# Tagged Questions

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### Parametric equation of a circle given starting point.

Find the parametric equations of a circle with radius of $5$ where you start at point $(5,0)$ at $v=0$ and you travel clockwise with a period of $3$. So, I know that I require to have a $x(v)$ and ...
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### Parametric equation of a circle with given radius and starting point

Find the parametric equations of a circle with radius of $5$ where you start at point $(5,0)$ at $v = 0$ and you travel clockwise with a period of $3$. So, I know that I require to have a $x(v)$ ...
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### How can I find the projection of a sphere onto a direction vector subject to constraints?

Given a direction defined by a unit vector, I can find the projection of a point on that direction by using the dot product. The projection interval of a set of points is the max and min values of the ...
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### After removing the parameter from $x=\sec \theta$ and $y=\cos\theta$, why does the domain become $|x|\geq1, |y| \leq1$?

For the parametric equations $x=\sec \theta$ and $y=\cos\theta$ with initial domain $0\leq\theta\lt\frac{\pi}{2}$, $\frac{\pi}{2}\lt\theta\leq\pi$, I understand that you arrive at $y = \frac{1}{x}$ ...
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### Line integrals and parametrization

I've just learned about line integrals, and I need some help understanding an example problem in my textbook. The question is supposed to be really easy. Integrate $f(x,y,z)=x-3y+z$ over the line ...
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### Find the area of the circle

Find the area of the circle defined by the parametric equations $x = \cos t$ and $y = \sin t$. I know this is circle defined by $x^2 +y^2 =1$ so i used $0 < t < 2\pi$ as my bounds, then ...
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### Sketching Parametrizations - how to get something more understandable?

So I have some parametric functions (of one variable) I'm trying to sketch. Generally I can do so by "reverse parametrizing" where I take $x(t)$ and make $t$ a function of $x$ and then substituting ...
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### 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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### 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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### parametrization of plane in $\mathbb R^3$

Parametrize the plane in $\mathbb R^3$ with direction vectors $\hat u$ and $\hat v$ and through the point $p$ as in representation as the range of a $C^1$ function $f:\mathbb R^2\to\mathbb R^3$. ...
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### Arc length paramatrizations satisfy original system of differential equations?

Say we have a system of differential equations $$\begin{cases} x'''(t)+f(t)x'(t)=0\\ y'''(t)+f(t)y'(t)=0 \end{cases}$$ on an interval $[a,b]$, along with the restriction that $$x'(t)^2+y'(t)^2=1$$ ...
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### Parametric Equations: Find $\dfrac{\mathrm d^2y}{\mathrm dx^2}$.

Find $\dfrac{\mathrm d^2y}{\mathrm dx^2}$, as a function of $t$, for the given the parametric equations: \begin{align}x&=3-3\cos(t)\\y&=3+\cos^4(t)\end{align} ...
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### distance between parametric line and a point (4,3,s)

I've tried solving this problem every way I know how and I just can't get it. I've looked at similar problems of this type, and I still cannot get an answer that seems right. Parametric Equations: ...
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### area under a parametric curve problem

so I have a parametric curve, x = cos(t) y = sin(2t) I found that I need the area from 0 to pi/2. put this into an integral in terms of t I get $$-\int_0^{\pi/2}sin(2t)sin(t)dt$$ But in my ...
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### Extend a vector field of normal vectors beyond the surface

I am not terribly well-versed in differential geometry, so please keep that in mind when answering the question. We are given a surface in ${R}^3$ defined parametrically by $\vec{r}(u,v)$ where ...
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### Parametric Equation of a Hyperbolic Paraboloid

I need to make two trace plots of the hyperbolic paraboloid $z=x^2-y^2$. In the first plot, we set $z$ equal to a constant $k$, $z=k$. How do I find the parametric equation for this representation of ...
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### Trouble with caculus problem, parametric equations, I don't know what I'm doing wrong

When a mortar shell is ﬁred with an initial velocity of v0 ft/sec at an angle α above the horizontal, then its position after t seconds is given by the parametric equations $x = (v0 \cos \alpha)t$ , ...
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### Parallel Curves to a Parabola

I have been modelling parallel curves to a parabola and realise if the parallel curve to a parabola is offset enough then the curve will overlap. I came across this research paper to explain why a ...
Find parametric equations for the tangent line to the curve with the given parametric equations at the speciﬁed point. <1+2$\sqrt{t}$ ,$t^{3}$-1 , $t^{3}$+1 > at point P(3,0,2)
### Possible $x$ values for a parametric equation
$x=\sin t$ and $y=3\cos 2t$ over the interval $-\pi/2 \leq t \leq \pi/2$. I know how to eliminate $t$, but I was asked to determine the possible $x$ values for the parametric equation and the ...