Page:Calculus Made Easy.pdf/53

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

Differentiate the following:

(1) $y=ax^{3}+6$.	(2) $y=13x^{\tfrac {3}{2}}-c$.
(3) $y=12x^{\tfrac {1}{2}}+c^{\tfrac {1}{2}}$.	(4) $y=c^{\tfrac {1}{2}}x^{\tfrac {1}{2}}$.
(5) $u={\frac {az^{n}-1}{c}}$.	(6) $y=1.18t^{2}+22.4$.

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 $a=0.5\times 10^{-10}$ 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

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

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