- #1
pivoxa15
- 2,255
- 1
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.
quetzalcoatl9 said: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!
pivoxa15 said:This could be it but I don't follow it very well.
jostpuur said: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.
jostpuur said: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.
pivoxa15 said:THe problem is they still didn't mention how the mean value theorem works in the proof.
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.
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.
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.
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.
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.