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

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Parametric Eqn / Differentiation

Parametric eqns of a curve are $x = t + \frac{1}{t}$ , $y = t - \frac{1}{t}$, where $t$ cannot be $0$. At point $P$ on curve, $t = 3$ and the tangent to curve at $P$ meets the $x$-axis at $Q$. The ...
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29 views

Find all the values of real parameter “n”…

Let $S$ be the set of real solutions for the following equation:$$\log_2(1-x-x^2)=n\log_{1-x-x^2}2+2$$ Determine all the values of real parameter $n$ for which $S\cap(0;{1\over2})\neq\emptyset$.
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Mass and density function Calculus II Problem

A thin metal plate lies over the portion of the cylindrical surface $y^2 + z^2 = 4$ for $z ≥ 0$ between $1 ≤ x ≤ 4$. The density of the plate is given by $f(x,y,z) = z$. How do I calculate the mass ...
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58 views

Find the real parameter so as the equation has no real solutions…

My question is: For which values of parameter $a\in \mathbb{R}$ the following equation $$25^x+(a-4) \,5^x-2a^2+a+3=0$$ has no real solutions? My idea is: First of all we should transform the ...
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28 views

when is it not possible to eliminate parameter t from parametric equations?

We can eliminate parameter t from a set of parametric equations to convert to a Cartesian equation. My book mentions that it is sometimes impossible to do that. What would be an example of not being ...
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1answer
38 views

Surface area generated by revolving about the y-axis

I have to find the surface area which is generated by revolving the curve about the y-axis found below: $$x=\frac{1}{2}(e^{y} + e^{-y}) \ ; 0<=y<=ln2 $$ I know how to solve the question, when ...
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1answer
118 views

Rotation of conics sections using linear algebra

When given an equation of the form $$Ax^2+Bxy+Cy^2 + Dx + Ey + F$$ where $B \not= 0$ and it is not a degenerate conic, then you can use $\Delta = B^2 -4AC $ to see what type of conic it is, and then ...
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81 views

How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
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2answers
121 views

Curve on a basketball

The sewing pattern on a basketball is composed of two great circles and a single curve that intersects each great circle twice. Does this curve have a name? Are there any parametric descriptions of ...
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46 views

Find the point on an ellipse by angle.

How do I find the point on the ellipses at 45'. I found this, which answers part of it, but I need to know how to calculate for (x,y) at 45'. I could also use a good explanation for the ...
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1answer
49 views

Tangent Planes and Surfaces (Calc 3)

I am wondering if I am on the right track for the following question: Find a for the plane $x+y+z=-1$ so that it is a tangent plane to the surface $z=x^2+ay^2$ I figured since you are given a ...
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1answer
167 views

Umbilic Points of an Ellipsoid

I have an ellipsoid given by $S = \{ (x,y,z): \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$, for some fixed $a,b,c \in \mathbb{R}^{+} \}$. I need to find the umbilic points of ...
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1answer
31 views

Checking a solution of a PDE

I have the following PDE: \begin{equation} -yu_x + xu_y = 0 \quad\text{where } u(0, y) = f(y) \end{equation} I derived a solution as follows: \begin{align} -yu_x + xu_y =& 0 \\ \iff& ...
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3answers
697 views

Equation of a torus

First I am a newbie in maths so please forgive me if I am not as rigorous as you would like, but do not hesitate to correct me. I want to find the equation of a torus (I mean the process, not just ...
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1answer
56 views

Area of part of parametric function

I need to get area of function: $x= 2\sqrt{2}\cos ^3 t$ and $y= 4\sqrt{2}{\sin ^3 t}$, but only the part when $x\geq1$. How can I do that? I know that area of full function would be $$S= \int_a^b ...
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1answer
28 views

Evaluate $\iint_s\text{curl}\textbf F\cdot \textbf {n}dS$

Let $\textbf F=<xy,yz,zx>$ and $S$ be the upper half of the ellipsoid $\displaystyle \frac {x^2}{4}+\frac {y^2}{9}+z^2=1$. Evaluate $\iint_s\text {curl}\textbf F\cdot \textbf {n}dS$ I know the ...
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104 views

How to draw stick trefoil knot

http://commons.wikimedia.org/wiki/File:Stick_number_trefoil.png I am interested in plotting the stick trefoil knot. I don't know where to start. I am looking for equations or co-ordinates of ...
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1answer
63 views

Find all the intersection points of a vector parabola (in R3) and a sphere

Given that I have a vector in R3 (7t, 10t - 2t^2, 5t) | (These numbers are arbitrary for the sake of the process) A sphere centered at the point ( 15, 25, 10) with a radius of 20 There is a ...
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1answer
48 views

What is being represented by this 2 images?

image 1 image 2 It's possible that image 1 is showing some kind of methods for building polygons out of trigonometric functions ? It's also possible that image 2 is a quadratic bezier curve ?
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1answer
22 views

How can I reduce the interval of this parametric equation:$ x=t/(1+t^4) ; y=t^3/(1+t^4)$ to the simplest possible domain.

How can I reduce the interval of this parametric equation: $x=t/(1+t^4) ; y=t^3/(1+t^4)$ to the simplest possible domain? Can you explain this solution?
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1answer
30 views

Values of a limit with parameter

Evaluate the values of the limit as the changes of $\lambda \in \mathbb{R}$ $$\lim\limits_{x \to +∞} e^{x-x^3+\lambda\ln(x)}f^{(n)}x$$ with $$f^{(n)}(x)=p_n(x)e^{x^3-x}$$ where $p_n(x)$ is a generic ...
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29 views

Graphing an inverse parametrically

My calculus book has the following question: Graph the one - to - one function $ f(x) = x^2$ , where x is greater than or equal to zero, with its inverse. Now my answer is that the inverse is the ...
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27 views

Parametric Function for Coloring

I'm trying to write a parametric straightline function that changes its values between 0.529 and 0.933. Usually what I would do is: $r = 0.529+(0.933-0.529)*v$ where parameter $v= [0,1]$ This ...
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1answer
27 views

Parametrization Semicircle On Sphere

I need to find a parametrisation in terms of $t$ for a half circle on a sphere with radius $R$. The circle goes from $(R,0,0)$ to $(-R,0,0)$ and is going through the point $(0, ...
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1answer
28 views

Parametric equation for a curve $C_1$ in $\mathbb{R}^3$ such that the angle of its tangent $T$ and the $Y axis$ equals the angle of $T$ and a vector

I know that to find the angle between two curves in $\mathbb{R}^3$ at their intersection I differentiate both curves and evaluate at the intersection to find the slope of each tangent line and then do ...
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1answer
125 views

Find the area of the surface obtained by rotating the curve of parametric equations

Rotate about the $x$ axis $x = 2t-2/3t^3$ $y = 2t^2$ $0 \leq t \leq 1$ I did the integral of $\sqrt{(2-2t^2)^2+(4t)^2}$ and got $(2x(x^2+3))/3$ and then I did the integral of $2\pi 2t^2 ...
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55 views

Parametrizing a 3D surface

Find a parametrization of the surface $x^3 + 3xy + z^2 = 2$, $z > 0$, and use it to find the tangent plane at $x = 1$, $y = \dfrac{1}{3}$, $z = 0$. I know how to find the tangent plane once I have ...
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91 views

How to parametrize this region surface

$S$ is the portion of the plane $$x+2y-3z=3$$ in the octan bounded by the positive direction of the $x$ and $y$ axis and the negative direction of the $z$ axis. How can I parametrize this crazy ...
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43 views

Is it possible that $f_1,f_2,\dots,f_n$ are not all differentiable, but $\alpha:I\to \Bbb{R^n}$ is differentiable?

Consider the parametrized curve $\alpha:I\to \Bbb{R^n}$. These notes say that $f_1,f_2,\dots f_n$ being differentiable $\implies$ $\alpha$ is differentiable. I wonder why the converse is not true. Is ...
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90 views

parametrize surface region

S is the elliptic region of the plane $y+z=1$ inside the cylinder $4x^2+4(y-0.5)^2=1$. First parametrize $S$ using $(x,y,z)=G(u,v)$ and then calculate $\displaystyle \frac{dG}{du}\times ...
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177 views

Parametrization of the lemniscate

All over the net it is stated that the parametrization of the lemniscate with Cartesian equation: $(x^2 + y^2)^2 = 2a^2 (x^2 - y^2)$ is: $$\varphi: t \mapsto ...
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1answer
58 views

Normal line to cycloid

A Cycloid is given by $$\left\{\begin{matrix} & x(t) = 3 \cdot (t-\sin t)\\ & y(t) = 3\cdot(1-\cos t) \end{matrix}\right.$$ I need to find the parametrized curve for the Normal line ...
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39 views

Find the parametric equations of the line of intersection…

Find the parametric equations of the line of intersection of the planes x - z = 1 and x + 2y + 3z = 1. I'm assuming it's something to do with cross product? Here's what I've set up: x y z 1 ...
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1answer
51 views

Evaluating a surface integral of a paraboloid

Calculate the average value of $(1+4z)^{3}$ on the surface of the paraboloid $z=x^{2}+y^{2}$,$x^{2}+y^{2} \leq 1$ I'm not sure on how to start this problem. I have already found the area of the ...
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3answers
44 views

eliminate parametric parameter to determine the Cartesian equation.

$$x = \sin^2(t), y = \cos(t)$$ I know that to eliminate parameter involving $\sin$ and $\cos$, we should reduce it to $x^2 + y^2 = r^2$. So, $x^2 = \sin^4(t)$, $y^2 = \cos^2(t)$ But I can't make ...
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1answer
57 views

Two curvature formulas when equal arc-length

all. So with a parametric curve $\vec{r}=\langle x(t),y(t)\rangle$, curvature is given by $$\kappa=\frac{|x'y''-x''y'|}{(x'^2+y'^2)^{3/2}}.$$ When we have constant arc-length, an alternate ...
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1answer
24 views

Find the parametric curver from points (2,1) and (5,4)

I already found $x=2+3t$ and $y=1+3t$ but I don't know how to get the whole equation of $r(t)=$? and what the boundaries are $?<t<?$
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1answer
683 views

Find a parameterization for the circle of radius 2 in the xy-plane, centered at the origin, clockwise

Find a parameterization for the circle of radius $2$ in the $xy$-plane, centered at the origin, clockwise. I know to use $2\cos(t)$ and $-2\sin(t)$ but I'm not sure what to do after that
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81 views

System of parametric inequalities

I'm still struggling with the parametric inequalities.I'm doing some systems of parametric inequalities, but I can not understand how to proceed.This is the system: $x-2a<1+a$ $\frac x2 ...
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1answer
54 views

Equation of a polygon

I need a parametric equation for a filled polygon defined by 3 or more points. The closest I've got is by using 3 points in this equation - $polygon = p1 + u(p2-p1) + v(p3-p1)$. But by using points ...
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101 views

Find Vector and Parametric Equation

I'm having some trouble finding answers to these problems. When i try to find help online, all i find are (x,y,z) problems and I'm simply looking for a PreCalculus (x,y) problem solving technique: ...
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1answer
40 views

Parametric inequality problem

I would ask for help regarding a problem with the parametric inequalities. $\dfrac{(x+a)}{(a-1)}+\dfrac{(x-a)}{(a+1)}-\dfrac{x}{(a+1)}-\dfrac{2(x-1)}{(a-1)}\ge 0\ \text{for}\ a<-1$ Since ...
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83 views

Supremum Definition in context of a Rectifiable/Differentiable Curve

I am trying to prove that if I have 2 paramterizations of the same curve $\gamma$ and $\sigma$ (i.e. there is continuous bijective map $\phi$ such that $\sigma = \gamma \circ \phi$) then if the curve ...
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0answers
55 views

Inverse ease-in-out parametric function

I'm trying to create an inverse ease-in-out function that given values from 0 to 1, produces values from 0 to 1. Opposite of a typical ease-in-out function, though, I want it to start accelerated, ...
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83 views

Surface integral

Problem statement $$ \mbox{Calculate the surface integral}\quad \int_{Y}\ y\,\sqrt{z\,}\,\sqrt{4x^{2} + 4y^{2} + 1\,}\,\,{\rm d}S $$ where $Y$ is the surface $\left\{\left(x,y,z\right)\ \ni\ ...
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114 views

Paramtrizing a counterclockwise circle vs. a clockwise one

Does it make a different when you parametrize a counterclockwise full circle and a clockwise circle in the complex plane? For example, I am looking at computing an integral $\int_\gamma ...
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380 views

Wolfram and solids of revolution

I'm looking for the easiest method of having WolframAlpha calculate the volume of a solid of revolution. I've been working on a particular Project Euler problem for a long time. So far, I think I ...
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1answer
36 views

ratio of tangent to the ellipse

The tangent at point $P = ( a \cos \phi, b \sin \phi)$ on the ellipse $\frac{x^2} {a^2} + \frac{y^2}{b^2}=1$ meets the $x$ and $y$ axes at the points $X$ and $Y$, respectively. Find in terms of ...
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1answer
52 views

Parabolic flight. h(t) vs 2 parametric equations.

Often you have something like: $$h(t)=-16t^2+V_0t+C$$ I have little experience with parametric equations, but I have also seen parabolic functions represented this way: $$x=x_0 + V_{0_x}*t$$ ...
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1answer
139 views

How to construct a parametric cubic B spline?

If I am given n+1 control point Pi(xi,yi), Po .... Pn , how do I construct a parametric relationship to draw a curve ? From what I understand , a parametric relationship is that you can express x and ...