Questions tagged [parametric]
For questions about parametric equations, their application, equivalence to other equation types and definition.
1
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0answers
23 views
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.
$$\sum_{n=1}^{\infty} {n^{-\alpha}\cdot(\ln{n}...
0
votes
0answers
11 views
Spearman of Pearson in dataset with various distributions
I want to correlate several biomarkers with clinical parameters. Some are normally distributed (with or without log-transformation) and some are not. Can I use Pearson for the normally distributed ...
0
votes
1answer
23 views
Rolle's and Lagrange's theorems in parametric equations
I was trying to solve some calculus problems and I came across with some doubts related to two of them.
Given the parametric equation
$$
(x,y)=(3-3\cos^2 (t),3-3\cos(t)\sin(t)) \quad 4 \pi /3 < t ...
2
votes
0answers
21 views
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 ...
0
votes
0answers
7 views
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 ...
0
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0answers
8 views
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 ...
-1
votes
0answers
19 views
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/...
0
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2answers
19 views
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 ...
0
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5answers
22 views
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 ...
0
votes
2answers
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 ...
0
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0answers
17 views
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 ...
-1
votes
0answers
17 views
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 ...
1
vote
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 ...
0
votes
1answer
23 views
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.
...
1
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2answers
44 views
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 ...
0
votes
0answers
18 views
Eliminating parameter
How do you solve for "$t$" in either equation to eliminate the parameter and solve in terms of $x$ and $y$?
$$
\begin{split}
x&=\frac{ 1+t }{ \sqrt{1+t^2}}\\
\\
y &= \frac{1-t}{\sqrt{1+t^2}}
\...
0
votes
1answer
20 views
Parametrize the line in $\Bbb R^3$ that is determined by the intersection of the planes:
I am given two equations
\begin{alignat}{2}
x&+\phantom{0}y&\phantom{0}+z&=\phantom{0}4 \\
2x&+\phantom{0}y&\phantom{0}-2z\phantom{0}&=\phantom{0}4 \\
\end{alignat}
I am ...
1
vote
1answer
31 views
How to eliminate parameter of parametric equations?
If the parameter $t$ of $x$ and $y$ in a plane is given in the interval $(- ∞ , ∞)$ and if $x = t cos (t)$ and $y = t sin (t)$ How can one eliminate the parameter t and write a single equation using ...
0
votes
0answers
15 views
trying to plot a gear curve in gnuplot
Im looking at this gear curve: thinking wow I'd like to plot that!
so then I tried, in c++, with gnuplot.
...
0
votes
0answers
14 views
Using the Jacobian matrix to calculate a parametrized area?
$
x = \cos(\theta)r\\
y = \sin(\theta)r\\
z = r
$
I have done this parametrization, and now I want to integrate to get the area of a cone. According to my textbook, I'm supposed to multiply with the ...
0
votes
1answer
38 views
Integration, Area under curve
The question goes by : Find the finite region bounded by the curve
$$x=5t^2$$
$$y=2t^3$$
and the line $x = 5$. Find also the volume of the solid formed when this region is rotated through $\pi$ ...
-1
votes
0answers
9 views
Parametric functions
How can I plot a parametric function using Graphing Calculator 3D?
I am studing parametric equations and sometimes it would be very useful to plot this equations to help visualization, but I have no ...
1
vote
1answer
28 views
Matrixes with common parameters to result in no inverse
I've been given three matrices $A, B \ \& \ C$ which are defined as follows:
$$ A = {
\left[
\begin{array}{ccc}
b & 5 & 8 \\
c & 1 & 3 \\
a & 4 & 3 \\
\end{array}
\right]...
0
votes
0answers
21 views
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}\...
0
votes
0answers
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)...
1
vote
2answers
26 views
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 ...
0
votes
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 ...
0
votes
4answers
22 views
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 ...
0
votes
1answer
19 views
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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votes
2answers
20 views
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$ = $\...
1
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0answers
28 views
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 ...
0
votes
1answer
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$.
$...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
27 views
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) +...
0
votes
0answers
13 views
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 ...
0
votes
0answers
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 ...
1
vote
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:
$x^2+y^2=1$
$\cos^2(t)+\sin^2(t)=1$
$x^2+y^2=\cos^2(t)+\sin^2(t)$
$x^2=\cos^2(t)$
$...
1
vote
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 (...
0
votes
0answers
21 views
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.
0
votes
0answers
16 views
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 ...
0
votes
2answers
29 views
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 ...
2
votes
0answers
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 ...
2
votes
1answer
69 views
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 ...
0
votes
0answers
16 views
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 ...
1
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0answers
42 views
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 ...
0
votes
0answers
34 views
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}...
0
votes
0answers
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 ...
1
vote
0answers
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 ...
0
votes
0answers
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 ...
