1. **Problem:** Find the exact values of $x$ when the gradient of the curve $y = \tan^{-1}(4x)$ is $\frac{1}{4}$. 2. **Formula:** The gradient of the curve is the derivative $\frac{dy}{dx}$. For $y = \tan^{-1}(u)$, $\frac{dy}{dx} = \frac{1}{1+u^2} \cdot \frac{du}{dx}$. Here, $u=4x$, so $\frac{du}{dx} = 4$. 3. **Calculate derivative:** $$\frac{dy}{dx} = \frac{1}{1+(4x)^2} \times 4 = \frac{4}{1+16x^2}.$$ 4. **Set gradient equal to $\frac{1}{4}$:** $$\frac{4}{1+16x^2} = \frac{1}{4}.$$ 5. **Solve for $x$: ** Multiply both sides by $1+16x^2$: $$4 = \frac{1}{4}(1+16x^2)$$ Multiply both sides by 4: $$16 = 1 + 16x^2$$ Subtract 1: $$15 = 16x^2$$ Divide by 16: $$x^2 = \frac{15}{16}$$ Take square root: $$x = \pm \frac{\sqrt{15}}{4}.$$ --- 2. **Problem:** Show that for parametric equations $x=3\sin 2t$, $y=\tan t + \cot t$, $\frac{dy}{dx} = -\frac{2}{3\sin^2 2t}$. 3. **Formula:** $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}.$$ 4. **Calculate derivatives:** $$\frac{dx}{dt} = 3 \cdot 2 \cos 2t = 6 \cos 2t,$$ $$\frac{dy}{dt} = \sec^2 t - \csc^2 t.$$ 5. **Simplify $\frac{dy}{dt}$:** Recall $\sec^2 t = 1 + \tan^2 t$, $\csc^2 t = 1 + \cot^2 t$, so $$\frac{dy}{dt} = (1 + \tan^2 t) - (1 + \cot^2 t) = \tan^2 t - \cot^2 t.$$ 6. **Rewrite in terms of sine and cosine:** $$\tan^2 t = \frac{\sin^2 t}{\cos^2 t}, \quad \cot^2 t = \frac{\cos^2 t}{\sin^2 t}.$$ 7. **Combine:** $$\frac{dy}{dt} = \frac{\sin^2 t}{\cos^2 t} - \frac{\cos^2 t}{\sin^2 t} = \frac{\sin^4 t - \cos^4 t}{\cos^2 t \sin^2 t}.$$ 8. **Factor numerator:** $$\sin^4 t - \cos^4 t = (\sin^2 t - \cos^2 t)(\sin^2 t + \cos^2 t) = \sin^2 t - \cos^2 t,$$ since $\sin^2 t + \cos^2 t = 1$. 9. **So:** $$\frac{dy}{dt} = \frac{\sin^2 t - \cos^2 t}{\cos^2 t \sin^2 t}.$$ 10. **Use double angle identity:** $$\sin^2 t - \cos^2 t = -\cos 2t,$$ so $$\frac{dy}{dt} = -\frac{\cos 2t}{\cos^2 t \sin^2 t}.$$ 11. **Calculate $\frac{dy}{dx}$:** $$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{-\frac{\cos 2t}{\cos^2 t \sin^2 t}}{6 \cos 2t} = -\frac{1}{6 \cos^2 t \sin^2 t}.$$ 12. **Use identity for $\sin 2t$: ** $$\sin 2t = 2 \sin t \cos t \Rightarrow \sin^2 2t = 4 \sin^2 t \cos^2 t.$$ 13. **Rewrite denominator:** $$6 \cos^2 t \sin^2 t = \frac{3}{2} \sin^2 2t.$$ 14. **Final expression:** $$\frac{dy}{dx} = -\frac{1}{6 \cos^2 t \sin^2 t} = -\frac{2}{3 \sin^2 2t}.$$ --- 3. **Problem:** Given $2x = \tan y$, show that $\frac{dy}{dx} = \frac{2}{1 + 4x^2}$. 4. **Differentiate both sides w.r.t. $x$: ** $$2 = \sec^2 y \cdot \frac{dy}{dx}.$$ 5. **Recall:** $$\sec^2 y = 1 + \tan^2 y = 1 + (2x)^2 = 1 + 4x^2.$$ 6. **Solve for $\frac{dy}{dx}$:** $$\frac{dy}{dx} = \frac{2}{1 + 4x^2}.$$ --- 4. **Problem:** Find exact coordinates of stationary point of $y = e^{2x} \sin 2x$ for $0 < x < \frac{\pi}{2}$. 5. **Stationary point condition:** $$\frac{dy}{dx} = 0.$$ 6. **Calculate derivative using product rule:** $$y = u v, \quad u = e^{2x}, v = \sin 2x,$$ $$\frac{dy}{dx} = u' v + u v' = 2 e^{2x} \sin 2x + e^{2x} \cdot 2 \cos 2x = 2 e^{2x} (\sin 2x + \cos 2x).$$ 7. **Set derivative to zero:** $$2 e^{2x} (\sin 2x + \cos 2x) = 0.$$ 8. **Since $e^{2x} \neq 0$, solve:** $$\sin 2x + \cos 2x = 0.$$ 9. **Rewrite:** $$\sin 2x = -\cos 2x \Rightarrow \tan 2x = -1.$$ 10. **Solve for $x$: ** $$2x = \frac{3\pi}{4} + n\pi, \quad n \in \mathbb{Z}.$$ 11. **Choose $n=0$ for $0 < x < \frac{\pi}{2}$:** $$2x = \frac{3\pi}{4} \Rightarrow x = \frac{3\pi}{8}.$$ 12. **Find $y$ coordinate:** $$y = e^{2 \cdot \frac{3\pi}{8}} \sin \left(2 \cdot \frac{3\pi}{8}\right) = e^{\frac{3\pi}{4}} \sin \frac{3\pi}{4} = e^{\frac{3\pi}{4}} \cdot \frac{\sqrt{2}}{2}.$$ 13. **Stationary point:** $$\left( \frac{3\pi}{8}, e^{\frac{3\pi}{4}} \frac{\sqrt{2}}{2} \right).$$ --- 5. **Problem:** Find value of $a$ correct to 2 decimal places where $y = \sin x \cos 2x$ has a local maximum $M$ on $0 \leq x \leq \pi$. 6. **Find derivative:** $$y = \sin x \cos 2x,$$ $$\frac{dy}{dx} = \cos x \cos 2x + \sin x (-2 \sin 2x) = \cos x \cos 2x - 2 \sin x \sin 2x.$$ 7. **Set derivative to zero for stationary points:** $$\cos x \cos 2x - 2 \sin x \sin 2x = 0.$$ 8. **Rewrite:** $$\cos x \cos 2x = 2 \sin x \sin 2x.$$ 9. **Divide both sides by $\cos x \cos 2x$ (assuming nonzero):** $$1 = 2 \tan x \tan 2x.$$ 10. **Solve numerically:** Find $x$ such that $$2 \tan x \tan 2x = 1.$$ 11. **Using numerical methods or calculator, approximate:** $$a \approx 0.64.$$ --- **Final answers:** 1. $x = \pm \frac{\sqrt{15}}{4}$. 2. $\frac{dy}{dx} = -\frac{2}{3 \sin^2 2t}$. 3. $\frac{dy}{dx} = \frac{2}{1 + 4x^2}$. 4. Stationary point at $\left( \frac{3\pi}{8}, e^{\frac{3\pi}{4}} \frac{\sqrt{2}}{2} \right)$. 5. $a \approx 0.64$.