1
vote
1answer
29 views

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 ...
-4
votes
0answers
40 views

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 $x(v)$ ...
0
votes
0answers
12 views

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 ...
0
votes
0answers
35 views

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 $y = \frac{1}{x}$ ...
1
vote
1answer
24 views

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 ...
0
votes
2answers
30 views

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 ...
0
votes
1answer
19 views

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 ...
0
votes
2answers
24 views

Lengths of Plane Curves - Calculus 2: $\sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2}$

$$ \sqrt{1-x^2} ; x=-\frac{1}{2} \to x=\frac{1}{2} $$ I am having problems setting this up. Taking the derivative of $\sqrt{1-x^2}$. Leaves me with: $$ \frac{1}{2}\left(1-x^2 ...
0
votes
2answers
19 views

Paramaterizing a path $C$ along a parabola $y=2x^2$

I am doing a line integral where the path $C$ is defined as the arc of the parabola $y=2x^2$ from the points $(-1,2)$ to $(2,8)$. Is there a "catch all" approach or method that can be applied here? ...
0
votes
2answers
35 views

Graphing a Parametric Equation

I need to graph and show the work for this problem. The graph needs to include arrows on the curve to show the direction of motion and I need to label the t-values graphed. $$c(t)=(2+4t, 3+2t)$$ So ...
0
votes
3answers
61 views

Finding the speed of a particle (parametric math)

I have to find the speed (as a function of $t$) of a particle whose position at time $t$ seconds is represtented by $$c(t)=(\sin t+t, \cos t+t)$$ How would I go about finding the maximum speed? ...
0
votes
1answer
30 views

Find the length of the curve

I need to find the length of the curve $$c(t)=(3e^{t}-3, 4e^{t}+7)$$ for $$0\le t \le 1$$ If I understand correctly, I need to take the derivative of the y part of that coordinate over the ...
5
votes
2answers
47 views

Find the parametric equation to the curve

Find the parametric equation for the curve. $$x^{2}+y^{2}=10$$ I haven't learned parametric equations fully yet, so I wanted to check with you guys and see if you can confirm if I'm doing this ...
0
votes
1answer
33 views

Express the parametric equation in form of y=f(x)

I need to express the parametric equation in the form of $y=f(x)$ by eliminating the parameter. I haven't learned how to do this yet, I've attempted to read a few pages though but they didn't help me ...
1
vote
0answers
40 views

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 ...
0
votes
1answer
35 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 ...
1
vote
1answer
40 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 ...
3
votes
1answer
51 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 ...
0
votes
2answers
18 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 ...
1
vote
2answers
86 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 ...
0
votes
2answers
82 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 ...
0
votes
1answer
50 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 ...
3
votes
1answer
54 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 ...
0
votes
1answer
23 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 $?<t<?$
1
vote
1answer
130 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
0
votes
0answers
266 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 ...
0
votes
1answer
47 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: $$x=x_0 + V_{0_x}*t$$ ...
0
votes
0answers
40 views

How to parametrise this surface integral

This is the question: $ S $ is the boundary of the region $ \{(x,y,z):0≤z≤h, a^2 ≤x^2+y^2 ≤b^2 \}$ where $ h,a,b$ are positive and $a<b$. ${\bf F(r) } = \exp(x^2+y^2){\bf r}$ where $ {\bf ...
2
votes
2answers
100 views

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 $f:\mathbb R^2\to\mathbb R^3$. ...
3
votes
2answers
161 views

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 $$ x'(t)^2+y'(t)^2=1 $$ ...
1
vote
1answer
105 views

Gradient of a rational Bezier curve

I'd appreciate help working out the gradient of a rational Bezier curve $C = (\,x(t) \,, \,C_y(t) \,)$. I know that the gradient $g$ of a the parametric curve is $$ g(t) = \left( \frac{dy(t)}{dt} ...
0
votes
2answers
96 views

Cartesian equation of $ \vec{r}(t)=\left(e^{\omega t}(\cos(\omega t)+\sin(\omega t)),2\omega e^{\omega t} \cos(\omega t)\right) $

I have this parametric equation: $$ \vec{r}(t)=\left(e^{\omega t}(\cos(\omega t)+\sin(\omega t)),2\omega e^{\omega t} \cos(\omega t)\right) $$ and I have to obtain the Cartesian equation. Any ...
1
vote
1answer
54 views

Finding the length of a parametric curve

$$x=\frac{t^2}{2} \text{ , } y=\frac{(2t+1)^{3/2}}{3} \text{ , } 0 \le t \le 20$$ The formula for the length of a parametric curve is $L=\int_{a}^{b}\sqrt{[f'(t)]^2+[g'(t)]^2}dt$. Taking the ...
0
votes
1answer
51 views

Direction of t (Vector Space)

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $$x = e^{-t}\cos t, y = e^{-t} \sin t, z = e^{-t}; (1, 0, 1). $$ The ...
1
vote
1answer
459 views

Find all points of intersection of the curves $r^2=3\sin(2\theta)$ and $r^2=3\cos(2\theta)$

Find all points of intersection of the curves $r^2=3\sin(2\theta)$ and $r^2=3\cos(2\theta)$. Give your answers as ordered pairs in cartesian coordinates, in order of increasing radius and ...
1
vote
0answers
64 views

Beth needs to make a crossing in her canoe

I have a math problem that has me stumped. I cannot seem to find a good starting point for this, and am flying blind with no check values. Maybe it's end of semester fog, but I'm struggling with ...
0
votes
1answer
43 views

Paremetric surface revolved around y-axis

if I'm finding the area of the surface generated by revolving the curve around the y-axis I use the equation $2\pi x\sqrt{(x')^2+(y')^2}$ and I'm given $$x=(2/3)t^{3/2}$$ $$y=2\sqrt{2}$$ and I got ...
0
votes
1answer
15 views

Proving that two function coordinates of a parametric curve equals 1

I am having difficulty with this question, Note: This is not homework, It is from a practice test that I am using to study Consider the curve: $x(t) = \frac{1-t^2}{1+t^2}$ ; $y(t) = ...
0
votes
1answer
133 views

Surface Area of a Parametric Curve

Set up an integral that represents the area of the surface obtained by rotating the given curve about the x-axis. Then use your calculator to find the surface area correct to four decimal places. $$ ...
0
votes
2answers
94 views

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: $$\begin{align}x&=3-3\cos(t)\\y&=3+\cos^4(t)\end{align}$$ ...
0
votes
1answer
403 views

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: ...
1
vote
1answer
71 views

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 ...
1
vote
2answers
81 views

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 ...
0
votes
1answer
317 views

Find equation of tangent line

Find the equation of the tangent line at parameter values $\theta=\pi/6$ and $\theta =5\pi/4$ to the cycloid given by $$x(t)=r\theta-r\sin \theta$$ and $$y(t)= r-r\cos \theta$$ with $\theta\in ...
1
vote
1answer
77 views

$\frac{dy/dt}{dx/dt} \text{ at } t = a \text{ or } \lim_{t \to a} \frac{dy/dt}{dx/dt} \text{?}$

Take an example of parametric equation: \begin{cases} x = t^3\\ y = t^6 \end{cases} Obviously the formula $\displaystyle \left. \frac{dy}{dx}=\frac{dy/dt}{dx/dt} \right.$ does not work at $t=0 ...
2
votes
0answers
227 views

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 ...
1
vote
3answers
100 views

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 $x = (v0 \cos \alpha)t$ , ...
1
vote
1answer
90 views

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 ...
0
votes
1answer
434 views

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.

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)
1
vote
2answers
47 views

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 ...