# Mean-Value Theorem, Taylor's formula, and error estimation

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• mcastillo356
In summary, Taylor's formula provides a formula for the error in a Taylor approximation, similar to that provided for linear approximation. The case n=0 of Taylor's formula is just the Mean-Value Theorem, which states that for some s between a and x, the slope of the chord line joining the points (0,1) and (2,7.3890) is equal to the slope of the tangent line to the curve y=e^x at the point (1.1614,3.1944). For a quadratic, this point will always be exactly the midpoint between x and a on the x-axis. For higher order functions, the error depends on the delta and the higher order terms in the function.
mcastillo356
Gold Member
TL;DR Summary
Can't conclude any error estimation from MVT, i.e., Taylor's formula for ##n=0##
Hi, PF

Taylor's formula provides a formula for the error in a Taylor approximation ##f(x)\approx{P_{n}(x)}## similar to that provided for linear approximation.

Observe that the case ##n=0## of Taylor's formula, namely,

##f(x)=P_{0}(x)+E_{0}(x)=f(a)+\dfrac{f'(s)}{1!}(x-a)##,

is just the Mean-Value Theorem

##\dfrac{f(x)-f(a)}{x-a}=f'(s)## for some ##s## between ##a##

and ##x##

The question is: to what extent, i.e. how, MVT provides a formula for the error in a Taylor approximation?

Attempt: let ##f(x)=e^{x}##, ##x=2##, ##a=0##: for ##n=0##

1- ##\dfrac{e^{2}-1}{2}=e^{s}##
2- ##\ln{\dfrac{e^{2}-1}{2}}=s\cdot{\ln{e}}\Rightarrow##
3- ##\Rightarrow{s\approx{1.1614}}##
4- ##\therefore{f(s)\approx{3.1944}}\Rightarrow##
5- All I can conclude is that the slope of the chord line joining the points ##(0,1)## and ##(2,7.3890)## is equal to the slope of the tangent line to the curve ##y=e^x## at the point ##(1.1614,3.1944)##, so the two lines are parallel.
6- Which is here the error estimation, or how can I find out?

Greetings

PS: I post with no preview. Twice edited

Last edited:
MVT says there is at least one point between x and a that has a tangent egual to the slope of the line through f(x) and f(a) (the secant). For a quadratic, that point will always be exactly be the mid point between x and a (on the x axis). For higher order functions, the error depends on the delta and the higher order terms in the function.

mcastillo356
valenumr said:
MVT says there is at least one point between x and a that has a tangent egual to the slope of the line through f(x) and f(a) (the secant). For a quadratic, that point will always be exactly be the mid point between x and a (on the x axis).
Didn't know that fact for quadratics. Nice
valenumr said:
For higher order functions, the error depends on the delta and the higher order terms in the function.
Sorry, could this be explained furthermore?

Love, thanks @valenumr

mcastillo356

## 1. What is the Mean-Value Theorem?

The Mean-Value Theorem is a fundamental theorem in calculus that states that for a continuous and differentiable function f on an interval [a, b], there exists a point c in (a, b) where the slope of the tangent line at c is equal to the average rate of change of f over the interval [a, b]. In other words, there exists a point where the instantaneous rate of change is equal to the average rate of change.

## 2. How is Taylor's formula used in calculus?

Taylor's formula is a mathematical tool used to approximate a function with a polynomial. It is especially useful for approximating complicated functions that are difficult to integrate or differentiate. By using Taylor's formula, we can break down a function into simpler polynomial terms, making it easier to analyze and calculate.

## 3. What is the purpose of error estimation in calculus?

Error estimation is used in calculus to determine the accuracy of an approximation. It helps us understand how close our estimated value is to the actual value. In other words, it allows us to quantify the error or the difference between the approximation and the exact value.

## 4. How is the error in Taylor's formula calculated?

The error in Taylor's formula is calculated using the remainder term, also known as the Lagrange remainder. This term takes into account all the higher-order derivatives of the function and their corresponding coefficients. The error can be estimated by finding the maximum value of the remainder term within the interval of interest.

## 5. Can the Mean-Value Theorem be applied to all functions?

No, the Mean-Value Theorem can only be applied to continuous and differentiable functions. This means that the function must have no breaks or jumps and must have a well-defined derivative at all points on the interval of interest. If these conditions are not met, the Mean-Value Theorem cannot be applied.

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