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amoodarya
 one year ago
I want to make short tutorial for parametric equation (not advanced )
amoodarya
 one year ago
I want to make short tutorial for parametric equation (not advanced )

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amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17In mathematics, parametric equations of a curve express the coordinates of the points of the curve as functions of a variable, called a parameter for example : x=t+1 y=t2 is a line equation if we want to find explicit equation first we find "t" from x or y and put it to other t=x1 y=(t)2=(x1)2 so y=x3

Kash_TheSmartGuy
 one year ago
Best ResponseYou've already chosen the best response.0Cool, so make a tutorial!

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17example 2: find explicit equation \[x=t1\\y=t^2+t+2\\ \rightarrow x=t1 \rightarrow \\t=x+1\\ \rightarrow y=(x+1)^2+(x+1)+2\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17example 3:find explicit equation x=sint +1 y= cos t 2 we know \[\sin^2t+\cos^2t=1\] so \[x=sint +1\rightarrow sint =x1\\y=cost2\rightarrow \cot =y+2\\sin^2t+\cos^2t=1\\(x1)^2+(y+2)^2=1\\\] it is a circle

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17example 4:find explicit equation \[x=\sqrt{t}\\y=2t1\\\] note that \[t \geq 0\] \[x=\sqrt{t} \rightarrow t=x^2\\y=2t1\\ \rightarrow y=2(x^2)1\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17example 5:\[x=2cost \\y=3sint \\\]find explicit equation \[x=2cost \rightarrow cost =\frac{x}{2}\\y=3sint \rightarrow sint=\frac{y}{3}\\weknow \\sin^2t+\cos^2t=1\\so\\(\frac{x}{2})^2+(\frac{y}{3})^2=1\\\frac{x^2}{4}+\frac{y^2}{9}=1\] it is an ellipse

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17example 6: x=cost +sint y=3 sint find explicit equation \[x=cost +sint\\ y=3 sint \rightarrow sint =\frac{y}{3} \\ \rightarrow x=cost +sint =cost +\frac{y}{3} \\x\frac{y}{3} =cost \\put into \\sin^2t+\cos^2t=1\rightarrow \\(\frac{y}{3})^2+(x\frac{y}{3})^2=1\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17example 7: x=cost +sint y=cost sint find explicit equation \[x=cost +sint \\ y=cost sint \\ \rightarrow \\x^2=\cos^2t+\sin^2t+2sint cost\\y^2=\cos^2t+\sin^2t2sint cost\] if we find sum of them it will be a circle \[x^2=\cos^2t+\sin^2t+2sint cost \rightarrow x^2=1+2sint cost\\y^2=\cos^2t+\sin^2t2sint cost \rightarrow y^2=x^2=12sint cost\\ \rightarrow \\x^2+y^2=1+2sint cost+12sint cost=2 \\ \rightarrow x^2+y^2=2\] \[radius=\sqrt{2} ,center=(0,0)\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17example 8 : x=cost y=cos (2t) find explicit equation note : you have to find a relation between cos t, cos 2t \[x=\cos t\\y=\cos2t\\ \left\{ \cos2t=2\cos^2t1 \right\}\rightarrow \\y=\cos2t=2\cos^2t1 \\y=2x^21\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17example 9: \[x=t+\frac{1}{t}\\y=t\frac{1}{t}\] find explicit equation 2 method will show first use \[(a+b)^2+(ab)^2=2(a^2+b^2)\\(a+b)^2(ab)^2=4ab\] \[x=t+\frac{1}{t} \rightarrow x^2=t^2+(\frac{1}{t})^2+2t \frac{1}{t}=t^2+(\frac{1}{t})^2+2\\y=t\frac{1}{t} \rightarrow y^2=t^2+(\frac{1}{t})^22t \frac{1}{t}=t^2+(\frac{1}{t})^22\\find \\x^2y^2\\x^2y^2=(t^2+(\frac{1}{t})^2+2)(t^2+(\frac{1}{t})^22)\\ \rightarrow x^2y^2=4\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17second method : it easy to find "t" \[x=t+\frac{1}{t}\\y=t\frac{1}{t}\\x+y=t+\frac{1}{t}+t\frac{1}{t}=2t\\t=\frac{x+y}{2}\] then put t in 1st or 2nd equation

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17example 10: \[x=2tant+3 \\y=3\cot t 5\\\] find explicit equation we have tan , cot and we know tan t * cot t=1 so \[x=2tant+3 \rightarrow \tan t=\frac{x3}{2}\\y=3\cot t 5 \rightarrow \cot t =\frac{y+5}{3}\\ \rightarrow \tan t \times \cot t=1\\\frac{x3}{2} \times \frac{y+5}{3}=1\\y+5=\frac{6}{x3}\\y=\frac{6}{x3}+5=\frac{6+5x15}{x3}=\frac{5x9}{x3}\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17example 11: \[x=a_0+b_0t\\y=a_1+b_1t\] find explicit equation it is a line equation :

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17\[if \\b_0 \neq 0 \\t=\frac{xa_0}{b_0} \\ \rightarrow y=a_1+b_1t\\y=a_1 +b_1(\frac{xa_0}{b_0})\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17Now draw parametric curve if we do not eliminate the parameter we can put some value for parameter and find x, y as a point of (x,y) then with some point draw the curve for example \[x=\frac{t}{2}\\y=t+1\\t=0 \rightarrow x=0 ,y=1 \rightarrow (x,y)=(0,1)\\t=2 \rightarrow x=1 ,y=3 \rightarrow (x,y)=(2,3)\\...\\\] they are polynomial in degree 1 : so this is a line equation dw:1436394739768:dw

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17we need to draw \[x=\sqrt{t}\\y=t+1\] put some t into x,y equations \[x=\sqrt{t}\\y=t+1\\t \ge 0\\t=0 \rightarrow x=0 , y=1 \rightarrow point=(0,1)\\t=1 \rightarrow x=1 , y=2 \rightarrow point=(1,2)\\t=4 \rightarrow x=2 , y=5 \rightarrow point=(2,5)\\t=9 \rightarrow x=3 , y=10 \rightarrow point=(3,10)\] dw:1436395050876:dw note that \[t \geq 0 \rightarrow x \geq 0\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17Now : Parametric derivative Parametric derivative is a derivative in calculus that is taken when both the x and y variables (traditionally independent and dependent, respectively) depend on an independent third variable t, usually thought of as "time". for example : \[x=2t+cost\\y=t^2+sint+5\\\frac{dy}{dx}=?\] \[\frac{dy}{dx}=\frac{dy}{dt}*\frac{dt}{dx}=\\\frac{dy}{dt}*\frac{1}{\frac{dx}{dt}}=\\\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{y'_t}{x'_t}\\\] so ,in this case \[y'_x=\frac{dy}{dx}=\frac{y'_t}{x'_t}=\frac{(t^2+sint+5)'_t}{(2t+cost)'_t}=\frac{2t+cost}{2sint}\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17@Donya2222 @dolloway97 @Donblue @Mehek14 @Mindblast3r @Marcorie @Moe95 @JFraser @superhelp101 @slade @sammixboo @Destinyyyy @Deeezzzz @Nicoleegilmoree @aaronq @AeroSmith @AG23 @Elsa213 @wil476003 @Rubyblades @razor99 @raggedy_roo @ritesh_asu @Rubyblades @Tinkerbell2001 @bruno102 @bunny256 @vera_ewing @Vocaloid @KAKES1967 @Hero @hendersonjen02 @iloveyou;* @IrishBoy123

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17we are in ,to help you

Empty
 one year ago
Best ResponseYou've already chosen the best response.1I was wondering if you've heard of the Leminscate curve: \[(x^2+y^2)^2=2a^2(x^2y^2)\] Can we parametrize this?

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17a simple Leminscate curve: \[x=\frac{a \cos t}{1+\sin^2t}\\y=\frac{asint \cos t }{1+\sin^2 t}\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17Leminscate curve equation is nice in polar coordinate "Empty"

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17@YanaSidlinskiy @yomamabf @yingpeng @yacobtewolde @uybuyvf @uybuyvf @otaylor19 @pooja195 @peachpi @pattycake1 @pinkros @AbdullahM @xavierbo2 @Xaze @xamr @xitsaliciaa @calculusxy @valiant1 @brebre5564 @EmilyD22 @esam2 @Emilyf29 @radar @GeniousCreation @Ghostedly @gabgurl @GloGangg_Jayy @LegendarySadist @lizajune @lofi @lorrainetocuteherrera

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17I think ,maybe useful not for promoting I beg you ,if bother you

YanaSidlinskiy
 one year ago
Best ResponseYou've already chosen the best response.0*ahem* Mass tagging?;) Never knew anything about this. To me, this is like a whole new Italian language.

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17\[x=t\\y=2t1\\0 \leq t \leq 2\\\] draw it note about domain of "t" \[x=t\\y=2t1\\0 \leq t \leq 2\\ \rightarrow \left\{ (x,y):x=t,y=2t1,0 \leq t \leq 2 \right\}\\0 \leq x \leq 2\\0 \leq t \leq 2 \rightarrow 0 \leq 2t \leq 4 \rightarrow 01 \leq 2t1 \leq 41\\ 1\le y \le 3\] it is a line ,but restricted \[0 \leq t \leq 2\\t=0 \rightarrow x=t=0 ,y=2t1=1 \rightarrow (0,1)\\t=2 \rightarrow x=t=2 ,y=2t1=1 \rightarrow (2,3)\] dw:1436398201172:dw

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17Trick to solve ,system of equation like this \[\frac{x}{2}=\frac{y1}{3}=\frac{z}{5}\\xy+z=11\] we can use the parameter to show all of variable by ,one variable \[\frac{ x }{ 2 }=\frac{ y1 }{ 3 }=\frac{ z }{ 5 }=t\\x=2t\\y=3t+1\\z=5t\\\] now put them in xy+z=11 \[x=2t\\y=3t+1\\z=5t\\xy+z=11\rightarrow \\(2t)(3t+1)+(5t)=11\\4t=12\\t=3\\\] now we can easily find x,y,z \[x=2t \rightarrow x=2(3)=6\\y=3t+1\rightarrow y=9+1=10\\z=5t \rightarrow z=15\\\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17some parametric curve Cycloid\[x=r(t\sin t)\\y=r(1cost)\] The cycloid represents the following situation. Consider a wheel of radius r. Let the point where the wheel touches the ground initially be called P. Then start rolling the wheel to the right. As the wheel rolls to the right trace out the path of the point P. The path that the point P traces out is called a cycloid and is given by the equations above. In these equations we can think of θ as the angle through which the point P has rotated.

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17Here is a cycloid sketched out with the wheel shown at various places. The blue dot is the point P on the wheel that we’re using to trace out the curve.

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17Circle A more sophisticated example is the following. Consider the unit circle which is described by the ordinary (Cartesian) equation \[x^2+y^2=1\] This equation can be parameterized as follows \[ (\cos(t),\; \sin(t))\quad\mathrm{for}\ 0\leq t < 2\pi.\, \] With the Cartesian equation it is easier to check whether a point lies on the circle or not. With the parametric version it is easier to obtain points on a plot. In some contexts, parametric equations involving only rational functions (that is fractions of two polynomials) are preferred, if they exist. In the case of the circle, such a rational parameterization is \[x=\frac{1t^2}{1+t^2}\\y=\frac{2t}{1+t^2}\] or \[x=\pm \sqrt{1t^2}\\y=t\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17ellipse \[x=a sint \\y=b cost \\a \neq b \neq 0 \\(\frac{x}{a})^2+(\frac{y}{b})^2=\sin^2 t+\cos^t=1\] or \[ x=a(\frac{2t}{1+t^2})\\ y=b(\frac{1t^2}{1+t^2})\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17Parabola The simplest equation for a parabola \[y=x^2\] can be parameterized by using a free parameter t, and setting \[x=t\\y=t^2\]

amoodarya
 one year ago
Best ResponseYou've already chosen the best response.17dw:1436438108955:dw note that \[t \geq 0\]
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