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Mean Value Theorem Calculator
Apply the mean value theorem step by step
The calculator will find all numbers $$$c$$$ (with steps shown) that satisfy the conclusions of the mean value theorem for the given function on the given interval. Rolle's theorem is a special case of the mean value theorem (when $$$f(a)=f(b)$$$).
Solution
Your input: find all numbers $$$c$$$ (with steps shown) to satisfy the conclusions of the Mean Value Theorem for the function $$$f=e^{- x} \sin{\left(x \right)}$$$ on the interval $$$\left[0, \pi\right]$$$.
The Mean Value Theorem states that for a continuous and differentiable function $$$f(x)$$$ on the interval $$$[a,b]$$$ there exists such number $$$c$$$ from the interval $$$(a,b)$$$, that $$$f'(c)=\frac{f(b)-f(a)}{b-a}$$$.
First, evaluate the function at the endpoints of the interval:
$$$f \left( \pi \right) = 0$$$
$$$f \left( 0 \right) = 0$$$
Next, find the derivative: $$$f'(c)=- e^{- c} \sin{\left(c \right)} + e^{- c} \cos{\left(c \right)}$$$ (for steps, see derivative calculator).
Form the equation: $$$- e^{- c} \sin{\left(c \right)} + e^{- c} \cos{\left(c \right)}=\frac{\left( 0\right)-\left( 0\right)}{\left( \pi\right)-\left( 0\right)}$$$
Simplify: $$$- e^{- c} \sin{\left(c \right)} + e^{- c} \cos{\left(c \right)}=0$$$
Solve the equation on the given interval: $$$c=\frac{\pi}{4}$$$
Answer: $$$\frac{\pi}{4}\approx 0.785398163397448$$$