*Q1* (a) The curve spirals as it converges to the origin. ![[Pasted image 20240124223538.png|500]] (b) Velocity: $\underline{r}'(t)=(-e^{-t}(\sin t+\cos t),e^{-t}(\cos t-\sin t))$. Hence its speed is given by $||\underline{r}'(t)||=\sqrt{ 2 } e^{-t}$. So arc length $=\int_{0}^{\infty } \sqrt{ 2 }e^{-u} \, du=\lim_{ n \to \infty }[-\sqrt{ 2 } e^{-n} + \sqrt{ 2 } ]= \sqrt{ 2 }.$ (c) Arc length function: $s(t)=\int_{0}^{t} \sqrt{ 2 } e^{-u } \, du=\sqrt{ 2 }- \sqrt{ 2 } e^{-t}$. Now $t=-\ln\left( 1 - \frac{\sqrt{ 2 }}{2} s \right)$ so the arc-length parametrisation of the curve is given by $\underline{r}(s)=\left(\left( 1 - \frac{\sqrt{ 2 }}{2} s \right)\cos\left(\ln\left( 1 - \frac{\sqrt{ 2 }}{2} s \right) \right) ,\; \left( \frac{\sqrt{ 2 }}{2} s -1\right)\sin\left(\ln\left( 1 - \frac{\sqrt{ 2 }}{2} s \right) \right) \right), \;s\in(0,\sqrt{ 2 }).$ (d) $\underline{r}'(t)=(-e^{-t}(\sin t +\cos t), e^{-t}(\cos t-\sin t) )$. Suppose $\underline{r}'(t)=\underline{0}$, equating the first component to zero gives $\tan t=-1$ while equating the second component to zero gives $\tan t=1$ which leads to a contradiction. Hence $\mathcal{C}$ is regular since $\underline{r}(t)$ is a regular parametrisation. *Q2* (a) The curve is parametrised by $\underline{r}(t)=\left( \frac{\sinh t}{9}, \cosh t \right), \; t\in\mathbb{R}$ The derivative is given by $\underline{r}'(t)=\left( \frac{\cosh t}{9} , \sinh t \right)\neq \underline{0}$ since $\cosh t\geq 1$ so this is a regular parametrisation of the curve. (b) $y^{2}-9x^{2}=1\implies \frac{y^2}{x^{2}}-9=\frac{1}{x^2}$ so when $|x|$ is sufficiently large, $y\approx \pm 3x$ since $\frac{1}{x^{2}} \approx0$. ![[Pasted image 20240124231920.png|500]] (c) Let $u(t)= \frac{1}{\pi} \tan^{-1}(t) + \frac{1}{2}$ then $0<u<1$ and $t=\tan (\pi \left( u-\frac{1}{2} \right))$ so the curve is parametrised by $\underline{r}(u)=\left( \frac{\sinh\left( \tan\left( \pi\left( u- \frac{1}{2} \right) \right) \right)}{9}, \cosh \left( \tan\left( \pi\left( u - \frac{1}{2} \right) \right) \right) \right), \;u\in(0,1).$ *Q3* (a) ![[Pasted image 20240125010327.png]] ![[Pasted image 20240125010056.png]] ![[Pasted image 20240125010115.png]] ![[Pasted image 20240125010403.png]] ![[Pasted image 20240125010146.png]] (b) The curve is not closed if $r$ is irrational.