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 Find the differential coefficients of (11) The temperature $$t$$ of the filament of an incandescent electric lamp is connected to the current passing through the lamp by the relation $C = a + bt + ct^2$. Find an expression giving the variation of the current corresponding to a variation of temperature.

(12) The following formulae have been proposed to express the relation between the electric resistance $$R$$ of a wire at the temperature $$t^\circ$$ C., and the resistance $$R_0$$ of that same wire at $$0^\circ$$ Centigrade, $$a$$, $$b$$, $$c$$ being constants.

$$R = R_0(1 + at + bt^2)$$. $$R = R_0(1 + at + b\sqrt{t})$$. $$R = R_0(1 + at + bt^2)^{-1}$$.

Find the rate of variation of the resistance with regard to temperature as given by each of these formulae.

(13) The electromotive-force $$E$$ of a certain type of standard cell has been found to vary with the temperature t according to the relation $E=1.4340\bigl[1-0.000814(t-15) + 0.000007(t-15)^2\bigr]$ volts.

Find the change of electromotive-force per degree, at $$15^\circ$$, $$20^\circ$$ and $$25^\circ$$.