Page:Calculus Made Easy.pdf/183

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

(4) The following equations give very similar curves:

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

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

(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

$$t$$ 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.