I integrated the quadratic formula
ฝัง
- เผยแพร่เมื่อ 2 มิ.ย. 2024
- Ultimate math for fun! Last year we differentiated the quadratic formula, so today we will integrate the quadratic formula! This is my first time integrating the quadratic formula and I was surprised that the answer turned out to be pretty nice! This is a super fun problem to challenge any Calculus 2 student. In the end, we will check if ChatGPT can integrate the quadratic formula for us.
Check out how to differentiate the quadratic formula: • I differentiated the q...
0:00 We will integrate the quadratic formula
7:48 ChatGPT vs integral of the quadratic formula
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Thank you all!
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Check out how to differentiate the quadratic formula: th-cam.com/video/JEcE-wDRMCk/w-d-xo.htmlsi=Uq1p1KYNj-Kpu5g2
Integrating the quadratic equation = god mode! You sir are a badass!
Well everything has physical meaning so what does it physically mean?
@@abishworpandit5200 🤔
dont rely on AI and dont ask AI to any math question.. it is very dumb in that area and still thinks that it knows the answer.. but in near future it will eventaully learn to do math and most likely beats any human being on earth :) wolfram is around for a long time and it is best tool to check results or solve something for the timebeing.
If there’s one thing that ChatGPT tells us it’s that AI won’t be replacing math professors anytime soon.
Well that may not be true in the future....
@@aashsyed1277won't be soon 😅
there's already multiple AI's online that can integrate any integratable function and give you the full meticulous steps idk if you count that as replacing math professors.
@@acuriousmind6217 a math professor does a lot more than integrating functions, they deal with a lot more abstract concepts that don't have a set way to do them. Until those AI's can understand abstract concepts, math professors won't be going anywhere
ChatGPT write me a witty youtube comment reply.
If you change the last integral (1/(u^2-b^2)) with diff of two squares you can split it up with partial fractions and use natural log… although it turns out the end result is actually an identity for inverse hyperbolic tan
They aren’t entirely the same because of the domain difference.
tanh^-1 (x) is only valid for -1
@Ninja20704 if x>1 or x
w pfp
@@Ninja20704bro can you explain why sometimes we get two different answers for the same integrant?
How to decide which one is correct?
@@mrfarooqkhan8454 Usually it is because the two different answers are actually just off by a constant, so they are essentially the same as far as the indefinite integral is concerned. One example is secx*tanx.
In this case, the different answers we are getting is more of domain, but the functions have the same value for all x in the common domain.
The integral of 1/(1-x^2) has 3 seemingly different answers: tanh^-1 (x), coth^-1 (x) and ln|(1-x)/(1+x)|
tanh^-1 (x) = 1/2*ln[(1-x)/(1+x)], which is only good for -1
I love the grin at 6:03 with the inverse hyperbolic tangent. Math is fun and your subtle humour like this makes it fun.
😅
There is absolutely no reason for me to have watched this, I'm not in -or ever going to take- any calculus class where I have to integrate the quadratic formula. Yet here I am, loving every second of this video :)
Thank you!
Bro, I just graduated from high school but I can't ignore your video titles... Great as USUAL!
Thanks!!
A much more interesting approach is differentiating each quadratic formula with respect to each variable (a, b, c) and observing how each variable contributes to the fluctuation and the identification of the sweet spot for every variable that results in a more diverse set of solutions.
Sure, a lot of cool things could be done with this. Take the solution set as a function with 3 variables, include complex numbers if you like, and look for all sorts of patterns.
I'm thinking of stuff like maximizing the distance between 2 solutions for all vectors (a,b,c) of fixed length and shit like that. It's a heavily unexplored topic!
5:28 You can also use partial fractions for this integration u^2 - b^2 = (u - b)(u + b)
1/(u - b)(u + b) = 1/2b(u - b) - 1/2b(u + b)
And that way you don’t end up with inverse hyperbolic trig
I was gonna write sameee
Nerd
@@perfectman3077 stupid Sigma 10 year old kid , your place is not here go somewhere else dude 🤡
If you want to learn something than learn, if don't than get out ..
so u will get 1=A(u-b) +B(u+b)
so you cant find the value of A and B since you dont know neither the value of u nor b
@@joyboy69lffy My bad. You can still use partial fractions since b is constant with respect to the integration but you need to divide by b. I updated my answer so it should be right now.
The last part reminded me of when I asked 2 different AI models (not ChatGPT) about whether they can prove Euler's Sum of Powers Conjecture . . . both of them had factual inaccuracies in their responses and one of them even claimed that it's currently unknown
its the same as asking google a long question... at the end of the day its trained on pattern recognition you have to de-personify the ai so to speak when approaching it with complicated questions
All they are is token prediction systems, just a more advanced version of the predictive text in your phone. They can't "do" math or anything so no, that's not surprising
Lol
Because only GPT 4 is AGI. Above human level.
Yeah ai is dogshit at medium hard or slightly complex
Its been 30 years since I took this in college and I can still fall asleep halfway through.
Who said you would forget how to do the things you learned to do in your freshman calculus course?
i love how we can just see all of those dry erase markers on standby in the bottom right. video’s fantastic as always, keep up the great work!
Great stuff, bprp! I love your videos. You teach but you do not talk down to the audience and you bring a catchy sense of humor and fun. Here's to your continued success!
Thank you very much for your nice comment!
Its a fact that ChatGPT has cracked the toughest exams but does dumbest mistakes in new type of questions....
Those exams aren’t usually logical ones, that and it will have many resources that allow it to “cheat” on those exams by pulling data from sites that have very identical questions/answers
Just ask it to write a python program to symbolically integrate.
Me holding a stack of answers to every exam ever published online:
It's a fact that I can answer the toughest exam questions, but am unable to apply any reasoning skills to "new types of questions".
@@elquesohombre9931Yep, I’d do pretty good on a lot of exams too if I was able to copy and paste answers from the internet
Given it is only trained on language, that it even attempts to work out the answer is impressive.
Love the video. Easy to follow; well explained. I also like how you show the technology failure at the end.
Teacher: Maybe there is a mistake.
Please verify the answer by differentiate the answer.
Really enjoyed this one !
Glad to hear. Thanks!!
Math Professors are the real JOD people on Earth....
😂😂😂 the last part of your amazing video was amazing. Wow...
Nice Explanation Sir...
Is there a physical interpretation to integrating the quadratic formula with respect to the polynomial? Does it tell us anything special?
If we were to create a graph with an a axis and an x axis, where x is vertical (where y usually is) and a is horizontal (where x usually is), this would tell us the area under the curve of the graph. x = (-b+sqrt(b^2-4ac))/2a in terms of b and c which are constants. That’s really it, it’s not too special. If you wanted to see what this graph would look like you could go on Desmos and substitute x for y and a for x, and then create sliders for B and C.
@@spearmintlatios9047 Well that it can be viewed as an an area under a curve is one interpretation, however this is not what I am asking. He is doing something much more fun here, see, the quadratic formula is a function in polynomial of 2 degree space which has 3 degrees of freedom, a,b,c. It therefore maps a unique polynomial represented by the point (a,b,c) in the space and maps it onto two values which are it's roots.
Now, when he integrates the quadratic formula, with respect to a, this is the question on how it changes the mapping and what does it tell us now. I am 99% sure there is quite a bit more going on here.
@@ivanrodionov9724I kinda get what you’re saying but I’m rusty on calc 3 stuff tbh.
If you want, you can represent the quadratic equation as a graph mapping the function of a, b, and c to a value of x. or f(a, b, c)=x. this would be a 4 dimensional graph which is somewhat beyond our imagination, but everything should still apply.
Since we only integrated with respect to A here, B and C are still able to be changed within the integral. pretend that B is equal to 1 for now. Then we can have a 3D graph of f(a, c) = x. In this case, we have a the a and c axis representing our domain, and an x axis representing our range.
So for any point on the (a, c) plane, the output of this function is some “height” in the x axis creating the point (a, c, x).
If this is the case, and we assume B is constant, then this integral represents the area summation of the height from the x output to the flat ac plane from a given start and end point a0 and a1, So it would still just be a 2D area found from an infinitesimal slice of a 3D graph.
It is cool that we could move this slice by changing our bounds of a0, a1, and C. You could imagine changing these bounds as moving and stretching a tall piece of paper through the 3D space of the function, and calculating the area of that paper if you slice off all negative values and everything above the x points.
To calculate the volume of the shape this graph makes, we would have to integrate by another bounds as well, a double integral.
But the same logic works as well when we include B again. The integral would simply be a slice in a 4D shape that has X as the output.
Unfortunately, we can’t just set X to 0 and have a graphical interpretation of 0 = integral (g(a, b, c))da. I don’t think there actually any representation there? If we wanted to find an integral that works here we would probably have to move some terms around before hand and treat it as a differential equation. Correct me if I’m wrong
@@ivanrodionov9724 after reading what you’re saying again I see you’re talking about something entirely different. Is there any tool that could create a mapping similar to what you’re talking about? I think what I was discussing might relate when you consider he only integrated using a plus and not a minus
@@spearmintlatios9047any applications in Engineering could you think of? Aerodynamics maybe
that trick of adding in a -b and a +b blew my mind. it makes perfect sense in retrospect, -b+b=0 so if you have something that is also something + 0 and thus you can put anything in so long as it cancels out to 0...
Yes 😃
Fantastic video, thank you.
Very good lesson, thanks.
If you want ChatGPT to do advanced stuff you tell it to write a python program that does what you want like plot a function, do a symbolic integration or even get it to create a rotating globe in OpenGL. I've done all of these things with it and more... a subway station mapper using google maps API and TKinter etc. Some, like the maps API require a bit if tweaking but not the integrator.
"I integrated the quadratic formula" has the same energy as "I sawed this boat in half!".
And since you differentiated this already, "I sawed ANOTHER boat in half!"
Its giving the same energy as that flex tape ad where he cuts a boat in half and then tapes it back together: th-cam.com/video/0xzN6FM5x_E/w-d-xo.html
Beautiful, but can we find a geometric interpretation of this integral? Or at least use it to gain some insights on the regions where the determinant is positive or negative as a function of the first coefficient of the trinome?
Maybe f(a)=all the formula
Someone correct me if I'm wrong, but utilizing the intuition of the mean value theorem I think this means: if you vary a from some value a0 to a1, then the mean value of x which solves ax^2+bx+c=0 will be [f(a1)-f(a0)]/[a1-a0], where f(a)=-(b/2)ln|a|+sqrt(b^2-4ac)-(1/b)tanh^-1(sqrt((b^2-4ac)/b)* . I tried running it on desmos, I'm tired so I might have did this wrong, but here's an example of what I mean: from a=1 to a=2, the mean value of one of the roots of ax^2+3x+5=0 is -2.66324ish. I did have to throw in a negative sign at the end to make it all work, which means I'm made some small error along the way, but visual check of a few examples seem to confirm that it does work.
* Important caveat, this will be the average value of one of the roots as you very a, the other one would have a slightly different definition of f(x) for the choice of the negative root of b^2-4ac, and thus give you a different average value corresponding to that root. @blackpenredpen ?
Last part of integration with u^2-b^2 can be done with writing it as( u+b)(u-b) and then matching the numerator to get 1/2b^2 ln(u-b)/(u+b)
Thanks so much, it was fun!
Muchas gracias. Muy bonita esta integral.
So definitely an academically interesting exercise. Is there any insight about quadratics and their solutions that can be learned from this result?
No!
You are doing good things !!
Would love to see triple integration with respect to all variables there!
Legit the first question i asked from that title was "with respect to what?"
Nice solution - great👍
Thank you!
I am always fan of this great person, brilliant brain
@@English_shahriar1
Thanks
It would be interesting to compare the integrals of da, db, and dc for the same curve parameters to see what happens.
6:09 If u is in interval (-1,1)
but if u is outside this interval you have b/u as argument of inverse hyperbolic tangent
I’m looking forward to learning hyperbolic tangent!
Last part was very meaningful 😂😂❤
This is wild, man!
I literally just watched the differentiating video of the quadratic formula yesterday
That was fun to watch.
the sequel we needed
amo suas questões 🫶 um abraço do Brasil
I think you can also expand the denominator of the last integral into partial fractions and then express the result as a difference of two logarithms
Another one that's fun is the limit as a goes to 0, It gives an interesting result that makes sense when you think about it. If you really want a challenge you could do the limit of the cubic formula as the x^3 coefficient approaches 0 to see if you can get the quadratic formula
I did that like 2 month ago and thought it was very nice how the limit works out at the end. I don't know what you mean with the coefficient of x^3, what equation will you apply the limit to?
@@euler1 there is a cubic formula but it is kind if complicated
@@euler1 The solutions to ax^3+bx^2+cx+d=0 is given by the following: let u=b^2-3ac, let v=2b^3-9abc+27a^2d, let w=((v+-(v^2-4u^3)^(1/2))/2)^(1/3)
where we can use any root for each cube or square root. Then x=(-1/(3a))(b+w+u/w) gives the roots
@@adrienanderson7439
It's the first I see this thanks for the explanation, I am curious to try it out but I think I will just look up a proof for it as I don't have that much time xd
@@euler1 Yeah honestly I haven't done it myself and I tried to today but it really gets into the weeds
Last segment is a perfect example of ChatGPT just making stuff up lmao
Applied maths: Useful things that would probably come handy in life
Pure maths:
you know what, im startin to like this guy
I love your t-shirt.
Im glad I woke up at 3am to enjoy this 😂
The ending had my rolling.
for the inverse hyperbolic tan part,you've better to go with a partial fraction decoposition ,since we don't now the domain of integration , it's ok to go with the inverse hyperbolic tan only if a is from (-1,1)
Instead of using inverse hyperbolic function as the integral of 1\(u^2-b^2), you may also use ln{(u-b)/(u+b)}/2b. That's how GPT might have also got the answer in terms of natural log
Came for blackpenredpen got blackpenbluepenredpen what a day to be alive
You can also use trig sub to avoid hyperbolic tan
ive figured out a long time (by curiosity) dat chatgpt absolutely sucks with math problems. it can fail in something as simple as a quadratic equation, let alone a full integral
i retried asking the integral to gpt couple times and with some hand it eventually integrated the formual exactly as in the video eventually
Hi Dr. Pen!
Very cool!
thanks!
Brilliant !
can we multiply both the num and denm with sqrt(b² - 4ac) when we split the integral -b/2a da +... and see what will happen?
I don't understand a lot but I'm getting interested again to learn the basics of Calculus
Very charming, thanks 😃😃😃😃😃😃😃
We got the integral but can you explain the solution graphically as integration with limits gives area and what would the area of this graph be
It's been a while since I last saw one of your videos
Wouldn't it be better to do trigonometric substitution in the 2nd integral instead of substituting the root as an algebraic variable?
Thats hectic! Love the definitive proof at thr very end that chatgpt clearly has discalculia 😂
Just ask it to write a symbolic integrator in Python.
Have you ever tried differentiating/integrating quadratic formula?
Yes, basically as soon as I learned how to do basic derivatives from limits and noticing that circle/sphere calculations look like they have been integrated/differentiated using "power rule" (I think that's what it's called). 2*pi*r, pi*r^2 and 4/3 pi r^3 looked too much like coincidence. So I looked at quadratic formula (I also wondered back then what would happen if I did look for zeros when "delta" is "negative" - of course without imaginary numbers it is "harder" to do. Still doable in some cases and if you don't mind going a bit outside real numbers for rules). Sadly my knowledge was insufficient - had I tried it when I was more proficient it would likely help.
I'm just amazed you found 3 working whiteboard markers.
very interesting video, blackpenredpenbluepen! :)
Thank you!!
I got another (perhaps complicated solution: Suppose the integrand is x, where y=ax^2+bx+c. Then dy/dx=2ax+b, and dy/da=x^2. Hence da/dx=(2ax+b)/x^2. Then integral=int[(x)(2ax+b)/x^2]=int[2a+b/x]=2ax+b lnx +C
5:39 but the integrals of the form dx/(x²-a²)= 1/2a*ln(x-a)/(x+a) where ln is natural log
Excellent
The last integral
dU÷U²-b²
Is equal to (1÷2b)×ln|(u-b)÷(u+b)|
The above result is definitely okay, idk about the one you used(i am nkt saying yours one is incorrect, its just that i never saw that formula😅)
My eyes are burning! Too many knowlegde! This is insane!!!!
Basically integration is getting the area under the curve or easier formula to understand and so what is the area under the curve graphically or does it make it easier to realize after doing integration?
I just noticed the hundreds of black and red markers in the bottom right corner.
😆
Cubic formula next, please!
😆
please do triple integration on this next
Great video !! I have an interesting problem for you:) Infinite series from k =2 of 1/(ln(k)^(ln(ln(k))) we are only allowed to use limit comparison or comparison. :)
lim (k=2 .. inf) f(k) = A + lim(n=27..inf) f(n) where A is some number and f(n) = 1/(ln(n)^(ln(ln(n))).
Each term in the expansion of f(n) is now strictly positive, since ln(ln(27)) > ln(ln(e^3)) = ln(3)>ln(e) = 1, i.e. positive.
So f(n) series terms are bounded above by 1, so only need to consider a monitonic decreasing series.
Oh and also determine the leading constant A value.
@@tomctutor numerically this series diverges, so it is not bounded by 1.
@@tomctutor oh sorry misinterpreted your statement, it is, youre talking about the terms :)
@@teddy05p Yes I did not comment on wether or not the series converges. If alternating then it would converge, but it isn't alternating as I shown.
Do the triple integral over a b and c
Wolframalpha still has a lock on the freshmen calculus student market.
This integral is basically the unbounded interval is basically the terms for the "missle knows where it is" problem, isnt it?
I have a question, for u^2 -b^2 what if you replaced the constant b with say ib so that (ib)^2 was -b^2,
then substituted it back it? That is use u^2 + (ib)^2 for u^2 -b^2.
The solution for either case (+/-) excluding hyperbolic functions when integrating with respect to "a": -b*ln|√((b^2)-4*a*c) ± b| ± √((b^2)-4*a*c) + C
Please do a video where you triple integrate with respect to all 3 im genuinely intrigued what the result would be (is it even possible??)
awesome!
Worth noting this function has a domain restricted to the interval 0 =< 4ac =< b^2. The second inequality comes from the discrimant (existence of roots). The first is a statement that either a and c are both negative or both positive. This is weird to me!
Love how he doesn’t say everything but says everybody
Yes I have integrated the sridharacharya formula
Really nice
i pay for gpt 4 and its all in how you ask it, so far its done everything you have as well. and was very close to the same answer. AI can and will replace useless professors, as professors are not available to answer students questions at the scale of universities. we NEED ai to help students lean and ask questions when they are lost or FEEL lost. gpt has helped me immensly in my exploration and understanding of calculus. so far, it hasnt been wrong once.
Sir we can instead use 1/2u . ln[x-u]/[x+u] + c for the integration of
1/u^2 -b^2 du
Bro will integrate my life
I am drunk right now but dude this video is absurd, it would never get to my mind this solution! Great! Have a good day :D
Doing math with a smile!
You should do it with respect to b and c next :3
Integrate the qubic formula next
So what can this be applied to specifically? Curious.
Could you do double integration to integrate based of two variables?
You can see my 100 integrals part 2. I did about 20 double integrals there toward the end.