$\frac{\int_{a}^{b} f(x)\,dx}{b-a}$
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Mean Value of a Function Formula
The mean value (or average value) of a function $f(x)$ over a closed interval $[a, b]$ provides the average height of the function across that interval. It is formally defined using definite integration.
Deriving the Mean Value Formula
To find the mean value of $f(x)$ from $x = a$ to $x = b$, we calculate the definite integral of the function over the interval, which represents the total area under the curve, and then divide this area by the length of the interval, $(b-a)$.
The formula is:
$ f_{\text{avg}} = \frac{1}{b-a} \int_{a}^{b} f(x)\,dx $
Analysis of Options
We examine the provided options to identify the correct formula for the mean value:
- Option 1: $\frac{f(a)+f(b)}{2}$
This represents the average of the function's values solely at the endpoints $a$ and $b$. It does not account for the function's behavior within the interval $(a, b)$. - Option 2: $\frac{f(a) + 2f\left(\frac{a+b}{2}\right) + f(b)}{4}$
This expression resembles formulas used in numerical integration methods, like Simpson's rule approximations, rather than the exact definition of the mean value. - Option 3: $\int_a^b f(x)dx$
This is the definite integral, which calculates the net area under the curve of $f(x)$ between $a$ and $b$. While related, it is not the average value itself. - Option 4: $\frac{\int_{a}^{b} f(x)\,dx}{b-a}$
This option correctly represents the mean value. It divides the definite integral (total area) by the width of the interval $(b-a)$, giving the average function value or average height.
Therefore, the correct formula for the mean value of a function $f(x)$ from $x = a$ to $x = b$ is $\frac{\int_{a}^{b} f(x)\,dx}{b-a}$.
Important Questions from Mean Value Theorem
A series expansion for the function sin θ is
If f is the derivative of some function on [a, b], then there exists a number c in (a, b) such that Integral of f with respect to x =
Which condition is not required in checking for Taylor's theorem?
- What is the interval of Taylor series expansion of tan(x)?
According to the Mean Value Theorem, for a continuous function f(x) in the interval [a, b], there exists a value ξ in this interval such that \(\mathop \smallint \limits_a^b f\left( x \right)dx =\)
