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

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2
votes
4answers
303 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 ...
0
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0answers
36 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 ...
1
vote
2answers
24 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∘. ...
1
vote
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 ...
0
votes
2answers
43 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
votes
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
votes
1answer
30 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
vote
0answers
18 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,...
1
vote
2answers
49 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
votes
0answers
29 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
vote
1answer
43 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
votes
0answers
44 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
votes
1answer
34 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
votes
0answers
24 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
votes
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
votes
1answer
29 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
41 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 ...
-1
votes
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?
-1
votes
1answer
41 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
votes
1answer
24 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
30 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
24 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
0
votes
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
votes
0answers
14 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}\...
-1
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
votes
1answer
22 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 ...
1
vote
1answer
42 views

What's the name of this simple, closed, planar curve?

A simple, closed, planar curve can be given by the following parametric function: $$ \gamma(t)=\left(\cos t,\sin t+\frac{\sin^2t}{2}\right) $$ This function on $t=0$ to $t=2\pi$ gives the following ...
0
votes
0answers
32 views

How to determine the smallest period of a parametric curve?

Consider the polar function $r(\theta) = \sin(3\theta)$, and the parameterization of its graph given by $x = \sin(3\theta)\cos(\theta), \;y=\sin(3\theta)\sin(\theta)$. Upon inspection, one can observe ...
11
votes
4answers
391 views

Horizontal tangent line of a parametric curve

Suppose $x=t^2,y=t^3$ is a parametric curve. Here's a quote from my textbook: The origin, which corresponds to $t=0$, is a singular point of the parametric curve, because $dx/dt=2t,dy/dt=3t^2$ are ...
1
vote
1answer
28 views

Finding parametric equations of rectangular equation

Is there a general process to follow when finding the parametric equations of a normal rectangular equation ? I know that one rectangular equation might have many parametric equations, but are there ...
0
votes
2answers
76 views

Parametric Surface

A surface is given by $$r(u,v) = \langle u, v^2, uv\rangle$$ (a) Evaluate the unit normal vector, $\vec n$, to the surface at the point corresponding to $u=2$ and $v=1$. I've done this by ...
0
votes
1answer
25 views

finding cartesian equations of parametric equations

find the Cartesian equation for the parametric equation $$x=\frac{1}{\sqrt{1-t}} \text{ and } y=\frac{t}{1-t}$$ I tried cross multiplying but I cant seem to find the equation in terms of $t$ to ...
1
vote
0answers
44 views

Does anyone known the parametric equations for Cloud Gate?

I would like to use Mathematica to plot the famous Chicago "Bean." I couldn't find parametric equations anywhere and was wondering if anyone knew them. Thanks!
0
votes
1answer
38 views

Find arc length of $(x-1)^{2/3}+(y-2)^{2/3}=1$

I'm trying to find the arc length of the curve defined by $$(x-1)^{2/3}+(y-2)^{2/3}=1$$.My first approach was try to set 'y' in terms of 'x' and then apply the formula $$L=\int\limits_a^b\sqrt{1+y'}...
2
votes
0answers
27 views

Calculate $\int\int_sf(x,y,z)dS$ for $x^2+y^2=9$, $f(x,y,z)=e^{-z}$

Calculate $\int\int_sf(x,y,z)dS$ for $x^2+y^2=9$, $0<z<6$; $f(x,y,z)=e^{-z}$ I am completely confused on this. I know I can parameterize $x^2+y^2=9$ into... $x(r,\theta)=rcos\theta$ and $y(r,\...
0
votes
0answers
31 views

Minimizing arc length on unit sphere (geodesics)

I just completed a Calculus IV course and taught myself basic Calculus of Variations, and wanted to extend some of the basic principles of optimization from planes to surfaces. The arc length ...
1
vote
1answer
56 views

How can I draw a Bézier Curve through a set number of points?

For high school Mathematics Pre-Specialist, I have been given the task of writing a mathematical investigation based on the following three questions: Quadratic Bezier curve enables a smooth curve to ...
2
votes
0answers
75 views

Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
0
votes
1answer
32 views

Question about calculating the area underneath a “3d” curve.

I'm trying to calculate the area underneath a curve after a $z$-component has been added. Suppose we have the equation: $$y = -x^4 - x^3 + 3x^2 -x + 4$$ on the interval $[-2.38, 1.76]$ (the roots ...
1
vote
0answers
18 views

How could I determine the form of a function that chases another function?

This is a problem that a teacher told me about that's been bothering me for a while. I'm positive that this has been explored before because it seems way too useful for physicists to not have come up ...
0
votes
1answer
29 views

Projecting a parametric curve to a plane

I have a parametric equation: x = t^3 and y = t + 2t. I would like to do a line integral of this curve up to the plane z = 5. Basically, I would like to find the area of the "walls" formed when ...
1
vote
1answer
39 views

Tangent line for a parametric curve

I am given that $x=ae^{-t}$ and that $y=be^{2t}.$ I'm asked to find the tangent line at $t=0.$ I have said that $$\frac{dx}{dt}=-ae^{-t}, \frac{dy}{dt}=2be^{2t}$$ Thus $$\frac{dy}{dx}=\frac{-2b}{a}e^...
2
votes
3answers
47 views

Function parameters and cartesian curves

Given $$x = \cos t + \cos 2t \,,\; y = \sin t + \sin 2t ,$$ find the tangent line for the parameter at point $(-1, 1),$ and draw a graph of the curve. To find the point you could simply do $\...
1
vote
2answers
35 views

Finding parameterization of surface that maps parameter distance to arc length

I have a surface in $\mathbb{R}^3$ defined by four corner points $p_i$ and with known normals at each corner $n_i$. I've also constrained the contour of each edge to be a circular arc, which can be ...
0
votes
1answer
16 views

Need help understanding what the curve made by two or three intersecting surfaces looks like

I have trouble visualizing what curves are traced out by the intersection of multiple surfaces in $R^3$. for example take the parametric equations $ <cos(t),sin(t),sin(t)$ > Clearly this would ...