A particle moves along the curve \( \mathbf{r}=3 u \mathbf{i}+3 u^{2} \mathbf{j}+2 u^{3} \mathbf{k} \) in the direction corresponding to increasing \( u \) and with a constant speed of 6 . Find the velocity and acceleration of the particle when it is at the point \( (3,3,2) \).

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{:[r=3ui+3u^(2)j+2u^(3)k],[v=(du)/(dt)(3i+6uj+6u^(2)k)],[a=(d^(2)u)/(dt^(2))(3i+6uj+6u^(2)k)+((du)/(dt))^(2)(6j+12 uk)]:}Since u is increasing and the speed of the particle is 6 ,6=|v|=3(du)/(dt)sqrt(1+4u^(2)+4u^(4))=3(1+2u^(2))(du)/(dt). Thus (du)/(dt)=(2)/(1+2u^(2)), and(d^( ... See the full answer