Chapter 12 - Vector-Valued Functions - 12.4 Exercises - Page 848: 27

$$\eqalign{ & {\bf{T}}\left( {\frac{\pi }{2}} \right) = \frac{{{\bf{i}} + {\bf{j}}}}{{\sqrt 2 }} \cr & {\bf{N}}\left( {\frac{\pi }{2}} \right) = - \frac{1}{{\sqrt 2 }}{\bf{i}} + \frac{1}{{\sqrt 2 }}{\bf{j}} \cr & {a_{\bf{T}}} = \sqrt 2 {e^{\pi /2}} \cr & {a_{\bf{N}}} = \sqrt 2 {e^{\pi /2}} \cr} $$

$$\eqalign{ & {\bf{r}}\left( t \right) = {e^t}\cos t{\bf{i}} + {e^t}\sin t{\bf{j}},{\text{ }}t = \frac{\pi }{2} \cr & {\text{Calculate }}{\bf{v}}\left( t \right){\text{ and }}{\bf{a}}\left( t \right) \cr & {\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) \cr & {\bf{v}}\left( t \right) = \frac{d}{{dt}}\left[ {{e^t}\cos t{\bf{i}} + {e^t}\sin t{\bf{j}}} \right] \cr & {\bf{v}}\left( t \right) = \left( {{e^t}\cos t - {e^t}\sin t} \right){\bf{i}} + \left( {{e^t}\sin t + {e^t}\cos t} \right){\bf{j}} \cr & {\bf{a}}\left( t \right) = {\bf{v}}'\left( t \right) \cr & {\bf{a}}\left( t \right) = \frac{d}{{dt}}\left[ {\left( {{e^t}\cos t - {e^t}\sin t} \right){\bf{i}} + \left( {{e^t}\sin t + {e^t}\cos t} \right){\bf{j}}} \right] \cr & {\bf{a}}\left( t \right) = - 2{e^t}\sin t{\bf{i}} + 2{e^t}\cos t{\bf{j}} \cr & {\bf{a}}\left( {\frac{\pi }{2}} \right) = - 2{e^{\pi /2}}\sin \left( {\frac{\pi }{2}} \right){\bf{i}} + 2{e^{\pi /2}}\cos \left( {\frac{\pi }{2}} \right){\bf{j}} \cr & {\bf{a}}\left( {\frac{\pi }{2}} \right) = - 2{e^{\pi /2}}{\bf{i}} \cr & {\text{Find the unit Tangent Vector }}{\bf{T}}\left( t \right) \cr & {\bf{T}}\left( t \right) = \frac{{{\bf{v}}\left( t \right)}}{{\left\| {{\bf{v}}\left( t \right)} \right\|}},{\text{ }}{\bf{v}}\left( t \right) \ne 0 \cr & {\bf{T}}\left( t \right) = \frac{{\left( {{e^t}\cos t - {e^t}\sin t} \right){\bf{i}} + \left( {{e^t}\sin t + {e^t}\cos t} \right){\bf{j}}}}{{\sqrt {{{\left( {{e^t}\cos t - {e^t}\sin t} \right)}^2} + {{\left( {{e^t}\sin t + {e^t}\cos t} \right)}^2}} }} \cr & {\bf{T}}\left( t \right) = \frac{{\left( {{e^t}\cos t - {e^t}\sin t} \right){\bf{i}} + \left( {{e^t}\sin t + {e^t}\cos t} \right){\bf{j}}}}{{\sqrt {{e^{2t}} - 2{e^{2t}}\sin t\cos t + {e^{2t}} + 2{e^{2t}}\sin t\cos t} }} \cr & {\bf{T}}\left( t \right) = \frac{{\left( {{e^t}\cos t - {e^t}\sin t} \right){\bf{i}} + \left( {{e^t}\sin t + {e^t}\cos t} \right){\bf{j}}}}{{\sqrt {2{e^{2t}}} }} \cr & {\bf{T}}\left( t \right) = \frac{{\left( {{e^t}\cos t - {e^t}\sin t} \right){\bf{i}} + \left( {{e^t}\sin t + {e^t}\cos t} \right){\bf{j}}}}{{\sqrt 2 {e^t}}} \cr & {\bf{T}}\left( {\frac{\pi }{2}} \right) = \frac{{{\bf{i}} + {\bf{j}}}}{{\sqrt 2 }} \cr & {\text{Find }}{\bf{T}}'\left( t \right) \cr & {\bf{T}}'\left( t \right) = \frac{d}{{dt}}\left[ {\frac{{\left( {{e^t}\cos t - {e^t}\sin t} \right){\bf{i}} + \left( {{e^t}\sin t + {e^t}\cos t} \right){\bf{j}}}}{{\sqrt 2 {e^t}}}} \right] \cr & {\text{Differentiate by using a CAS }}\left( {{\text{WolframWebsite}}} \right) \cr & {\bf{T}}'\left( t \right) = - \frac{{\sin t + \cos t}}{{\sqrt 2 }}{\bf{i}} + \frac{{\sin t + \cos t}}{{\sqrt 2 }}{\bf{j}} \cr & {\text{Evaluate }}{\bf{T}}'\left( t \right){\text{ at }}t = \frac{\pi }{2} \cr & {\bf{T}}'\left( {\frac{\pi }{2}} \right) = - \frac{1}{{\sqrt 2 }}{\bf{i}} + \frac{1}{{\sqrt 2 }}{\bf{j}} \cr & {\text{Finding the Principal Unit Normal Vector}} \cr & {\bf{N}}\left( t \right) = \frac{{{\bf{T}}'\left( t \right)}}{{\left\| {{\bf{T}}'\left( t \right)} \right\|}} \cr & {\text{Evaluate }}{\bf{N}}'\left( t \right){\text{ at }}t = \frac{\pi }{2} \cr & {\bf{N}}\left( {\frac{\pi }{2}} \right) = \frac{{{\bf{T}}'\left( {\pi /2} \right)}}{{\left\| {{\bf{T}}'\left( {\pi /2} \right)} \right\|}} = \frac{{ - \frac{1}{{\sqrt 2 }}{\bf{i}} + \frac{1}{{\sqrt 2 }}{\bf{j}}}}{{\left\| { - \frac{1}{{\sqrt 2 }}{\bf{i}} + \frac{1}{{\sqrt 2 }}{\bf{j}}} \right\|}} \cr & {\bf{N}}\left( {\frac{\pi }{2}} \right) = - \frac{1}{{\sqrt 2 }}{\bf{i}} + \frac{1}{{\sqrt 2 }}{\bf{j}} \cr & {\text{Find }}{a_{\bf{T}}}{\text{ and }}{a_{\bf{N}}}{\text{ at }}t = \frac{\pi }{2} \cr & {a_{\bf{T}}} = {\bf{a}} \cdot {\bf{T}} = \left( { - 2{e^{\pi /2}}{\bf{i}}} \right)\left( { - \frac{1}{{\sqrt 2 }}{\bf{i}} + \frac{1}{{\sqrt 2 }}{\bf{j}}} \right) \cr & {a_{\bf{T}}} = \sqrt 2 {e^{\pi /2}} \cr & {a_{\bf{N}}} = {\bf{a}} \cdot {\bf{N}} = \left( { - 2{e^{\pi /2}}{\bf{i}}} \right)\left( { - \frac{1}{{\sqrt 2 }}{\bf{i}} + \frac{1}{{\sqrt 2 }}{\bf{j}}} \right) \cr & {a_{\bf{N}}} = \sqrt 2 {e^{\pi /2}} \cr & \cr & {\text{Summary}} \cr & {\bf{T}}\left( {\frac{\pi }{2}} \right) = \frac{{{\bf{i}} + {\bf{j}}}}{{\sqrt 2 }} \cr & {\bf{N}}\left( {\frac{\pi }{2}} \right) = - \frac{1}{{\sqrt 2 }}{\bf{i}} + \frac{1}{{\sqrt 2 }}{\bf{j}} \cr & {a_{\bf{T}}} = \sqrt 2 {e^{\pi /2}} \cr & {a_{\bf{N}}} = \sqrt 2 {e^{\pi /2}} \cr} $$