This is the first time I felt that I started to get this, thank you so much! It would be so much appreciated if you did more videos on Lagrange Error Bound!!
when maximizing f(z) for sin are you maximizing it on any interval or only the given one (from 0 to 0.1)? Why did you plug in sin(pi/2) to get 1 instead of plugging in sin(0.1)?
Does anyone notice that Patrick's voice has changed? I watched his calculus videos(which he uploaded 8 years ago) and I'm watching this video now and his voice changes are so obvious. A life spent in help of humanity. Thanks Patrick! You are saving our lives :P
Minor point and I hate bringing these up, but, if (2n+1)=3, then n=1. Then also we're considering R_1, & f^(1+1), or f''(x). The process otherwise is similar. Other than that, for me, the hardest part of this is maximizing the z, or Xi in some texts. The rest is more or less plug-n-play.
Patrick, considering that you estimated sin(0.1) to be at most 1 in order to get a bound on the error of sin(0.1), does that mean you can do the process again and get a better bound? And if so, how much can we improve this by just iterating this process? I'm sure it will converge, but to what value? That is a cute thing from the fact you use a 3rd degree approximation and the 4th derivative is the same function. Very interesting. Seems like a similar argument would hold for the exponential function too
Hey Patrick! I've been watching your videos for about 6 years now! I'm heading off to grad school and was wondering if you could do some GRE math videos. Thanks!
yeep that's what I thought too, but I think he did that because we are trying to approximate the function therefore we cannot use the function to find the maximum. I think
As Nico said, sin(0.1) is the function you're approximating, so you can't use it's value in your approximation. If you could calculate it, you wouldn't need the approximation. Before you say "use a calculator/computer too calculate it", I'll ask you to think about how those devices calculate sin(0.1). (Hint: they are giving approximations using either a Taylor polynomial with several terms or another method like the CORDIC algorithm.) Also keep in mind that this is an error BOUND, not the exact amount of error in your approximation. That helped me become comfortable using values other than the exact max. We use 1, specifically, to maximize the derivative for sine (and cosine) problems, because |sinx|
Hi, the way I get it, is z is a value that exists between c and x and c and x can be figured out from what's given to you. Using the remainder theorem, sub in your n to the remainder formula and your x and c. It should leave you with a nice answer for the remainder except for the f^n+1(Z) part. if you were asked to approximate e.g ln(1.1), then we can say f(x) = lnx. if we need to find f^5(Z), then differentiate that 5 times and you will get an equation in terms of x (in this example, 24x^-5). If you imagine the remainder formula with a c instead of a z, its really just giving the value of the term with the n+1th power, which is your error in your approximation to the nth power (I think n+2th powers and so on are ignored for being comparatively smaller but im not entirely sure). If we are asked to find the MAXIMUM error, then we need to find the largest possible value for the equation (in my example, (24)/(x^5) ), as if we tried to find the minimum value of this, it would be like saying we want the value closest to zero, or equal to zero which means there is no error at all. So using your equation, find the largest output value by subbing in the correct x value between C and X. this value is z. you don't actually need to know z, as you can just find the largest value of the function and then sub the function f^n+1(Z) into the formula. Hope that helped! The issue i'm having, as stated above is that I don't understand why the formula only considers the remainder to the n+1 and not also n+2 and n+3, as they might be small, but are also errors. Maybe this max error is also an approximation in itself.
@@adam-jm1gq It is because following terms' effect are within the error. For example, you are calculating to nth power then n+1 will be error and we know this keeps getting smaller in order to converge. Hence you show error to be +R which is calculated using n+1 so n+2 and following terms will increase accuracy, however, won't effect the decimal place we needed because we already corrected it so they are not errors that you need to add what already we have but rather it means as you go further, you'll have a smaller error. I answered it but probably you've already got your answer by now (you posted this 6 months before me). I hope this can help who are having the same issue
Shouldn't we use 5th degree for remainder? I mean, since for sin approximation we use only odd (i.e . 2n + 1) degrees. Shouldn't we skip (2n+2)'nd degree since in this degree polynomial doesn't exists in c?
How would you answer a student who asks WHY the Taylor formula evaluated at the n+1 derivative for (and here's the kicker) any value between x and a gives the largest error? Is there a common sense, way to explain this? I know it's connected to the mean value theorem, but I can't find that explanation in a textbook. Students tend to comprehend the alternating series error calculation (which is much more intuitive) and then try (unsuccessfuly) to use it to make sense of the Lagrange error bound. I need help explaining this!
i think you made an error at the end. did you mean to write .099833? instead of .099883? the former is what you solved through the taylor polynomial. (and the latter makes the final statement wrong b/c sin(.1) is appx .099833417
Been watching your videos for years and would like to make videos like this for my students, but I don't know how to record. Could you let me know how you record your videos?
I'm only a student, but if I understand the concept right, the Lagrange can only be used if the series alternates. This is because the values are getting smaller and smaller and are being added and then subtracted, so the value of the taylor polynomial gets closer and closer to the real value.I probably haven't explained this right, I don't really know how else to explain it on paper, but I hope this helps
you did a transcribing error at 13.06 but that's only human but my main point is that your approach is mechanical you do not derive or explain the error term i.e why does z lie between 0 and 0.1 it would only require a few sentences would it not
This was my question exactly. How would you explain it? I know it has something to do with the mean value theorem, but I can't find a good explanation anywhere.
one question: what do you do with all the paper you use? I hope you recycle them. And a quick suggestion, try using your white board as much as possible instead of the paper sheets. Keep up the good work! Thanks :)
Unrelated to the video but do you see any hope for mankind considering the devastatingly dysgenic population growth? How is the world going to function when there are 4 billion sub-Saharan Africans with IQs mostly in the 70s and 80s? And how many more Oswald Teichmuellers will perish due to needless brother wars? Is this the end of history?
Dear PatrickJMT,
You have saved our asses in every math class for four years and counting. Thank u for ur beautiful brain.
Sincerely
Sofia and sasha
Thank you for a concise, clear explanation. Most calculus textbooks do not offer near enough explanation or examples. Take a bow, sir! Bravo!
thanks for continuing to make excellent videos for so many years. words can't describe how much you have helped me.
this is the real classic video on this subject, so mahy videos came after this one, and as years pass by, still the classic
This is the first time I felt that I started to get this, thank you so much! It would be so much appreciated if you did more videos on Lagrange Error Bound!!
when maximizing f(z) for sin are you maximizing it on any interval or only the given one (from 0 to 0.1)? Why did you plug in sin(pi/2) to get 1 instead of plugging in sin(0.1)?
He is wrong. It is supposed to be on the given interval, so 0 to 0.1. This video is not that good
Could you do another example, perhaps involving e?
@patricJMT dude, thanks for making these video that prep me for the ap test!! you are my hero!!
Who else is studying for AP Calculus BC and IB Math HL?
Loser* oh the irony.
The irony is just amazing.
me!
I AM
Yeah I’m taking it this May. It doesn’t seem too bad.
Does anyone notice that Patrick's voice has changed? I watched his calculus videos(which he uploaded 8 years ago) and I'm watching this video now and his voice changes are so obvious. A life spent in help of humanity. Thanks Patrick! You are saving our lives :P
MedamriHD Exactly my thoughts xD
MedamriHD Nope, did not notice any changes. His voice is somewhat adorable.
Minor point and I hate bringing these up, but, if (2n+1)=3, then n=1. Then also we're considering R_1, & f^(1+1), or f''(x). The process otherwise is similar. Other than that, for me, the hardest part of this is maximizing the z, or Xi in some texts. The rest is more or less plug-n-play.
Thank you. This was great and helps me so much on my homework
I dropped Degree in Biology for Math...thanks to my math doctor @patrickJMT. My math is surely salvated. Thanks.
Patrick, considering that you estimated sin(0.1) to be at most 1 in order to get a bound on the error of sin(0.1), does that mean you can do the process again and get a better bound? And if so, how much can we improve this by just iterating this process? I'm sure it will converge, but to what value? That is a cute thing from the fact you use a 3rd degree approximation and the 4th derivative is the same function. Very interesting. Seems like a similar argument would hold for the exponential function too
The 4th degree polynomial is the same as the 3rd: we can find a smaller error bound: 0.1^5/5!
50 times better than video's solution!
Thanks so much! Finally understood this
Hey Patrick! I've been watching your videos for about 6 years now! I'm heading off to grad school and was wondering if you could do some GRE math videos. Thanks!
Why is the max in sin (z) = 1 ? since z is max at 0.1 , doesn't that mean that max in sin is sin(0.1)?
yeep that's what I thought too, but I think he did that because we are trying to approximate the function therefore we cannot use the function to find the maximum. I think
As Nico said, sin(0.1) is the function you're approximating, so you can't use it's value in your approximation. If you could calculate it, you wouldn't need the approximation. Before you say "use a calculator/computer too calculate it", I'll ask you to think about how those devices calculate sin(0.1). (Hint: they are giving approximations using either a Taylor polynomial with several terms or another method like the CORDIC algorithm.)
Also keep in mind that this is an error BOUND, not the exact amount of error in your approximation. That helped me become comfortable using values other than the exact max.
We use 1, specifically, to maximize the derivative for sine (and cosine) problems, because |sinx|
Fantastic video, thank you 😊
I just have trouble grasping how to find the max value of z :(
Hi, the way I get it, is z is a value that exists between c and x and c and x can be figured out from what's given to you. Using the remainder theorem, sub in your n to the remainder formula and your x and c. It should leave you with a nice answer for the remainder except for the f^n+1(Z) part. if you were asked to approximate e.g ln(1.1), then we can say f(x) = lnx. if we need to find f^5(Z), then differentiate that 5 times and you will get an equation in terms of x (in this example, 24x^-5).
If you imagine the remainder formula with a c instead of a z, its really just giving the value of the term with the n+1th power, which is your error in your approximation to the nth power (I think n+2th powers and so on are ignored for being comparatively smaller but im not entirely sure). If we are asked to find the MAXIMUM error, then we need to find the largest possible value for the equation (in my example, (24)/(x^5) ), as if we tried to find the minimum value of this, it would be like saying we want the value closest to zero, or equal to zero which means there is no error at all. So using your equation, find the largest output value by subbing in the correct x value between C and X. this value is z. you don't actually need to know z, as you can just find the largest value of the function and then sub the function f^n+1(Z) into the formula.
Hope that helped! The issue i'm having, as stated above is that I don't understand why the formula only considers the remainder to the n+1 and not also n+2 and n+3, as they might be small, but are also errors. Maybe this max error is also an approximation in itself.
Bandito Donburrito Thank you so much!
@@adam-jm1gq It is because following terms' effect are within the error. For example, you are calculating to nth power then n+1 will be error and we know this keeps getting smaller in order to converge. Hence you show error to be +R which is calculated using n+1 so n+2 and following terms will increase accuracy, however, won't effect the decimal place we needed because we already corrected it so they are not errors that you need to add what already we have but rather it means as you go further, you'll have a smaller error.
I answered it but probably you've already got your answer by now (you posted this 6 months before me). I hope this can help who are having the same issue
Patrick do you have any videos on how to solve 3rd order Cauchy Euler equations? Thanks.
no, I'm pretty sure I don't! (There are so many I forget sometimes).
patrickJMT Darn, thanks anyways! Keep up the good work!
thank you so much! this was very helpful.
how do you see the stuff through ur hand when you are writing it..
Bro you so clutch thanks a lot!
Shouldn't we use 5th degree for remainder? I mean, since for sin approximation we use only odd (i.e . 2n + 1) degrees. Shouldn't we skip (2n+2)'nd degree since in this degree polynomial doesn't exists in c?
Thank you sir 😊
You are definetely my idol!!
Glad you like the videos, but I definitely ain't idol material! :)
patrickJMT thank you for video and information above. peace and love.
likewise my friend!
patrickJMT you are God!
How would you answer a student who asks WHY the Taylor formula evaluated at the n+1 derivative for (and here's the kicker) any value between x and a gives the largest error? Is there a common sense, way to explain this? I know it's connected to the mean value theorem, but I can't find that explanation in a textbook. Students tend to comprehend the alternating series error calculation (which is much more intuitive) and then try (unsuccessfuly) to use it to make sense of the Lagrange error bound. I need help explaining this!
right before aps♡!!
Haha not for ucs
is not the choice of x "also" determening the max value?
why is z trapped between 0 and 0.1? Is there a video on that?
your work is always neat and tidy. I believe an accompanying graph will bring out the lots though.
i think you made an error at the end. did you mean to write .099833? instead of .099883? the former is what you solved through the taylor polynomial. (and the latter makes the final statement wrong b/c sin(.1) is appx .099833417
Thnx!
that pen scraping on paper sound is absolutely annoying, but good content
its SATISFYING XD
What is bound of e^x
Very helpful
Why is the actual value not between the error bounds?
Actual value of sin(0.1) = 0.09983341664
Is this normal?
Been watching your videos for years and would like to make videos like this for my students, but I don't know how to record. Could you let me know how you record your videos?
what if it's not centered at 0
i thought lagrange error is used when the series isnt alternating? but sin(x) taylor series is alternating??
I'm only a student, but if I understand the concept right, the Lagrange can only be used if the series alternates. This is because the values are getting smaller and smaller and are being added and then subtracted, so the value of the taylor polynomial gets closer and closer to the real value.I probably haven't explained this right, I don't really know how else to explain it on paper, but I hope this helps
Shoutout to Mr. Achille’s class!
I do not understand the useful result, is the useful result used in this example?
what if I know error value but I want "n"
Can someone please send this man a better microphone?
what if (x - a) > 1, say 5? the error estimate would be very large
How can I contact U ?
sin(0.1)=9.983341664682815230681419841062e-2
error ????
Add another video pleaseeee!
Why don't you solve for Z?
Wait never mind bc it's strictly increasing
Thanks that helped a lot
😂😂😂😂 agreed
you did a transcribing error at 13.06 but that's only human but my main point is that your approach is mechanical
you do not derive or explain the error term i.e why does z lie between 0 and 0.1
it would only require a few sentences would it not
This was my question exactly. How would you explain it? I know it has something to do with the mean value theorem, but I can't find a good explanation anywhere.
Patrick>Khan academy
why did i pick math at uni ffs
Where that 0.000004 come from
[(0.1)^4]÷4!
I would use |sin(z)|
Very intelligent comment
one question: what do you do with all the paper you use? I hope you recycle them.
And a quick suggestion, try using your white board as much as possible instead of the paper sheets.
Keep up the good work!
Thanks :)
Ariana Legrange
Again.
Why are your fingers so weird looking?
Unrelated to the video but do you see any hope for mankind considering the devastatingly dysgenic population growth? How is the world going to function when there are 4 billion sub-Saharan Africans with IQs mostly in the 70s and 80s? And how many more Oswald Teichmuellers will perish due to needless brother wars? Is this the end of history?