Page:Calculus Made Easy.pdf/183

(3) Plot the curve $y=100(1-\epsilon ^{-2t})$.

(4) The following equations give very similar curves:

${\begin{aligned}{\text{(i)}}\ y&={\frac {ax}{x+b}};\\{\text{(ii)}}\ y&=a(1-\epsilon ^{-{\frac {x}{b}}});\\{\text{(iii)}}\ y&={\frac {a}{90^{\circ }}}\arctan \left({\frac {x}{b}}\right).\end{aligned}}$

Draw all three curves, taking $a=100$ millimetres; $b=30$ millimetres.

(5) Find the differential coefficient of $y$ with respect to $x$, if

(a) $\;y=x^{x}$; (b) $\;y=(\epsilon ^{x})^{x}$; (c) $\;y=\epsilon ^{x^{x}}$.

(6) For “Thorium $A$,” the value of $\lambda$ is $5$; find the “mean life,” that is, the time taken by the transformation of a quantity $Q$ of “Thorium $A$” equal to half the initial quantity $Q_{0}$ in the expression

$Q=Q_{0}\epsilon ^{-\lambda t}$;

$t$ being in seconds.

(7) A condenser of capacity $K=4\times 10-6$, charged to a potential $V_{0}=20$, is discharging through a resistance of $10,000$ ohms. Find the potential $V$ after (a) $0.1$ second; (b) $0.01$ second; assuming that the fall of potential follows the rule $V=V_{0}\epsilon ^{-{\frac {t}{KR}}}$.

(8) The charge $Q$ of an electrified insulated metal sphere is reduced from $20$ to $16$ units in $10$ minutes. Find the coefficient $\mu$ of leakage, if $Q=Q_{0}\times \epsilon ^{-\mu t}$; $Q_{0}$ being the initial charge and $t$ being in seconds. Hence find the time taken by half the charge to leak away.