Parametric vs. Cartesian: Graphing & Converting
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- Опубліковано 7 лют 2025
- How to graph parametric equaitons and convert it to cartesian
Parametric vs. Cartesian Full playlist : • Parametric Vs. Cartesian
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#blackpenredpen #math #calculus #apcalculus
What really helped parametrics “click” for me is just understanding that each coordinate direction is independent of the other; when you throw a ball, how high in the y direction (yes, really a, I know) is or far it goes in the x direction are independent of one another BUT both dependent on time (t). That’s the simple intuition that changed my life!
THANK YOU
blackpenredpen getting spooky with the bluepenredpen.
First!
second
It's a bit hard to understand, let me get help from my 11 year old friend, who studies calculus
#yay
blackpenredpen Can you please make a video showing how to calculate the circumference of an ellipse using integration of parametric form?
What is maximum acceleration possible in universe ?
Thanks to you my fascination of math is coming back to life, thank you man :')
You're very welcome. I am glad to hear!
This series of videos reminds me of my first class at uni :')
I like the direction this is going.
Omg the graph you made for y=2+sqrt(x+1) is so pretty it almost seems like it's made with cgi
As soon as you had drawn it I knew that the graph of y=2-sqrt(x+1) would look ugly and I was right
Great, but to avoid the 2 equations situation, I did t=y-1 then x(t)=x(y-1)=(y-2)^2-1 xD. So x=(y-1)*(y+3) if you want to find roots of x=0. If not, just x(y)=y^2-4y-3 :)
The implicit x=(y-2)² -1 is very convenient as you can sketch the curve in a couple of seconds.
9:27 just perfect
At 6:44 you say "take the square root of both sides."
Our math department has a long standing debate about the language we should use when "taking the square root" of both sides. (It get ls quite heated at times, honestly). This language often leads students to think sqrt (4) = +-2 when of course the principal square root of 4 is 2 only.
Instead of saying, "take the square root of both sides" in such instances we instead say something like, "this means that..." And then write the results, never placing square roots over each side of the equation.
So: x^2 = 4 means x = +- 2.
In your opinion are we making too big a deal over this subtlety or are we being appropriately careful math teachers?
John Zak Well, yes and no. Teach them what roots actually are (number of zero crossings). I didn’t learn that until I was 25 and it fundamentally changed math for me.
the intro!!
Him: “I’ll punch you in your face, do you wanna test me??!”
Also him right after: “Cause we can get the cops in here” 🤣
Your pen is blue. Your pen is blue! YOUR GODDAMN PEN IS BLUE!!
I know that it’s not related to the video but I’m wondering after watching 3blue1brown’s video. Why do common functions never use quaternions (like exponential or trigonometric functions) or why do proofs never need quaternion constructions?
Because we are working in a 2d space, not 3d
Jordan Saenz Quaternions are not 3d.
@@valatko yes they're 4d but they represent 3d rotations
@@valatko and octonions represent 4d rotations
Please do one where we are given the equation in cartesian coordinates and we have to parametrise it.
Please show us how to parametrize a line or any function.
The easiest way: for y=f(x),
Let x=t and y=f(t) Done!
For examples, y = 2+sqrt(x+1)
x=t
y=2+sqrt(t+1)
There are other solutions to it, depends on how complicated you would like.
blackpenredpen It makes a lot of sense. Thank you very much.
(½ - 1)^2 = 1?
I think he meant 1/2 multiplied by -2 (a.k.a -2/2). Probably should've put parentheses to clarify:
((1/2)(-2))^2=1
and ((1/3)(-3)^² = (-1)^² =1 @@kurashi3078
How do you parametrize an implicit curve in x and y?
make an example of what you'd like to parametrize
What is the functionality of parametric equations?
That Ragtime 🎹
Genial, las ecuaciones paramétricas son muy util, lo digo por geometria analitica.
Thanks for making cute videos .
I wish I never miss a cute video of yours teaching 😍
対数や微積をとる時の正負はいつもごにゃごにゃになりますん😊
x*e^x in parametrics?
x=ln(t)
y=t*ln(t)
...
why can't we just write x=y^2-4y+3
??
there is any technic for create a equation of a specific curve, for example, a curve that equals 0 in the intervale of 0 from 1 and after that becomes y=-x ? (sorry my english)
You'd just have to define the function as a piecewise function, or use Heaviside functions to show on what intervals the specific parts are to be nonzero.
@@CharlesPanigeo i untersdant, but i was thinking if there is other way so that i can make a unique equation for this function, for example, there is one video in this channel that shows a function with the heart shape tha are a simply equation
You're to complicated, you could just define t=y-1 and replace it. NOOOOOO, AM SUPPOUSED TO BE FIRST. SO OBVIOUS. UWU
But what is the kardashian form?
咱天朝就我一个人看这些视频吗？
Hello
1+1=2. This is why kindergarteners will never succeed in life if they do not know this.
Fifth
Well this problem was ridiculously easy. Boring
How I would do this is, I would first eliminate the t² from x by substracting by y²:
x - y² = (t² - 2t) - (t + 1)² = t² - 2t - t² - 2t - 1 = -4t - 1
Now we're still left with -4t which we can eliminate by adding 4y:
x - y² + 4y = (-4t - 1) + 4(t + 1) = -4t - 1 + 4t + 4 = 3
Now we have the answer in cartesian form:
x - y² + 4y = 3
You could get the same answer as in the video by just solving for y, or you could solve for y:
x = y² - 4y + 3
This is better way because it can be easily done mentally. The downside might be that this is not very insightful solution to the problem and can't always be used...
Or you do t=y-1 and replace in x=t^2-2t
Y=sin(t)
X=cos(t)
Now convert it into Cartesian coordinates!
y=sin(cos^(-1)(x))
Y^2 = sin^2(t)
X^2 = cos^2(t)
X^2 + Y^2 = cos^2(t) + sin^2(t)
X^2 + Y^2 = 1
That's the graph of a circle of radius 1.