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

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2
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1answer
13 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 ...
0
votes
1answer
32 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 ...
0
votes
1answer
40 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 ...
1
vote
2answers
21 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 ...
2
votes
2answers
32 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
votes
1answer
50 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 ...
0
votes
1answer
23 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 $\int_0^t||r'(u)||dt$
0
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0answers
6 views

solution to curve limits (concept question)?

I had a question about how the limits work in that 4pi would not give the correct circle distance. I understand that if it has a radius 1 that the distance would be farther but that is only for a ...
0
votes
1answer
13 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 ...
0
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3answers
28 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 - ...
0
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1answer
36 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 ...
0
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2answers
19 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?
-1
votes
1answer
17 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 ...
1
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2answers
55 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
votes
2answers
238 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: ...
0
votes
2answers
32 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 ...
1
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2answers
43 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
vote
2answers
141 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 ...
0
votes
2answers
24 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 + ...
0
votes
1answer
39 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 ...
0
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0answers
25 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) = ...
3
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0answers
39 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 ...
0
votes
1answer
35 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 ...
1
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0answers
84 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 around helix" / ...
0
votes
2answers
49 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
41 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: ...
1
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2answers
38 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 ...
0
votes
1answer
25 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: ...
0
votes
3answers
51 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
3
votes
3answers
131 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
votes
1answer
48 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
votes
1answer
40 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
votes
1answer
31 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: $x = t^2, y = 2t$
0
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0answers
19 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 ...
1
vote
2answers
37 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 ...
1
vote
1answer
29 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
votes
1answer
32 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 = ...
0
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0answers
38 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 ...
0
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0answers
159 views

What approach to use when matching parametric equations with graphs?

I have no idea what to do. It says to give reasons for my choices. Don't give me the answers, I just need some guidance. What to look for in the graphs and equations when making the choices?
0
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0answers
16 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: ...
-1
votes
1answer
308 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. $$y = (t+1)^{1/2},\quad y = (t-1)^{1/2}$$
1
vote
1answer
61 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 ...
1
vote
2answers
44 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
votes
2answers
24 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 ...
1
vote
4answers
39 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 ...
1
vote
3answers
55 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
votes
1answer
64 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 ...
1
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0answers
35 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 ...
1
vote
3answers
34 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 ...
0
votes
1answer
16 views

Check me- Speed(Arc Length) of Parametric Equations

A ferris wheel has height = 100 ft and completes 1 revolution in 3 minutes at a constant speed. Compute the speed of a rider in the ferris wheel. Ferris wheel = circle modeled by x= cos t ; y = sin t ...