## Calculus (3rd Edition)

The first positive value of ${t_0}$ is ${t_0} = \frac{{3\pi }}{4}$. The speed at ${t_0} = \frac{{3\pi }}{4}$ is $\frac{{ds}}{{dt}}{|_{t = 3\pi /4}} = {{\rm{e}}^{ - 3\pi /4}}\sqrt 2$
Since $c\left( t \right) = \left( {{{\rm{e}}^{ - t}}\cos t,{{\rm{e}}^{ - t}}\sin t} \right)$, we have $x\left( t \right) = {{\rm{e}}^{ - t}}\cos t$, ${\ \ }$ $x'\left( t \right) = - {{\rm{e}}^{ - t}}\cos t - {{\rm{e}}^{ - t}}\sin t$, $y\left( t \right) = {{\rm{e}}^{ - t}}\sin t$, ${\ \ }$ $y'\left( t \right) = - {{\rm{e}}^{ - t}}\sin t + {{\rm{e}}^{ - t}}\cos t$. Using Eq. (8) of Section 12.1, the slope of the tangent line is $\frac{{dy}}{{dx}} = \frac{{y'\left( t \right)}}{{x'\left( t \right)}} = \frac{{ - {{\rm{e}}^{ - t}}\sin t + {{\rm{e}}^{ - t}}\cos t}}{{ - {{\rm{e}}^{ - t}}\cos t - {{\rm{e}}^{ - t}}\sin t}} = \frac{{ - \sin t + \cos t}}{{ - \cos t - \sin t}}$ $\frac{{dy}}{{dx}} = \frac{{\sin t - \cos t}}{{\sin t + \cos t}}$ The tangent line is vertical if $\frac{{dy}}{{dx}}$ is infinite. This occurs when $\sin t + \cos t = 0$. So, $\tan t = - 1$. The solutions are $t = - \frac{\pi }{4} + \pi n$, for $n = 0, \pm 1, \pm 2, \pm 3,...$ The first positive value of ${t_0}$ is ${t_0} = \frac{{3\pi }}{4}$. By Theorem 2 of Section 12.2, the speed along $c\left(t\right)$ is $\frac{{ds}}{{dt}} = \sqrt {{{\left( { - {{\rm{e}}^{ - t}}\cos t - {{\rm{e}}^{ - t}}\sin t} \right)}^2} + {{\left( { - {{\rm{e}}^{ - t}}\sin t + {{\rm{e}}^{ - t}}\cos t} \right)}^2}}$ $\frac{{ds}}{{dt}} = \sqrt {{{\rm{e}}^{ - 2t}}{{\left( { - \cos t - \sin t} \right)}^2} + {{\rm{e}}^{ - 2t}}{{\left( { - \sin t + \cos t} \right)}^2}}$ $\frac{{ds}}{{dt}} = {{\rm{e}}^{ - t}}\sqrt {{{\cos }^2}t + 2\cos t\sin t + {{\sin }^2}t + {{\sin }^2}t - 2\sin t\cos t + {{\cos }^2}t}$ $\frac{{ds}}{{dt}} = {{\rm{e}}^{ - t}}\sqrt 2$ The speed of the curve at ${t_0} = \frac{{3\pi }}{4}$ is $\frac{{ds}}{{dt}}{|_{t = 3\pi /4}} = {{\rm{e}}^{ - 3\pi /4}}\sqrt 2$