Questão 1 (0.5 ponto cada item) Considerando a trajetória parametrizada pela seguinte função vetorial:
\vec{r}(t)=t^{2} \vec{i}+\cos \left(t^{2}\right) \vec{j}-\operatorname{sen}\left(t^{2}\right) \vec{k}, \quad t \geq 0
está correto:
(A) tangente unitário
T(t)= :
( )
\frac{2 t \vec{i}-\operatorname{sen}\left(t^{2}\right) \vec{j}-\cos \left(t^{2}\right) \vec{k}}{\sqrt{1+4 t^{2}}}
( )
\frac{\vec{i}-\operatorname{sen}\left(t^{2}\right) \vec{j}-\cos \left(t^{2}\right) \vec{k}}{\sqrt{2}}
( )
\frac{\vec{i}+\operatorname{sen}\left(t^{2}\right) \vec{j}-\cos \left(t^{2}\right) \vec{k}}{\sqrt{2}}
( )
\frac{2 t \vec{i}+\operatorname{sen}\left(t^{2}\right) \vec{j}-\cos \left(t^{2}\right) \vec{k}}{\sqrt{1+4 t^{2}}}
( ) Nenhuma das anteriores
(B) aceleração
\vec{a}(t)=\frac{d^{2} \vec{r}}{d t^{2}}= :
( )
2 \vec{i}-\left(2 \operatorname{sen}\left(t^{2}\right)+4 t^{2} \cos \left(t^{2}\right)\right) \vec{j}+\left(4 t^{2} \operatorname{sen}\left(t^{2}\right)-2 \cos \left(t^{2}\right)\right) \vec{k}
( )
2 \vec{i}-\left(2 \operatorname{sen}\left(t^{2}\right)+2 t \cos \left(t^{2}\right)\right) \vec{j}+\left(2 t \operatorname{sen}\left(t^{2}\right)-2 \cos \left(t^{2}\right)\right) \vec{k}
( )
2 \vec{i}-2 t \cos \left(t^{2}\right) \vec{j}+2 t \operatorname{sen}\left(t^{2}\right) \vec{k}
( )
2 \vec{i}+\left(4 t^{2} \operatorname{sen}\left(t^{2}\right)-2 \cos \left(t^{2}\right)\right) \vec{j}-\left(4 t^{2} \cos \left(t^{2}\right)+2 \operatorname{sen}\left(t^{2}\right)\right) \vec{k}
( ) Nenhuma das anteriores
(C) vetor normal unitário
\vec{N}(t)= :
( )
\frac{\vec{i}+\operatorname{sen}\left(t^{2}\right) \vec{j}-\cos \left(t^{2}\right) \vec{k}}{\sqrt{2}}
( )
\operatorname{sen}\left(t^{2}\right) \vec{j}-\cos \left(t^{2}\right) \vec{k}
( )
-\cos \left(t^{2}\right) \vec{j}+\operatorname{sen}\left(t^{2}\right) \vec{k}
( )
\frac{\vec{i}-\cos \left(t^{2}\right) \vec{j}+\operatorname{sen}\left(t^{2}\right) \vec{k}}{\sqrt{2}}
( ) Nenhuma das anteriores
(D) vetor binormal
\vec{B}(t)= :
( )
\frac{t \vec{i}+\cos \left(t^{2}\right) \vec{j}+\operatorname{sen}\left(t^{2}\right) \vec{k}}{\sqrt{1+t^{2}}}
( )
\frac{\vec{i}+\cos \left(t^{2}\right) \vec{j}+\operatorname{sen}\left(t^{2}\right) \vec{k}}{\sqrt{2}}
( )
\frac{-\vec{i}-\operatorname{sen}\left(t^{2}\right) \vec{j}-\cos \left(t^{2}\right) \vec{k}}{\sqrt{2}}
( )
\frac{-t \vec{i}-\operatorname{sen}\left(t^{2}\right) \vec{j}-\cos \left(t^{2}\right) \vec{k}}{\sqrt{1+t^{2}}}
( ) Nenhuma das anteriores
(E) curvatura em
t=\sqrt{\pi} :
( )
\kappa(\sqrt{\pi})=\frac{1}{\sqrt{2}}
( )
\kappa(\sqrt{\pi})=\sqrt{2}
( )
\kappa(\sqrt{\pi})=2 \sqrt{2}
( )
\kappa(\sqrt{\pi})=2
( ) Nenhuma das anteriores
(F) torção em
t=\sqrt{\pi} :
( )
\tau(\sqrt{\pi})=2
( )
\tau(\sqrt{\pi})=\frac{1}{2}
( )
\tau(\sqrt{\pi})=\frac{1}{\sqrt{2}}
( )
\tau(\sqrt{\pi})=\sqrt{2}
( ) Nenhuma das anteriores
(G) aceleração tangencial em
t=\sqrt{\pi} :
( )
2 \sqrt{2}
( ) 0
( )
\sqrt{\pi}
( )
\frac{2}{\sqrt{\pi}}
( ) Nenhuma das anteriores
(H) aceleração normal em
t=\sqrt{\pi} :
( ) 0
( )
2 \sqrt{\pi}
( )
4 \pi
( )
4 \sqrt{\pi}
( ) Nenhuma das anteriores