Page:Calculus Made Easy.pdf/281

Exercises XIV. (p. 173.)

(1) (i) ${\dfrac {dy}{d\theta }}=A\cos \left(\theta -{\dfrac {\pi }{2}}\right)$;

(ii) ${\dfrac {dy}{d\theta }}=2\sin \theta \cos \theta =\sin 2\theta$ and ${\dfrac {dy}{d\theta }}=2\cos 2\theta$;

(iii) ${\dfrac {dy}{d\theta }}=3\sin ^{2}\theta \cos \theta$ and ${\dfrac {dy}{d\theta }}=3\cos 3\theta$.

(2) $\theta =45^{\circ }$ or ${\dfrac {\pi }{4}}$ radians.

(3) ${\dfrac {dy}{dt}}=-n\sin 2\pi nt$.

(4) $a^{x}\log _{\epsilon }a\cos a^{x}$.

(5) ${\dfrac {\cos x}{\sin x}}={\text{cotan}}\;x$.

(6) $18.2\cos \left(x+26^{\circ }\right)$.

(7) The slop is ${\dfrac {dy}{d\theta }}=100\cos \left(\theta -15^{\circ }\right)$, which is a maximum when $(\theta -15^{\circ })=0$ or $\theta =15^{\circ }$; the value of the slope being then $=100$. When $\theta =75^{\circ }$ the slope is $100\cos(75^{\circ }-15^{\circ })=100\cos 60^{\circ }=100\times {\frac {1}{2}}=50$.

(8) $\cos \theta \sin 2\theta +2\cos 2\theta \sin \theta =2\sin \theta \left(\cos ^{2}\theta +\cos 2\theta \right)$

$=2\sin \theta \left(3\cos ^{2}\theta -1\right)$.

(9) $amn\theta ^{n-1}\tan ^{m-1}\left(\theta ^{n}\right)\sec ^{2}\theta ^{n}$.

(10) $\epsilon ^{x}\left(\sin ^{2}x+\sin 2x\right)$; $\epsilon ^{x}\left(\sin ^{2}x+2\sin 2x+2\cos 2x\right)$.

(11) (i) ${\dfrac {dy}{dx}}={\dfrac {ab}{\left(x+b\right)^{2}}}$;

(ii) ${\dfrac {a}{b}}\epsilon ^{-{\frac {x}{b}}}$;

(iii) ${\dfrac {1}{90^{\circ }}}\times {\dfrac {ab}{\left(b^{2}+x^{2}\right)}}$.

(12) (i) ${\dfrac {dy}{dx}}=\sec x\tan x$;

(ii) ${\dfrac {dy}{dx}}=-{\dfrac {1}{\sqrt {1-x^{2}}}}$;

(iii) ${\dfrac {dy}{dx}}={\dfrac {1}{1+x^{2}}}$;

(iv) ${\dfrac {dy}{dx}}={\dfrac {1}{x{\sqrt {x^{2}-1}}}}$;

(v) ${\dfrac {dy}{dx}}={\dfrac {{\sqrt {3\sec x}}\left(3\sec ^{2}x-1\right)}{2}}$.

(13) ${\dfrac {dy}{d\theta }}=4.6\left(2\theta +3\right)^{1.3}\cos \left(2\theta +3\right)^{2.3}$.