Page:Calculus Made Easy.pdf/53

 Exercises II. (See p. 254 for Answers.)

Differentiate the following:

Make up some other examples for yourself, and try your hand at differentiating them.

(7) If $$l_t$$ and $$l_0$$ be the lengths of a rod of iron at the temperatures $$t^\circ$$ C. and $$0^\circ$$ C. respectively, then $$l_t=l_0(1+0.000012t)$$. Find the change of length of the rod per degree Centigrade.

(8) It has been found that if $$c$$ be the candle power of an incandescent electric lamp, and $$V$$ be the voltage, $$c=aV^b$$, where $$a$$ and $$b$$ are constants.

Find the rate of change of the candle power with the voltage, and calculate the change of candle power per volt at $$80$$, $$100$$ and $$120$$ volts in the case of a lamp for which $$a=0.5\times10^{-10}$$ and $$b=6$$.

(9) The frequency $$n$$ of vibration of a string of diameter $$D$$, length $$L$$ and specific gravity $$\sigma$$, stretched with a force $$T$$, is given by

$n=\frac{1}{DL}\sqrt{\frac{gT}{\pi \sigma}}$.|undefined

Find the rate of change of the frequency when $$D$$, $$L$$, $$\sigma$$ and $$T$$ are varied singly.