Error Bounds for Taylor Polynomial Approximations

[@sarahbiebah]

Thank you so much. This was clear and easy to follow. I was getting frustrated with the error term because I couldn't understand what c was and what f(c) was and why they were unknowns and why we were trying to find their max value. Everything online and in my textbook said the same thing; "This is the formula for the error term and this is how we will use it; do this and do that and move that around and take this out, and bam we have a value. Easy peasy!" But it didn't make any sense to me. The small explanations you did on the side as you wrote the formula made such a difference. If they didn't give me the whole picture, they gave me the first clue ("C is a mystery value. F(c) is unknown because if we had known it we wouldn't need to calculate the error.") , from which I was able to track the next couple of clues easily ("Ah, okay, so F(c) is like a mysterious quantity we add to our approximation to reach the real value of F(b), and to make it more tangible and limit the worst case scenario to a number instead of leaving it lying in an infinite interval of numbers, we attempt to get the maximum value that this unknown F(c) can take.") It's always good to know how something came to be and why it is that way and it doesn't hurt to repeat and re-explain something that might seem obvious because more often than not, at least for me, numbers all of a sudden can become vague and obscure instead of self-explanatory. And words help dispell that.

[@JCCCmath]

Awesome. I'm glad my explanation helped.

[@ellaniamiriani]

Literally the EXACT same problem over here

[@gleon1602]

Very clear and organized presentation. Thank you very much.

[@MukelabaiSindila]

Thank you very much indeed this is really easy to follow and well explained

[@ClaireBeard]

How does he write like that?? Like that’s def backwards to him

[@JCCCmath]

Computer magic.

[@yiiiwiii2022]

reflect the video

[@ThanggousemKipgen]

take ur pad…. record selfie video a of writing…. simultaneously screenrecord ur writings and overlap the videos

[@billwindsor4224]

Excellent explanation, sir, *Thank You!*

[@Abhishek_9570]

Watching One day before exam 😢

[@fredross2327]

Great video! Is there an additional video that finishes the problem? Thanks!

[@JCCCmath]

What would we need to do next?

[@owi7326]

Yh like what else do you want?

[@yahya-yl3rd]

great video!

[@williamhu5425]

Thanks !! really help me a lot!

[@Gothfen]

Thanks sir, nicely explained, clear and also in a very creative way :)

[@JCCCmath]

Awesome. I'm glad you found it useful.

[@khadijaijaz2368]

Excellent🤗

[@JCCCmath]

Glad you found it useful.

[@michellew4634]

so helpful!

[@lolmnater101]

Why is 2 < e < 3?

[@michieldolfing]

Because e~2.7182

[@JCCCmath]

In the course that this video is for, the students prove that by finding upper and lower bounds on the integral of (1/x) from 1 to 2 and 1 to 3, respectively. This, lead to concluding that there is a location between 2 and 3 where ln(x)=1, therefore 2

[@dhiaelhakrouabah8663]

I just dont understand why the c is in between a and x0 (the point of our intrest) given that we're intrested in x0 then f^(n+1)(x) is supposed to be evaluated at x0 ...and other points of x are simply irrelavent . I understand that error bound should be as large as it can be by taking into account the worst case scenario and this is how we can bound it ..but x is supposed to be between a and x0 instead of c=x0 which is where we wanna know value of our approximation . What am i missing here?

[@fullfungo]

Realistically, you don’t need to write the error term as f(c)…, it has other forms as well. For example, “if |f^(n+1) (x)| < M on [a,b], then |R_n(b)| < M |b-a|^(n+1)/(n+1)!” In this formulation you just use the same bound. But if the function is continuous, then it has a point “c” where f^(n+1) (c) = M. So the form used in the video really only helps if f^(n+1) is continuous.

[@dhiaelhakrouabah8663]

@@fullfungo thanks for commenting im still struggling with that concept . Like i understand that term is bounded by some number M bcuz we dont really know real value so we want to know how big it can get . But we assume c to be in between a and x0 ... like why exactly ? I didnt come across any justification and i dont quite get why they its not simply right at the point of evaluation x0? It just doesnt make sense to me

[@fullfungo]

@@dhiaelhakrouabah8663 en.wikipedia.org/wiki/Extreme_value_theorem?wprov=sfti1

[@coolboy666pop]

Thanks

[@JCCCmath]

You're welcome.

[@fahrenheit2795]

What if I was trying to estimate f(1/17)?

[@JCCCmath]

Since "b" is the location where you would like to approximate the function, everything would stay the same except b=1/17. You will have a nasty e^(1/17) in your error bound, but could use a rational upper bound for that.

[@fahrenheit2795]

@@JCCCmath Thanks for the reply.

[@JCCCmath]

@@fahrenheit2795 You are welcome. I apologize it took so long. I have not been keeping up on this page lately.

[@syedatif3832]

Please tell me how can i make video like this

[@DailyCakeSlice]

Write on a piece of glass like normal, have the camera on the other side, then flip the video in editing.

[@syedatif3832]

@@DailyCakeSlice thanks

[@JCCCmath]

@@syedatif3832 Also, run a strip on LED lights around the perimeter of the glass. This will highlight the neon markers.

[@syedatif3832]

@@JCCCmath ok thanks 😊

[@JCCCmath]

Here is a good how-to: kzbin.info/www/bejne/gmLEpmSArbKGl6M

[@jacobsandy8539]

How tall are you

[@StarZihack]

Boring