0
votes
0answers
13 views

Partial derivative of straigh-line parametrized integral

I would like to evaluate the following $$ F(\mathbf{r}_1,\mathbf{r}_2) = \int_0^1 ds~f(\mathbf{r}_1 + (\mathbf{r}_2 - \mathbf{r}_1)s) $$ where $\mathbf{r}_{1/2} = (x_{1/2} , y_{1/2})$, i. e. a ...
0
votes
0answers
34 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 $y = \frac{1}{x}$ ...
0
votes
1answer
38 views

Prove the continuity and differentiability of parametric integration

$$F(\alpha )=\int_{0}^{+ \infty } \frac{\cos x}{1+(x+\alpha )^{2} } dx$$ Prove the function F is continuous and differentiable on the interval $[0, +\infty )$
0
votes
0answers
17 views

Parametrics Question Help Please [duplicate]

Would anyone be able to verify the answer of (iv) being $y=-a$? I assume since -a is a constant, the x value is irrelevant Thanks
0
votes
0answers
16 views

Calculate the convergence domain of parameter improper integral

$$\int^{+ \infty }_{1} x^{u} \frac{x + \sin x}{x - \sin x}dx$$ The answer is $u<-1$. I suppose we need to find the simplified equivalent form of it, but I stuck on my way.
0
votes
4answers
40 views

Showing that these two lines are parallel.

$$ \dfrac{x - 1}{2} = 2 - y = 5 - z \quad \text{and} \quad \dfrac{4 - x}{4} = \dfrac{3 + y}{2} = \dfrac{5 + z}{2}. $$ I was given this math problem as homework, and I have spent the past hour ...
0
votes
3answers
78 views

Find all values of $a$ for which there are two real solutions of $x^3-2ax^2+a^2x-3=0$

Find all values of $a$ for which there are two real solutions of the equation. $$x^3-2ax^2+a^2x-3=0$$ Ans = $1.5\sqrt[3]{6}$ I tried to research the function by dint of derivative, but it didn't ...
0
votes
3answers
33 views

Parameter “m” for which $P(x)=4(m+1)x^3+(m-3)x+1-m$ has a root with multiplicity two…

Can you please help me solve this parametric problem. So, we have to find all the values of real parameter $m$ for which the following equation has a solution with multiplicity ...
1
vote
1answer
24 views

How do I change parametric in t to cartesian when I can't re-arrage for t

I'm stuck looking at this parametric equation which I have to put in cartesian form $x=t^2+ \frac1t$, $y=t^2-\frac 1t$ Something to do with difference of two squares? I can't see how to ...
1
vote
3answers
38 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 ...
3
votes
2answers
48 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 ...
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
29 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 ...
0
votes
1answer
35 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 ...
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 $f:\mathbb R^2\to\mathbb R^3$. ...
1
vote
2answers
50 views

Find a parametrization of the intersection curve between surfaces

Find a parametrization of the intersection curve between the surfaces $−3x^2+2z=10$ and $4x^2+10y^2=5$. You should parametrize such that $y=k\sin(t)$ for some constant k. The answer should be in ...
2
votes
2answers
65 views

When does this parametric curve cross itself?

Find the points where the curve given parametrically by$$\mathbf{r}(t)=\left(2+\cos\frac{3}{2}t\right)\left(\begin{matrix}\cos t\\\sin t\end{matrix}\right)$$crosses itself. So, I understand that ...
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
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: $$\begin{align}x&=3-3\cos(t)\\y&=3+\cos^4(t)\end{align}$$ ...
4
votes
2answers
488 views

Parametric equations of cycloid on a Ramp

A small wheel of radius r is situated at the top of a ramp having an angle θ = π/3 rad as it appears in the figure below. At t = 0 the wheel is at rest and then it starts to rotate clockwise in the ...
2
votes
0answers
66 views

Parametric equation of cycloid on Ramp [closed]

A small wheel of radius r is situated at the top of a ramp having an angle θ = π/3 rad as it appears in the figure below. At t = 0 the wheel is at rest and then it starts to rotate clockwise in the ...
0
votes
1answer
372 views

Assignment: Find the number of parameters in the general solution to a system of linear equations

This is a question given in an assignment I'm working on: If the coefficient matrix $A$ in a homogeneous system of 33 equations with 28 unknowns is known to have rank 12, how many parameters are ...
0
votes
1answer
96 views

Parametric surfaces - Parameterization of torus

A rotational surface area is created when a curve in the $xz$-plane, with parameterization $\def\i{\pmb{i}}\def\k{\pmb k}$ $r=x(t)\i + z(t)\k$ , $t \in [a,b]$, rotates around the $z$-axis. This ...
0
votes
1answer
735 views

Parametrization for the curve $y = 7 - x^4$ that passes through the point $(0, 7, -3) $when t = 0 and is parallel to the xy-plane

Can you help me? So far I have turned $y = 7-x^4$ into $\langle1, 1, 0\rangle$ and used it to make the equation $L = (0, 7, -3) + t(1, 1, 0)$. I know this is wrong, but I just don't know what, and I ...
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 ...
2
votes
2answers
174 views

Find parametrizations for Circles and Ellipses

a) The portion of the circle $x^2 + y^2 = 4$ traversed clockwise from $(-2,0)$ to $(0,2)$ b) The part of the ellipse $(x^2)/(4) + (y^2)/(9) = 1$ that lies above the line $y = 0$, traversed clockwise. ...
2
votes
1answer
178 views

How to Parametrise a Parabola?

What is the method of find the parametric equations for all types of parabolas? And in both directions? So if I had 2 points: Parametrise from A to B Point $A = ( \frac{3}{\sqrt2} , 9 )$ and Point ...
2
votes
1answer
65 views

Parametric Equations (Basic) - Cartesian equation of curves

$x = 2 \cos t$, $y = 2 \sin t$, $0 \le t \le 2\pi$ Find the Cartesian equation of the curves. Please help i know it's basic but my problem is that $2 \cos t$ doesn't equal $1 - \sin^2 t$ and if it ...
4
votes
1answer
281 views

Using geometric arguments to solve an analysis problem

Im not good in geometric interpretations... any help is very welcome. Consider the unitary disc $$D=\{(x,y,0)\in\mathbb{R}^3, x^2+y^2\leq1\},$$ parameterized by ...
3
votes
2answers
72 views

Why do we need to find the intersection between these lines?

We have the functions $$ x = -1 + 2 \cos(t)$$ $$ y = 3 + 2 \sin(t)$$ They give P's orbit with $t$ on $\left[0, \dfrac{3}{2} \pi\right]$ Find (to 2 decimal places accurate) for which values of t ...
1
vote
0answers
84 views

Vector Tangent to Curve of Intersection

I am having problems solving this. Find a vector tangent to the curve of intersection of $z = 4x^2 + y^2$ and $z=(27-x^2-y^2)^{1/2}$ at the point $(1,1,5)$. I'm able to do this kind of thing using ...
1
vote
3answers
49 views

How to take parametric equations (x, y) to create a derivative formula?

I always thought that if I take the derivative of the y and x equation and divide y' by x', then that would be the derivative in formula form. Is this correct?
0
votes
0answers
71 views

parametric equation derivative question: can someone help me understand this question?

I am given $x$ and $y$ coordinates in parametric form with equations... $x(t)$ and $y(t)$. The questions asks to calculate $f'(x)$ for when $x = x(2\pi/5)$. Now am I first to calculate the ...
0
votes
1answer
55 views

Parametric problem: do these 2 comets collide. Am I solving this correctly?

$\text{comet1} = x_1(t), y_1(t)$ $\text{comet2} = x_2(t), y_2(t)$ set $x_1(t) = x_2(t)$ and solve for $t$. Since $t$ had a square, I had 2 possible values for $t$ ($t_1$ and $t_2$). substitute ...
4
votes
1answer
82 views

Parametric plots: Determine if 2 comets collide at a given time. Am I solving it correctly?

There are $2$ comets comet 1 $(x(t), y(t))$, comet 2 $(x_1(t), y_1(t))$ I need to determine if these two comets collide. From reading my steps below, is this the proper way to solve this? $1.$ set ...
2
votes
1answer
57 views

How to interpret this task?

I have a task given to me in my homework I can not figure out what asks of me. The task is worded like this: A curve in a plane is given by $$ x(t) = 3(t - \sin(t)) $$ $$ y(t) = 3(1 - \cos(t)) $$ ...
1
vote
1answer
134 views

Proper methods of solving parametric equations

I'm learning parametric equations in this section. Although I understand why the following works, I'm having difficulty understanding why the method employed for solving it is the correct one. I'm ...
0
votes
1answer
57 views

Line integrals of vector fields

Consider the vector field:$$\vec G = \left(\frac y{x^2+y^2}, \frac {-x}{x^2+y^2}\right)$$ compute $\int_\Gamma \vec G$ where $\Gamma$ is the proportion of a parabola $y=a(x-1)^2$ from (1,0) to (2,a). ...
1
vote
1answer
531 views

Representing A Plane Curve By A Vector Valued Function

I am given the function $x^2+y^2=25$, and I am suppose to write this as a vector valued function. I have always been awful at these sort of problems, even with parametric equations, which requires ...
1
vote
2answers
75 views

Parametrizing this curve

How can I parametrize the trajectory so that it is a smooth path $h:[-1,1]\rightarrow \mathbb{C}$? I think that I should use $$h=\left\{\begin{array}{ccl}t+i |t|&:&-1\leq t \leq 0\\ ? ...
2
votes
2answers
46 views

Parameterized curve describing trajectory of thrown object

We describe the trajectory of a thrown object (neglecting friction and similiar effects) with the curve $$k(t) = \left(v_0\cos(\beta)t,\,v_0\sin(\beta)t-\frac{g}{2}t^2\right)$$ with ...
1
vote
1answer
226 views

Arc Length Of Parametric Curve

I attached the problem as a file: Where did the trig functions go? I sifted through the different trig identities and formulas, but couldn't find anything that I could use. What should I do?
2
votes
2answers
2k views

Finding Where A Parametric Curve Intersects Itself

The problem I am working on is to find the where the curve intersects itself, using the parametric equations. These are: $x=t^2-t$ and $y=t^3-3t-1$ For the graph to intersect itself, there must be ...
2
votes
1answer
2k views

Finding Where A Parametric Curve Crosses Itself

The parametric functions I am dealing with are: $x=2\sin2t$ and $y=3\sin t$ I know for a parametric graph to cross itself, there must be two distinct $t$, $t_1$ and $t_2$, that when placed into the ...
1
vote
1answer
1k views

Finding Parametric Equations For A Rectangular Equation

I am trying to find a general way of finding parametric equations for a rectangular equation. The problem I am working on is $y=x^3$, and I have to find two examples of parametric equations. ...
1
vote
2answers
245 views

Parametric Equation Problem

The problem is, "to determine any differences between the curves of the parametric equations. Are the graphs the same? Are the orientations the same? Are the curves smooth? Explain." (a) $x=t;\quad ...
1
vote
1answer
330 views

converting a parametric R5 vector into a Cartesian form

How do you solve a problem like this. I'm completely stumped. it seems like there should be an easy solution but I'm obviously over looking it. any help would be greatly appreciated.
1
vote
1answer
432 views

Parameterize a straight line using polar coordinates… without angle.

I had to parameterize a straight line with starting point in $A=(-3,7)\\ $ and endpoint in $B=(4,1)$. My idea was to use the equation for the line that goes through two points. That is: $$ \frac { ...
3
votes
1answer
511 views

Summation of an Arithmetic, Parametric Sequence

I'm trying work on my ability to break complex patterns down, and in this case I'm trying to model the denominators of Lacsap's Fractions: I managed to get the sequence that represents the ...