- #1
pivoxa15
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Does anyone know the mean value theorem associated with the Taylor series. Representing the Taylor series a finite sum and an end term? I don't get how they get it to look that way.
Generalised mean value theorem?
- Thread starter pivoxa15
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In summary, the mean value theorem is used in the proof of the Taylor series by converting the integral into the remainder term. This is done by using the fact that the integral can be represented as (b-a)f(y), where y is an element of the interval (a,b). The generalised mean value theorem can also be used to prove the Taylor series, but it involves using induction. Furthermore, the fundamental theorem of calculus can be derived from the mean value theorem.
- #2
quasar987
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can you be more specific about what you'Re asking?
- #3
quetzalcoatl9
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i have seen the taylor series derived that way..you consider the n-th antiderivative of the n derivative. when you apply the fundamental thm of calculus you will get a series of terms that define the taylor series, with a remainder term. you use the mean value theorem to show that the remainder will go to 0 as long as your function grows no faster than n!
- #4
pivoxa15
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quetzalcoatl9 said:
This could be it but I don't follow it very well.
- #5
quetzalcoatl9
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pivoxa15 said:
i know, but i don't feel like typing out all of the latex. just write out taking N definite integrals over some interval for a function that has been differentiated N times.
Arfken does exactly this derivation in his book (pg. 260 in the 2nd edition)
- #6
jostpuur
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http://en.wikipedia.org/wiki/Taylor's_theorem
Quite good explanations. This stuff was left unclear to me when I was sleeping in the calculus classes, but now I feel like understanding the integral trickery that is done in the Wikipedia's article.
Quite good explanations. This stuff was left unclear to me when I was sleeping in the calculus classes, but now I feel like understanding the integral trickery that is done in the Wikipedia's article.
- #7
quetzalcoatl9
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jostpuur said:
that is exactly the method that i was referring to. as long as the function doesn't grow faster than n! you will be fine and the remainder term will go to zero
<happy 600 posts to me>
- #8
pivoxa15
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jostpuur said:
THe problem is they still didn't mention how the mean value theorem works in the proof.
- #9
quetzalcoatl9
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pivoxa15 said:
you invoke the mean value thm in order to convert the integral into the remainder term. in other words,
[tex]
\int_a^b f(x) dx = (b - a)f(y)
[/tex]
where y is an element of the interval (a,b)
- #10
pivoxa15
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I've seen a proof of the genearlised mean value theorem in books and its long but it finally explained it fully in terms of Taylor series.
- #11
quetzalcoatl9
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i seem to recall that the MVT can be proven all by itself, and that the fundamental theorem of calculus can be derived from it
- #12
pivoxa15
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But the generalised MVT is something else. It involves using induction.
Related to Generalised mean value theorem?
What is the Generalised Mean Value Theorem?
The Generalised Mean Value Theorem is a mathematical theorem that relates the average rate of change of a function to its derivative. It is a generalisation of the Mean Value Theorem and can be applied to a wider range of functions.
What is the significance of the Generalised Mean Value Theorem?
The Generalised Mean Value Theorem is significant because it allows us to determine the average rate of change of a function over a given interval. This can be useful in many applications, such as in physics, economics, and engineering.
How is the Generalised Mean Value Theorem different from the Mean Value Theorem?
The Generalised Mean Value Theorem is a generalisation of the Mean Value Theorem, meaning that it is a more general version that applies to a wider range of functions. Unlike the Mean Value Theorem, the Generalised Mean Value Theorem does not require the function to be continuous on the closed interval, but it does require the function to be differentiable on the open interval.
Can the Generalised Mean Value Theorem be applied to any type of function?
The Generalised Mean Value Theorem can be applied to any function that is differentiable on the open interval and continuous on the closed interval. However, it is important to note that the theorem may not always hold for every function, as there are some conditions that must be met for it to be applicable.
What are some real-life applications of the Generalised Mean Value Theorem?
The Generalised Mean Value Theorem has many real-life applications, such as in calculating average speeds in physics, finding average rates of return in economics, and determining average growth rates in population biology. It is also used in optimization problems in engineering and in the study of fluid dynamics.
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