Mean Value Theorem Application
The Mean Value Theorem (MVT) states that if a function $f$ is continuous on a closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then there exists at least one number $c$ in $(a, b)$ such that: $ f'(c) = \frac{f(b) - f(a)}{b - a} $
Applying MVT to the Problem
We are given $f'(x) = e^x$, $f(0) = 5$, and we need to find the interval for $f(1)$. Here, $a = 0$ and $b = 1$. Using the MVT formula:
$ f'(c) = \frac{f(1) - f(0)}{1 - 0} $Substituting the known values:
$ e^c = \frac{f(1) - 5}{1} $ $ e^c = f(1) - 5 $Rearranging to solve for $f(1)$:
$ f(1) = 5 + e^c $Determining the Range for $f(1)$
According to the MVT, the value $c$ must lie in the open interval $(a, b)$, which is $(0, 1)$. So, $0 < c < 1$. Since the exponential function $f(x) = e^x$ is an increasing function, applying it to the inequality $0 < c < 1$ gives:
$ e^0 < e^c < e^1 $ $ 1 < e^c < e $Now, substitute this range of $e^c$ into the expression for $f(1)$:
$ f(1) = 5 + e^c $Adding 5 to all parts of the inequality $1 < e^c < e$:
$ 5 + 1 < 5 + e^c < 5 + e $ $ 6 < f(1) < 5 + e $Therefore, the value of $f(1)$ lies between 6 and $(5 + e)$.
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 \(\mathop \smallint \limits_a^b f\left( x \right)dx =\)
