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Rekenmachine voor de middelwaardestelling
Pas de middelwaardestelling stap voor stap toe
De rekenmachine zal alle getallen $$$c$$$ vinden (met de stappen getoond) die voldoen aan de conclusies van de middelwaardestelling voor de gegeven functie op het gegeven interval. De stelling van Rolle is een speciaal geval van de middelwaardestelling (wanneer $$$f(a)=f(b)$$$).
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: $$$\frac{\pi}{4}\approx 0.785398163397448$$$