<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass[a4paper,12pt]{article}
\newcommand{\un}{\underline}
\usepackage{epsfig}
\begin{document}
\parindent=0pt

\begin{center}
\textbf{Vector Functions and Curves}

\textit{\textbf{One variable functions}}
\end{center}

\textbf{Question}

Find the velocity, speed and acceleration of the particle with
position given by $\un{r}(t)$ at time
$t$. Also determine the particles path.

$$\underline{r}= e^{-t}\cos(e^t)\underline{i}+
e^{-t}\sin(e^t)\underline{j}- e^t\underline{k}$$

\textbf{Answer}

Position: $\underline{r}=e^{-t}\cos(e^t)\underline{i}
+e^{-t}\sin(e^t)\underline{j} -e^t\underline{k}$

Velocity: $\underline{v}=-(e^{-t}\cos(e^t)+ \sin(e^t))\underline{i}
-(e^{-t}\sin(e^t) -\cos(e^t))\underline{j} -e^t\underline{k}$

Speed: $\it{v}=\sqrt{1+e^{-2t}+e^{2t}}$

Acceleration:
\begin{eqnarray*}
\underline{a} &amp; = &amp; ((e^{-t}-e^t)\cos(e^t) +\sin(e^t))\underline{i}\\
&amp; &amp; +((e^{-t}-e^t)\sin(e^t)-\cos(e^t))\underline{j}\\
&amp; &amp;  -e^t\underline{k}
\end{eqnarray*}

Path: a spiral on the surface $z\sqrt{x^2+y^2}=-1$.

\end{document}
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