📚 Finding the Remainder of a Taylor Polynomial – Example Using Taylor’s Remainder Theorem 📚

📚 Finding the Remainder of a Taylor Polynomial – Example Using Taylor’s Remainder Theorem 📚

In this video, I walk through Taylor's Remainder Theorem and show how to find an expression for the remainder when approximating f(x) = e^x using a fourth-degree Taylor polynomial centered at x = 1. We show how the remainder formula applies to this specific example.

🚀 What’s covered:

Explanation of Taylor's Remainder Theorem and its importance in approximating functions.
Step-by-step calculation of the Taylor polynomial for f(x) = e^x at x = 1.
How to find the remainder using the general remainder formula and a detailed breakdown of the process.
This is an essential technique for students studying calculus and higher-level math, helping you understand the accuracy of Taylor series approximations and how to handle errors in your estimates.

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#TaylorPolynomial #RemainderTheorem #Calculus #TaylorSeries #MathTutorial #AdvancedMath #Mathematics #TaylorSeriesRemainder #MathHelp #LearningMath #STEM #HigherMath #SeriesApproximation

[starchmonkey]

Thank you for showing an actual example rather than just the theory. They were really hard to find on the Internet.

[abdelrahmangamalmahdy]

but what is z and where are the terms remaining from the remainder ?!

[BenjaminFife]

Came in totally confused, left totally getting it. Took only two minutes, thank you so much.

[ImAllDatRemains]

Shoulda been watching these days ago not 12 hours before my exam :/

[maresfillies]

Hey Patrick, quick question. Why is it in some textbooks well a majority of my math college textbooks replace f^(n+1)(z) with capital letter M? Any significance? :)

[HDitzzDH]

So the remainder of a taylor polynomial/series is the absolute value of the next degree of the polynomial? So for example if we had a taylor polynomial of degree 3 the "rest term" would be the absolute value of degree 4? It just feels like different sources gives different answers to what a remainder is in this case.

[Jamony]

If we need some numerical answer, how do we know what Z should be? I know it needs to be in between x and c? Should we find the expression and choose the z such that we can get the targest possible error?

[frogmouth]

wait im still confused, what is x? Should the remainder not be a whole number instead of an expression?

[andfer]

what is e^z, u gotta find a value let's say M, which is the maximum value of f^(n)(x) so that you find the maximum error....

[karenduseau]

z is any number between x and a (1 in this case)

[ale-hlpg]

what are the bounds of z? I know it's between 0 and 1 in this case but you state that it's between x and c, I assume this means the interval would go z belongs to [min(x, c), max(x, c)] since there's no requirement for c to be larger than x and vice versa?

[maresfillies]

Oh and Patrick you are awesome man, your tutorials have help me brush up and clarify math concepts over and over again. What's was your major in college if you don't mind me asking? Emphasis in what?

[markkennedy]

Hi Patrick. You might be able to help me with a query.

If one was to calculate R_5, one would get a smaller value again than R_4. However, as you know, in the case of the cosine Taylor polynomials P_4 and P_5, they are essentially the same with P_5 just adding an extra zero term to P_4. So could you offer any explanation as to why P_5 for cosine does a better job at approximating cosine than P_4 (according to the Remainder formula at least) when they are essentially the same polynomial. Thanks

[LeafyBreezes]

should label that this is the Cauchy Form

[ianfresno]

Thanks. Is this also called the LaGrange Error Bound?

[cmfong]

Patrick, thank you so much for your clear demonstration. It certainly helps me a lot! I am an old man trying to learn more mathematics myself. I want to ask you how can we derive the remainder function Rn in Taylor series. Does it involve the Cauchy theorem or the first mean value theorem?

[PianoMastah]

Patrick: It'd be great if you could tag this video with Lagrange remainder or error, my friend tried searching through your videos for that and couldn't find it. thanks!

[razzvidya]

I think I take higher oder derivative from the taylor theorem ???

[pascalebouchahine]

What about the formula of the remainder of a multivariable funtion?

[coughcoughjosh]

that tutor guy is trying to steal your free business, pat