Page:Calculus Made Easy.pdf/281

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


 * (ii) $$\dfrac{dy}{d\theta} = 2\sin\theta \cos\theta = \sin2\theta$$ and $$\dfrac{dy}{d\theta} = 2\cos2\theta$$;


 * (iii) $$\dfrac{dy}{d\theta} = 3\sin^2 \theta \cos\theta$$ and $$\dfrac{dy}{d\theta} = 3\cos3\theta$$.

(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) $$\cos\theta \sin2\theta + 2\cos2\theta \sin\theta = 2\sin\theta\left(\cos^2 \theta + \cos2\theta\right)$$
 * $$ = 2\sin\theta\left(3\cos^2 \theta - 1\right)$$.

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

(10) $$\epsilon^x \left(\sin^2 x + \sin2x\right)$$; $$\epsilon^x \left(\sin^2 x + 2\sin2x + 2\cos2x\right)$$.

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