ for trochoid ...
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4answers
41 views

Find $\frac{dy}{dx}$ for $x=2\theta+sin2\theta$ and $y=1-cos2\theta$

The parametric equations of a curve are $$x=2\theta+\sin2\theta,\:y=1-\cos2\theta.$$ Show that $\frac{dy}{dx}=\tan\theta$. I can use the chain rule to get ...
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3answers
61 views

Proof that this surface is of revolution

I have a surface with parametric equation $$\mathbf{x}(u,v)=(u\cos(v),u\sin(v),u^2),$$ $u$ is any real number, $v$ is between $0$ and $2\pi$. I don't know how to show that this is surface of ...
0
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1answer
205 views

How to clip Bézier curves using Casteljau's algorithm?

I am attempting to approximate intersections of Bézier curves using iterative clipping. This common method is described here and here. It basically works like this: Find bounding lines outside one ...
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0answers
41 views

Finding the mean value of y

I don't understand how to obtain the limits for the $t$-values considering that they gave us the $x$-values in the first part of the equation. I've considered substituting the $x$-values into the ...
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3answers
46 views

Check if two vector equations of parametric surfaces are equivalent

Give the vector equation of the plane through these lines: $\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}4\\1\\1\end{pmatrix}+\lambda\cdot\begin{pmatrix}0\\2\\1\end{pmatrix}\,\,\,$ and ...