the cover-up method & why it works! (for partial fractions decomposition)
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- เผยแพร่เมื่อ 4 ต.ค. 2024
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Hey professor Chow, I wanted to come say thanks. I just won my school’s integral bee in 1st place, and in the final round I did partial fractions in my head in a few seconds. You taught me how to do that
That’s awesome! Glad to hear it and thanks for sharing!
yea but why does x have to be 1?
@@blackpenredpenI want to make myself like you sir ❤❤,pray for me please 🙏
Who needs calc 2 with this channel around? Amazing work.
Please consider expanding to calc iii.
Consider my channel if you want to expand your knowledge of integrals :)
yea but why does x have to be 1?
@@chill4r585 because we are dealing with A , and A has a denominator of (x-1) , we look at the denominator and equal the denominator to 0 ... so when you do x-1 = 0 , you get x = 1
similarly when you try to look for B, the denominator this time is x-2 and when we try to equal it to 0 , ie , x - 2 = 0 , we get x = 2, thus in the case of finding B we will substitute x as 2
hope it helps
3:35 OMG sir you just proved this, how this work. Thanks a lot.
Because of you, I'm going to save a lot of time in exam.
yea but why does x have to be 1?
Glad to see a new video! Thanks for the free lesson.
Your lessons are phenomenally great! I wish I could have he’d you as a teacher when I took all those math classes 50 years ago! I finally get it!
You explain nicely and understandably in this video about this method.
Thank you for helping us lost souls in calculus.
i think you literally just saved my life.
the way my teacher asked us to do was to find the lcm and equate it to the numerator and find values for A, B, C. aye, it works, but it's long and for the LIFE of me i couldn't figure out what values to use for x. i always thought my teacher was just subbing random values through guess work.
now i feel like i freed up a lot of time and effort. thank you.
thanks bprp im learning calc 2 now and my teacher didnt show me the proof so i was confused why the cover up rule works. now i understand it much much better
That's I love his videos. He not only explain things but also explain why it works
You're such a cool teacher. I whiich I had a teacher like you when I was young.
Gostei muito das tuas aulas e estou recomendando pra todo mundo aqui no Brasil.
You explain things better than my lecturer, and I'm studying at a QS100 university... I never expected this....
Wow 3 videos in a row ! Semi-automatic mode !!
yay!
Burst mode !
thank you so much prof chow to help me understand thisss
amazing, in differential equations right now and got introduced to this method but wasn't explained clearly. great vid
Thank you saved my interested in math and keep me motivated to keep going thanks again
Thank you sir "you not only give us a new method to calculate calculus but also you shows or visualize this new method"
just started calc 2, so glad i found this channel. love the explanations man
Wow, makes so much sense in so little time, truly impressive teaching
第一次看到這方法的解說
謝謝老師
不客氣！😄
Is it valid to multiply by (x-1) at the beginning of the first proof? I mean, before multiplying both sides by (x-1) you should mention that x isn’t equal to 1 isn’t it? If so, then you can’t plug in 1 later on
You can use The concept of limits,That is apply limit x tends to 1 on both sides,Though X cannot be 1 but it can be 0.999999999999999999... or 1.0000000000000000000000000....
So for both Left hand and right hand limit A tends to The value he calculated on left ,So it is Correct that we can simply put x=1 in the identity
@@NoMatterWhat279 just to add to the explanation, when you multiply by (x-1), the new function has a removable singularity at x = 1. Which means that you can "fix" it by putting in the limit (which will exist) in that "hole". Therefore, no issues with x = 1.
Yes, this is technically kinda hand wavy, we are assuming linear functions are continuous without a proof for it...
But why he replace the answer of the limit in the funtion?
I'm almost certain I never learned partial fractions in school - not middle, high school, nor college! It's a hugely useful technique, and I wonder if those of you in the US can confirm or deny you were taught this. Is this a notable hole in our mathematics education?
I was taught partial fractions in Calculus II in college and in Differential Equations in College, but I'm reviewing the coverup method because I'm not very good at partial fractions.
I'm in precalc in highschool and we are learning it right now!
Initially x couldn't have been 1 because that would mean we are multiplying both sides of the equation with zero( 3:07 ). The how can we suddenly put x = 1?( 4:26 )
thank you so much
currently learning integration by partial fractions and I'm here for quick relean
Wow this is beautiful, I hadn't realized this is where it came from!
My teacher for AP Calculus BC started his lesson on partial fractions today with this video (at my request).
This guy is gonna get me through Calc II 🤣.
Amazing teaching 👌
Thank you so much. As a 15-year-old in college, I was struggling quite a bit on math but this helped so much
They teach this in college? Isn't this middle school math?
i kinda don't get something
if we multiply both sides by (x-1), don't we have do provide that x-1 is not 0? Otherwise we could be accidentally multiplying both sides by 0 which breaks the equation
that said, 1 should be excluded from the domain and therefore we shouldn't be able to set x=1
what am I missing?
yes, my teacher has not described the reson behind it, now i know it , thank you !
Clear and easy to understand!!! Great explanation
You just made me flashback to, many decades ago, doing contour integrals by merely finding the sum of residues at the enclosed poles.
I suspect that there is similar 'trickery' as to why that 'works'.
Is this something you will ever touch upon in your videos? (hint, hint}
Such a fantastic teacher.
Thank u sir why this method is works nobody tell us, you explain it so nicely.
So, Thanking you sir
I like your starting audio blakpenredpen YAY!
sir your method is so easy thank you so much you save my day u made it awesome
helped a lot for my finals
Hey Professor Chow, thanks for the video! What about quadratic factors that will never equal 0?
This would actually be a great shot to learn in Precalc and then carry into calc II
You saved me again 😍😍
Good for Laplace transforms too, particularly reverse.
I'm not sure if I would agree on this. A, B and C may be the correct numbers, but the method to determine them lacks something, I think. Of course, x may not be equal to 1, 2 or 3 in the first equation because then you would divide by 0. That means that in every equation after that, the same x still can't be equal to 1, so you can't just let x = 1 and expect everything to work out. On the other hand, if you multiply by x-1 and let x = 1, you essentially multiplied by 0, so you end up with 0 = 0, but that yields no information about A.
Arteyyy I was thinking the same! Maybe it just works in this example because you may think of x going to 1, not exactly 1, so A will have a number (x-1) divided by itself, which is one. In case of B and C, you dont have the same situation, such that x is going to 1 and x-1 is going to 0, this way, you can think of it as B and C getting multiplied by a very small number and its result is 0. Anyway, its really strange indeed, i prefer the conventional and formal method.
Yes but the equation must hold for all x, so x = 1 will give you a valid equation, and you can rightfully solve for A. Your examples gives a true statement, though no information about A, it is valid and thus you know you have a valid equation.
Sure, it's about the numerator, but you must always make sure that everything is well-defined. In this case, you need to make sure that the denominator never equals 0 because if it does, you can't draw any conclusions about the numerator.
Yes, multiplying by x-1 cancels out another x-1, but it only does so if x != 1. If x = 1, you can't just say (x-1)/(x-1) = 1, it's undefined. If you multiply an equation by 0 (and let's assume you didn't divide by 0 before), you always end up with 0 = 0, even if there are still terms left that didn't get cancelled out. That's simply a property of 0.
It's nice to see a quicker way to determine A, B and C and like I said they are correct, but the formalities leave something to be desired.
No, it doesn't. At least not yet. I suppose my problem is this: It seems that x should have been defined as a real number except 1, 2 or 3 so that you don't divide by 0. That means that x != 1 is always true, even though x is a variable. Of course, letting x = 1 makes no sense then.
I wouldn't be surprised if the equation with no x-1 in the denominator is actually true if you plug in 1 for x, even though you couldn't do that before. It seems unlikely not to be, but how do you prove that?
Yes, it is kind of a half proof, but it works if you do it the long way.
Thank you very much
Thanks!
I love your videos :')
awesome as always 👍🌷
great method!
hello from greece ,first time not an indian
Wah this really make things simple
Thank-you. This was very well explained.
THANK YOU 🙏🙏🙏🙏
This works for all cases but we have advantages only for real distinct roots (if we dont want use complex numbers)
Thank you
But if x=1 isn’t it illegal to multiply by x-1?
Thank you sir 🖤
I vaguely remember this from my first semester of algebra in high school but they didn't do a whole lot of it
Okay so something I've never understood about partial fractions is why the numerator always to be one degree lower than the denominator. What's the reason?
Bc if they are the same degree, then you can do long division first.
For examples, x^2/(x^2+1)
you can just get 1-1/(x^2+1)
fair enough but that doesn't exclude having the numerator for instance two degrees lower.
Notice that you have as many coefficients as degrees down to 0 there are. Say the factor is linear, you only have A/factor. If the factor was quadratic, you have two possibilities, like you say, you'd have a linear answer, A*x/(factor) and a constant answer, B/(factor). If you add these up, you get (A*x + B) / (factor). Maybe the partial fraction is two degrees, lower, because A=0, but maybe not. We just make sure that all the possibilities are covered. I hope this makes sense.
That does make sense, thank you :)
You have not shown that there exit constants A, B and C that makes this equality true for all x. You argument would work just as well if the original numerator was e.g. sin x.
Amazing🔥🔥🔥🔥🔥🔥
Thanks sir ❤🔥 great respect towards you ❤
Idk why, but with the new beard you make math look cool
Thank u so very much Sir ❤️❤️❤️❤️
haha i like when you switching pens, i can see you are a master of chopsticks. i like your videos
What about complex numbers in the fraction? There is no way to make it equal to zero then in alot of cases
lifesaver
Does this work for a qudrartic denominator?
Thank You Sir .
i think it doesn't work on numerators that doesn't have constant, just x.
Wait, I've just seen a video exactly like this one. This is just a remake, right? Also, it doesn't answer my question if it works for same factors, but I've figured it out while watching this. If two (or more) denominators were the same, multiplying by it wouldn't leave only one constant, in fact, it would yield sum of two (or more) constants.
Ah, I will have to make another video on that. This only explain why cover up works for distinct linear factor
Finally got it
I am from Vietnam and it's good
Nice....new trick
Thanks a lot sir.☺
Does this work for any partial fraction?
5:10 is where i found out how the magic work
does this method not work in case the power is two or greater?
Thank you so much!
Another interesting use of the method can show that (3x+5)/(1-2x)^2 = (13/2)/(1-2x)^2 + (-3/2)/(1-2x)
Spencer Key so this method works for repeated factors?
Wait, you just multiplied by (x-1), and so x cannot be equal to 1, but then you set x=1. Why that's not a contradiction?
Thanks so much!!!
Isn't this dividing (x-1)/(x-1) on the left-hand side? Why are you able to cancel that if you're setting x-1 to 0? 0/0 doesn't equal 1, 0/0 is undefined
You aren't really calculating 0/0. What you really are doing, is setting up the partial fraction decomposition solution setup with undetermined coefficients. Then you multiply by one factor at a time, and then plugging in strategic x-values to make all the other terms equal zero, so that only one of the undetermined coefficients remains in the picture. This allows you to more directly solve for each undetermined coefficient.
Cool sir from Bangladesh
Super bro...
Thank u 👍🏻
Thanks 👍
WHY DID WE NOT LEARN THIS IN CALC 2 OMG
Great Stamina😂😂😂!!!Salute Sir😇😇
What if the B is B/x+2
But will you get working out marks?
Not often multiplying an equation by 0 yields good results.
Does this method work for partial fractions with repeated factors?
Yes.
As an example, consider:
(x^2 + 8)/((x + 1)*(x - 2)^2)
Set up Partial fractions:
A/(x + 1) + B/(x - 2)^2 + C/(x - 2)
I like assigning the first letters to what I can get with the cover-up method, which are the sole linear term (x + 1), and the highest power of (x - 2). I then descend the power for the repeated factor.
Set x = -1 to find A, and cover up (x + 1):
A = ((-1)^2 + 8)/(-1 - 2)^2 = 9/9 = 1
Set x = +2, and cover up (x - 2)^2 to find B:
B = (2^2 + 8)/(2 + 1) = 12/3 = 4
Construct what remains:
(x^2 + 8)/((x + 1)*(x - 2)^2) = 1/(x + 1) + 4/(x - 2)^2 + B/(x - 2)
We can't use H-cover-up in its simplest form to find B, but there is another strategic method to do so. Use infinity as a special value of x.
Multiply through by just one copy of (x - 2):
(x^2 + 8)/((x + 1)*(x - 2)) = (x - 2)/(x + 1) + 4/(x - 2) + B
Take the limit as x goes to infinity:
1 = 1 + 0 + B
Solution for B = 0
Thus, the solution is:
1/(x +1) + 4/(x - 2)^2
I don't fully understand; if you multiply both sides by x-1, then it couldn't be zero, but we're making it zero...
does this work on everything?
title says "why is works" instead of "why it works" :)
thanks, i will fix it now
@@blackpenredpen It's also in the description "why is works" haha
Thankyou❤️❤️❤️❤️
love it
does this method still work with, say, 5/(x-1)squared or cubed
why we use unknown arbitrary constants in the solution of partial fraction?
Because we recognize that the given fraction, is something that could've been formed by a sum of the partial fractions. The numerators are all unknown, and we'd like to solve for them.
For this example:
(2*x - 1)/((x-1)*(x-2)*(x-3))
We recognize that this could've been formed by the sum of three fractions, with (x-1), (x-2), and (x-3) as their denominators. We just don't know what the corresponding numerators are.
So when we add:
A/(x-1) + B/(x-2) + C/(x-3), we end up with:
(A*(x-2)*(x-3) + B*(x-1)*(x-3) + C*(x-1)*(x-2))/((x-1)*(x-2)*(x-3))
The solutions for A, B, and C are as follows:
A = 1/2, B = -3, C = 5/2
Plug them in and see how we can get our numerator to equal 2*x - 1, using this solution:
1/2*(x-2)*(x-3) - 3*(x-1)*(x-3) + 5/2*(x-1)*(x-2)
Expand:
x^2/2 - 5/2*x + 3 - 3*x^2 + 12*x - 9 + 5/2*x^2 - 15/2*x + 5
Add up the terms of the same exponent on x:
x^2/2 - 3*x^2 + 5/2*x^2 = 0
-5/2*x + 12*x - 15/2*x = 2*x
3 - 9 + 5 = -1
And you see we have our original numerator:
2*x - 1
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