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Questions tagged [parametric]

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

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Determining the Values of $\alpha$ for Which the Series is Conditionally and Absolutely Convergent

The task is to determine for which values of $\alpha$ is the following series is conditionally convergent and absolutely convergent. My attempt is below. $$\sum_{n=1}^{\infty} {n^{-\alpha}\cdot(\ln{n}...
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Spearman of Pearson in dataset with various distributions

I want to correlate several biomarkers with clinical parameters. Some are normally distributed (with or without log-transformation) and some are not. Can I use Pearson for the normally distributed ...
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1answer
23 views

Rolle's and Lagrange's theorems in parametric equations

I was trying to solve some calculus problems and I came across with some doubts related to two of them. Given the parametric equation $$ (x,y)=(3-3\cos^2 (t),3-3\cos(t)\sin(t)) \quad 4 \pi /3 < t ...
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Deriving logarithmic spiral equation from square corners

This is an interesting problem but I haven't been able to work it out (from Bender and Orszag, prob: 1.27) - any insight/assistance would be appreciated: Four caterpillars, initially at rest at the ...
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Determining if 2 variances are equal with a rule of thumb?

So yes I am familiar with the F-test and know how to use it. Though I remember there was a quick rule of thumb of determine if 2 variances are equal. By either subtracting or dividing the 2 and it ...
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Non-dimensionalize - Determine the dimensions of the parameters

I have the following assignment, I've made most of the assignment, but cannot seem to figure the rest out. The assignment: "Certain organisms have an effective growth, $\frac{\dot{n}}{n}$. A model ...
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Skewed conical spiral equation

I'm struggling to find a formula for a spiral tracing a skewed cone. I need to generate x, y, z coordinates. Wolfram alpha has parametric equations for a conical spiral: http://mathworld.wolfram.com/...
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System of linear equations with parameter

I have this system of linear equations with parameter: $ ax + 4y + z =0 $ $2y + 3z = 1$ $3x -cz=-2$ What I did was to put those equations into a matrix and transform that matrix it into a ...
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Finding cartesian equation of curve with parametric equations

A curve has parametric equations $x=a \sin(⁡t)+b \cos(⁡t)$ $y=a \cos⁡(t)-b \sin⁡(t)$ How do I eliminate t to find the Cartesian equation here? I've tried different weird approaches, i.e. squaring ...
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36 views

find $\lim _ {y\rightarrow + \infty } \int _ { 1 } ^ { 2 } \frac { \ln ( x + y ) } { \ln \left(x^{2}+y^{2} \right) } d x$

I need to find $$\lim _ {y\rightarrow + \infty } \int _ { 1 } ^ { 2 } \frac { \ln ( x + y ) } { \ln \left(x^{2}+y^{2} \right) } d x$$ Any hints? PS. i tried to convert problem to solving improper ...
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Calculate surface limited by curve

How to calculate the surface limited by the curve described with the two equations: $x = 3+cos(t)$ $y=4sin(t)$ The formula we use is the one shown below: $\int_a^b |y(t).x'(t)| \,dt$ Now I'm ...
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Convert $x(t)=2t\sin{t}+2\cos{t}$, $y(t)=2\sin{t}-2t\cos{t}$ to a Cartesian equation

I need help to convert following parametric equation in Cartesian: $$x(t)=2t\sin{t}+2\cos{t}$$ $$y(t)=2\sin{t}-2t\cos{t}$$ I've tried squaring both equations and adding first to second ...
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1answer
19 views

The prolate cycloid

A cycloid is given by the parametric equations: $ x = 2 - \pi \cos(t)$ and $ y = 2t - \pi \sin(t)$. The problem asks for the slope of the tangents on the cycloid at a point where the cycloid ...
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1answer
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How to eliminate $a$ from the parametric equations of a locus?

Stated question: The point T ($at^2$, $2at$) lies on the parabola $y^2=4ax$ and L is the point ($-a$, $2a$). M is the mid-point of TL. Find the equation of the locus of M as T moves on the parabola. ...
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2answers
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Find distance from line k given parametric equation of: $x=2+t, y=-3+2t, z=2-t, t\in\mathbb{R}$ from plane $\pi:2x+y+4z=0$.

My solution: $$2(2+t)+(-3+2t)+4(2-t)$$ $$4+2t-3+2t+8-4t$$ $$9=0$$ Contradiction, so no solutions, line and plane are parallel. Its first time where I have such an example where equation is ...
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Eliminating parameter

How do you solve for "$t$" in either equation to eliminate the parameter and solve in terms of $x$ and $y$? $$ \begin{split} x&=\frac{ 1+t }{ \sqrt{1+t^2}}\\ \\ y &= \frac{1-t}{\sqrt{1+t^2}} \...
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1answer
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Parametrize the line in $\Bbb R^3$ that is determined by the intersection of the planes:

I am given two equations \begin{alignat}{2} x&+\phantom{0}y&\phantom{0}+z&=\phantom{0}4 \\ 2x&+\phantom{0}y&\phantom{0}-2z\phantom{0}&=\phantom{0}4 \\ \end{alignat} I am ...
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1answer
31 views

How to eliminate parameter of parametric equations?

If the parameter $t$ of $x$ and $y$ in a plane is given in the interval $(- ∞ , ∞)$ and if $x = t cos (t)$ and $y = t sin (t)$ How can one eliminate the parameter t and write a single equation using ...
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trying to plot a gear curve in gnuplot

Im looking at this gear curve: thinking wow I'd like to plot that! so then I tried, in c++, with gnuplot. ...
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Using the Jacobian matrix to calculate a parametrized area?

$ x = \cos(\theta)r\\ y = \sin(\theta)r\\ z = r $ I have done this parametrization, and now I want to integrate to get the area of a cone. According to my textbook, I'm supposed to multiply with the ...
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1answer
38 views

Integration, Area under curve

The question goes by : Find the finite region bounded by the curve $$x=5t^2$$ $$y=2t^3$$ and the line $x = 5$. Find also the volume of the solid formed when this region is rotated through $\pi$ ...
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Parametric functions

How can I plot a parametric function using Graphing Calculator 3D? I am studing parametric equations and sometimes it would be very useful to plot this equations to help visualization, but I have no ...
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1answer
28 views

Matrixes with common parameters to result in no inverse

I've been given three matrices $A, B \ \& \ C$ which are defined as follows: $$ A = { \left[ \begin{array}{ccc} b & 5 & 8 \\ c & 1 & 3 \\ a & 4 & 3 \\ \end{array} \right]...
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Construct a differential equation whose solution in parametric form is the butterfly curve.

Is it possible, and if so, does anyone know how to construct a differential equation whose solution on parametric form is the butterfly curve: $$x=\sin (t)\left(e^{\cos (t)}-2 \cos (4 t)-\sin ^{5}\...
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21 views

Why do I get this diagonal? (normal of a curve at specific point)

I had to code how to draw the normal of a curve at a specific point; $t_0 = \frac{2\pi}{5}$ (https://stackoverflow.com/questions/55723461/how-to-plot-the-normal-at-a-point-for-a-given-parametric-curve)...
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2answers
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Incorrect sign when evaluating of bounds of integral

Apparently either I've forgotten some basic rule about integrals (it has been a while since I've taken a basic calc class) or something is wrong with this problem in pearson mylab. This was the ...
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1answer
36 views

Polar and parametric curves

I was solving a calculus problem on polar coordinates and I came across with some doubts, I don't know how to solve it. It says: "Given the curve $C: (x+1)^2+y^2=1$ parametrize the arc of a curve that ...
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4answers
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Parametrization of a line segment using angle as parameter

I know this is probably elementary level for most people here, but I've been stuck on this problem for no less than 4 hours and I am completely clueless as to how to figure this out. Is it possible ...
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1answer
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How to Integrate Parametric Curves With Algebraic Coefficient?

I have researched all over the place to get answers to this question; albeit to no avail. Most information that I could find focused on simple sin / cosine values - as such I make this request for ...
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2answers
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Eliminating the parameters of a cycloid [closed]

I was given two parametric equations and I need to eliminate the parameters, but I have no idea how. That pesky $\theta$ is hard to get rid of. Any suggestions? I tried many trig identities $x$ = $\...
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What's the difference between a parametric equation and a level curve of a function of multiple variables?

For example, the level curve of a function that takes two variables ($x$ and $y$ for example) and a parametric equation involving two functions ($x=f(t)=...$ and $y=g(t)=...$). Both of the resulting ...
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30 views

Finding an area of parametric curve that lies above and below x axis.

Find an area from $t \in <0, 2\pi>$ bounded by parametric curve: $\begin{cases} x(t) = \cos^3{t} \\ y(t) = \sin^3{t}\end{cases}$ I believe the formula is: area $=\int^{2\pi}_{0}y(t)x'(t)dt$. $...
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1answer
25 views

Area bounded by parametric curve and $y=x$

I am familiar with how to find an area bounded by the parametric curve if I am given some $t \in <0,1>$ or such interval. I can also find the area if I am told that the parametric curve is ...
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1answer
18 views

Parametric equation second derivative proof

Below is the question: If $x=\sin t$ and $y=\cos 2t$, find $\frac{dy}{dx}$ in terms of $x$ and prove that $\frac{d^2y}{dx^2} + 4 =0$. I found $\frac{dy}{dx}$ first (using the chain rule, since ...
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I want to know if I have done this excercise on parametric equations correctly.

I was solving the following parametric equation: $$r(t)=(x(t);y(t))=(sin^2 (t), cos(t)) t \varepsilon [0; 2 \pi )$$ I thought that I could do this so I could eliminate the parameter: $$1=sin^2 (t) +...
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How to give a parameterisation of $u(x_0,y)=y$ in the form $\Gamma(s)$?

I am working to solve a PDE with the given initial condition $$u(x_0,y)=y$$ where $x_0$ is a constant. In order to solve the PDE I need to parameterise the curve in the form $\Gamma(s)$. I have ...
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23 views

Calculating an area of parametric function with wolframalpha?

Using wolframalpha, how can I calculate an area between some parametric function like: $\begin{cases} x(t) = t^2 - t \\ y(t) = t^3\end{cases}$ and: some other function like: $y = x^2$ ? I've ...
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1answer
25 views

Logic of Step to Convert Rectangular Equation of Circle to Parametric

I recently learned in math class how to convert the rectangular equation of a circle into a parametric equation: $x^2+y^2=1$ $\cos^2(t)+\sin^2(t)=1$ $x^2+y^2=\cos^2(t)+\sin^2(t)$ $x^2=\cos^2(t)$ $...
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1answer
32 views

Parameterization of the initial condition of nonlinear PDE

(NOTE: I am not asking for the solution to this PDE) I have the PDE, $$u_x u_y + ln(x^2)=0,$$ with the condition that, $$u(x_0,y)=y,$$ where $x_0$ is a constant. I am to find the explicit solution (...
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Parametric equation of a hill shape surface

What is a simple parametric equation to plot a surface shaped like a hill? like the one made by the 3D gauss bell $e^{-(x^2 +y^2)}$ Any help is welcome.
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Surface area of revolution when revolving a partial arc of a circle?

I am really struggling on this particular concept. Consider the following question; Let C be the arc of the circle $x^2+y^2=9$ from $(3,0)$ to $(3/2, 3\sqrt\frac{3}{2})$. Find the exact area of the ...
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Solve for y within a parametric equation

The ellipse $$\frac{x^2}{2^2} + \frac{y^2}{3^2} = 1$$ can be drawn with parametric equations. Assume the curve is traced clockwise as the parameter increases. If $x=2\cos t $ then $y =$ _____ I ...
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55 views

What is the formula for the area between a curve and a parallel curve?

I recently took a liking to parallel curves and tried to find the area between them. Possible applications could be for making geometric swimming pools or some other area/volume based problem. The ...
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1answer
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How do I show the full solution for this?

My lecturer gave us a potential bonus to the grade if we can fully solve those equations: $y^x=9 $ $x^y=8$ easy to see that the value of $x$ and $y$ are $2$ & $3$ but what is the correct full way ...
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Rulebook on how to eliminate variables in parameter (Cartesian equation of curve)

I was tutoring a student in Calculus II, who was working on the subject of "Curves Defined by Parametric Equations." The excellent example of a problem that the student was facing was to eliminate ...
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What is the name of this type of related equation?

So I was playing with this idea of taking any equation ($f(x)$) and be able to have a parametric equation so that both equations slopes are parallel. I was able to generate the said parametric ...
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Continuity of a special map between topological sets

Let $B_k \subset {[0,1]^{k+1}}$ and define the map: $$ \phi_k:B_k\mapsto C^k[0,1]:(\beta_0,\beta_1, \ldots,\beta_k)\mapsto\sum_{i=0}^k\beta_i b_{i,k}, $$ where $\{b_{i,k}(t)=\binom{k}{i}t^i(1-t)^{k-1}...
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55 views

Envelopes In Mathematics

How can I make sure that when we eliminate the parameter from the curve \begin{align*} F(t,x,y) &= 0 \\ \frac{\partial F}{\partial t}(t,x,y) &= 0\,, \end{align*} the equation obtained is the ...
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35 views

How does one find a parameter representation for bounded region?

I need help with this question. I have been stuck at it for a few days. My main problem is how I use the curve $K_r$ to find the parametric representation. I have a curve $K_r$ in the $(x,y)$-plane ...
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23 views

Supply and Demand Functions with Tax

I've been given the below supply and demand functions: $q^s(p)=50p~~~~~~q^d(p)=100(\frac{12}{p}-1)$ I've answered the first few questions, which include finding the equilibrium etc, and inverting ...