Background RC Time Constants

Consider the first order RC circuit given in Figure $7\mathrm{.}1$

Preliminary Work

Calculate the time constant for the series $R-$C circuit in Figure $7\mathrm{.}1$ for $C=1F$ and $R=470,1k$ and $2\mathrm{.}2k.$

Calculate the time constant for the same circuit for $R=1k$ and $C=0\mathrm{.}470F,1F$ and $2\mathrm{.}2F.$

For a series RC low$-$pass filter, component values are given as $R=10k$ and $C=0\mathrm{.}01F.$ The amplitude of the source voltage is $10$ V peak$-$to$-$peak.

a$)$ Calculate the cut$-$off frequency in Hz $.$

b$)$ Calculate $V_o$ in magnitude when the source voltage has frequencies of $0\mathrm{.}4,0\mathrm{.}8,1\mathrm{.}2,1\mathrm{.}6,2,4,8kHz.$