The rate (as a percentage) of $\mathrm{CO}_2$ contained in a room after $t$ minutes of hood operation is modelled by the function $f$ defined for all real $t$ in the interval $[0;20]$ by: $$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$ Let $V_m$ denote the average rate (as a percentage) of $\mathrm{CO}_2$ present in the room during the first 11 minutes of operation of the extractor hood. a. Let $F$ be the function defined on the interval $[0;11]$ by: $$F(t) = (-1{,}6t - 3{,}6)\mathrm{e}^{-0{,}5t} + 0{,}03t.$$ Show that the function $F$ is an antiderivative of the function $f$ on the interval $[0;11]$. b. Deduce the average rate $V_m$, the average value of the function $f$ on the interval $[0;11]$. Round the result to the nearest thousandth, that is to $0.1\%$.
The rate (as a percentage) of $\mathrm{CO}_2$ contained in a room after $t$ minutes of hood operation is modelled by the function $f$ defined for all real $t$ in the interval $[0;20]$ by:
$$f(t) = (0{,}8t + 0{,}2)\mathrm{e}^{-0{,}5t} + 0{,}03.$$
Let $V_m$ denote the average rate (as a percentage) of $\mathrm{CO}_2$ present in the room during the first 11 minutes of operation of the extractor hood.\\
a. Let $F$ be the function defined on the interval $[0;11]$ by:
$$F(t) = (-1{,}6t - 3{,}6)\mathrm{e}^{-0{,}5t} + 0{,}03t.$$
Show that the function $F$ is an antiderivative of the function $f$ on the interval $[0;11]$.\\
b. Deduce the average rate $V_m$, the average value of the function $f$ on the interval $[0;11]$. Round the result to the nearest thousandth, that is to $0.1\%$.