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

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44 views

Relation between $\sin(t)$($\cos(t)$) and $\sin(at)$ ($\cos(at)$) when both are rational

This question relates to Parametric equations where sin(t) and cos(t) must be rational. Suppose it is given that $\cos(t)$ and $\sin(t)$ are both rational and also $\cos(at)$ and $\sin(at)$, where ...
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1answer
38 views

Calculating curvature of a curve on a the surface $x^2+y^2=1$. [closed]

Find a curve on the cylinder surface $x^2+y^2=1$ in $\mathbb R^3$ such that its curvature is equal to $\frac1{100}$ at each point of this curve. Does this easily generalize to different surfaces?
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2answers
70 views

Parametric equations where sin(t) and cos(t) must be rational

Suppose there are parametric equations $$ x(t) = at - h\sin(t) $$ $$ y(t) = a - h\cos(t) $$ and it is required that both $\sin(t)$ and $\cos(t)$ should be rational. What the values of $t$ should be ...
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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? ...
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1answer
29 views

What is the non-piecewise curve that resembles the following roller coaster track?

I want to create an animation about roller coaster. One track I want to use looks like the following figure. I am looking for the simplest non-piecewise parametric equation for both $x(t)$ and ...
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2answers
37 views

How to determine the coordinate of roller coaster's wheels?

I want to create an animation about roller coaster. For a simple track, for example, a circle, I can determine the position of the center of its wheel easily. However, for any parametric curve, I ...
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1answer
28 views

Help find the equation of two planes

I have the question Consider the line L through the distinct points A = (a,b,c) and D = (d,e,f) Find the equations of the two planes which intersect at right angles along L MY ATTEMPTED SOLUTION I ...
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1answer
42 views

When and why must we parameterise $f(x, y) = …$ with variables besides $x, y$?

For 10C, my choice of parameterisation $\mathbf{r} (x,y) = ( x, y, z(x, y))$ fails to effect the right answer, but that of user ellya does function. Yet for 9C, the parameterisation $\mathbf{r} (x,y) ...
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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 ...
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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? ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
27 views

Find a right angle triangle in with 3 vertices and one parameter

Given three coordinates, which could be $A=(7,3)$, $B=(2,4)$, $C=(k,-2)$ I want to find the values of $k$ that make a right angle diagram out of the three points. So I initially was thinking to find ...
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1answer
33 views

$2\pi^2(x-1)^2+4a\cos(2\pi x)-9a^3=0$ For which $a$ has only one solution…

For which values of real parameter $a$ the following equation has only one solution: $$2\pi^2(x-1)^2+4a\cos(2\pi x)-9a^3=0$$ Frankly I have no idea and I hope you'll give me some understandable hint ...
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0answers
61 views

Intersection between sphere and ellipsoid

I am failing since two days to compute and to plot the intersection of an ellipsoid in parametric notation ...
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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 ...
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2answers
26 views

Parametric inequation…

Supppose we have $a$ a real positive number that's not equal to $1$. Solve the following inequation: $$\log_a(x^2-3x)>\log_a(4x-x^2)$$ If it's known that $x=3.75$ is one solution of it.
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1answer
14 views

Find the values of parameter $a$ so that…

Determine all the values of real parameter $a$ so that the equation:$$(x-a)[log_4(x-5)-1]=0$$ admits a maximum number of real solutions. Thank you!
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2answers
18 views

Parametric Eqn / Differentiation

Parametric eqns of a curve are $x = t + \frac{1}{t}$ , $y = t - \frac{1}{t}$, where $t$ cannot be $0$. At point $P$ on curve, $t = 3$ and the tangent to curve at $P$ meets the $x$-axis at $Q$. The ...
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0answers
24 views

Find all the values of real parameter “n”…

Let $S$ be the set of real solutions for the following equation:$$\log_2(1-x-x^2)=n\log_{1-x-x^2}2+2$$ Determine all the values of real parameter $n$ for which $S\cap(0;{1\over2})\neq\emptyset$.
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0answers
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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 ...
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3answers
39 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 ...
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0answers
21 views

when is it not possible to eliminate parameter t from parametric equations?

We can eliminate parameter t from a set of parametric equations to convert to a Cartesian equation. My book mentions that it is sometimes impossible to do that. What would be an example of not being ...
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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 ...
2
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1answer
88 views

Rotation of conics sections using linear algebra

When given an equation of the form $$Ax^2+Bxy+Cy^2 + Dx + Ey + F$$ where $B \not= 0$ and it is not a degenerate conic, then you can use $\Delta = B^2 -4AC $ to see what type of conic it is, and then ...
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1answer
73 views

How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
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2answers
90 views

Curve on a basketball

The sewing pattern on a basketball is composed of two great circles and a single curve that intersects each great circle twice. Does this curve have a name? Are there any parametric descriptions of ...
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2answers
42 views

Find the point on an ellipse by angle.

How do I find the point on the ellipses at 45'. I found this, which answers part of it, but I need to know how to calculate for (x,y) at 45'. I could also use a good explanation for the ...
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1answer
41 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 ...
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1answer
102 views

Umbilic Points of an Ellipsoid

I have an ellipsoid given by $S = \{ (x,y,z): \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$, for some fixed $a,b,c \in \mathbb{R}^{+} \}$. I need to find the umbilic points of ...
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1answer
29 views

Checking a solution of a PDE

I have the following PDE: \begin{equation} -yu_x + xu_y = 0 \quad\text{where } u(0, y) = f(y) \end{equation} I derived a solution as follows: \begin{align} -yu_x + xu_y =& 0 \\ \iff& ...
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3answers
191 views

Equation of a torus

First I am a newbie in maths so please forgive me if I am not as rigorous as you would like, but do not hesitate to correct me. I want to find the equation of a torus (I mean the process, not just ...
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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 ...
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1answer
28 views

Evaluate $\iint_s\text{curl}\textbf F\cdot \textbf {n}dS$

Let $\textbf F=<xy,yz,zx>$ and $S$ be the upper half of the ellipsoid $\displaystyle \frac {x^2}{4}+\frac {y^2}{9}+z^2=1$. Evaluate $\iint_s\text {curl}\textbf F\cdot \textbf {n}dS$ I know the ...
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1answer
63 views

How to draw stick trefoil knot

http://commons.wikimedia.org/wiki/File:Stick_number_trefoil.png I am interested in plotting the stick trefoil knot. I don't know where to start. I am looking for equations or co-ordinates of ...
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1answer
39 views

Find all the intersection points of a vector parabola (in R3) and a sphere

Given that I have a vector in R3 (7t, 10t - 2t^2, 5t) | (These numbers are arbitrary for the sake of the process) A sphere centered at the point ( 15, 25, 10) with a radius of 20 There is a ...
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1answer
39 views

What is being represented by this 2 images?

image 1 image 2 It's possible that image 1 is showing some kind of methods for building polygons out of trigonometric functions ? It's also possible that image 2 is a quadratic bezier curve ?
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1answer
21 views

How can I reduce the interval of this parametric equation:$ x=t/(1+t^4) ; y=t^3/(1+t^4)$ to the simplest possible domain.

How can I reduce the interval of this parametric equation: $x=t/(1+t^4) ; y=t^3/(1+t^4)$ to the simplest possible domain? Can you explain this solution?
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1answer
28 views

Values of a limit with parameter

Evaluate the values of the limit as the changes of $\lambda \in \mathbb{R}$ $$\lim\limits_{x \to +∞} e^{x-x^3+\lambda\ln(x)}f^{(n)}x$$ with $$f^{(n)}(x)=p_n(x)e^{x^3-x}$$ where $p_n(x)$ is a generic ...
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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 ...
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2answers
27 views

Parametric Function for Coloring

I'm trying to write a parametric straightline function that changes its values between 0.529 and 0.933. Usually what I would do is: $r = 0.529+(0.933-0.529)*v$ where parameter $v= [0,1]$ This ...
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1answer
24 views

Parametrization Semicircle On Sphere

I need to find a parametrisation in terms of $t$ for a half circle on a sphere with radius $R$. The circle goes from $(R,0,0)$ to $(-R,0,0)$ and is going through the point $(0, ...
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1answer
25 views

Parametric equation for a curve $C_1$ in $\mathbb{R}^3$ such that the angle of its tangent $T$ and the $Y axis$ equals the angle of $T$ and a vector

I know that to find the angle between two curves in $\mathbb{R}^3$ at their intersection I differentiate both curves and evaluate at the intersection to find the slope of each tangent line and then do ...
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1answer
79 views

Find the area of the surface obtained by rotating the curve of parametric equations

Rotate about the $x$ axis $x = 2t-2/3t^3$ $y = 2t^2$ $0 \leq t \leq 1$ I did the integral of $\sqrt{(2-2t^2)^2+(4t)^2}$ and got $(2x(x^2+3))/3$ and then I did the integral of $2\pi 2t^2 ...
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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 ...
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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 ...
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2answers
43 views

Is it possible that $f_1,f_2,\dots,f_n$ are not all differentiable, but $\alpha:I\to \Bbb{R^n}$ is differentiable?

Consider the parametrized curve $\alpha:I\to \Bbb{R^n}$. These notes say that $f_1,f_2,\dots f_n$ being differentiable $\implies$ $\alpha$ is differentiable. I wonder why the converse is not true. Is ...
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2answers
85 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 ...