$D _ { t } [ 2 \mathbf { r } ( t ) - 5 \mathbf { u } ( t ) ] \text { given the following }$ Use the properties of the derivative to find vector-valued functions. $$\begin{array} { l }
\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t ^ { 3 } \mathbf { j } + 2 t ^ { 2 } \mathbf { k } \
\mathbf { u } ( t ) = 2 \mathbf { i } + 4 t ^ { 2 } \mathbf { j } + 3 t ^ { 3 } \mathbf { k }
\end{array}$$
A) $4 \mathbf { i } + \left( 36 t ^ { 3 } - 40 t \right) \mathbf { j } + \left( 8 t - 45 t ^ { 2 } \right) \mathbf { k }$
B) $4 \mathbf { i } + \left( 36 t ^ { 2 } - 40 t \right) \mathbf { j } + \left( 8 t ^ { 2 } - 45 t \right) \mathbf { k }$
C) $4 t \mathbf { i } + \left( 36 t ^ { 2 } + 40 t \right) \mathbf { j } - \left( 8 t + 45 t ^ { 2 } \right) \mathbf { k }$
D) $4 \mathbf { i } + \left( 36 t ^ { 2 } + 40 t \right) \mathbf { j } + \left( 8 t + 45 t ^ { 2 } \right) \mathbf { k }$
E) $4 \mathbf { i } + \left( 36 t ^ { 2 } - 40 t \right) \mathbf { j } + \left( 8 t - 45 t ^ { 2 } \right) \mathbf { k }$

Question

$D _ { t } [ 2 \mathbf { r } ( t ) - 5 \mathbf { u } ( t ) ] \text { given the following }$ Use the properties of the derivative to find vector-valued functions. $$\begin{array} { l } 
\mathbf { r } ( t ) = 2 t \mathbf { i } + 6 t ^ { 3 } \mathbf { j } + 2 t ^ { 2 } \mathbf { k } \
\mathbf { u } ( t ) = 2 \mathbf { i } + 4 t ^ { 2 } \mathbf { j } + 3 t ^ { 3 } \mathbf { k }
\end{array}$$
A) $4 \mathbf { i } + \left( 36 t ^ { 3 } - 40 t \right) \mathbf { j } + \left( 8 t - 45 t ^ { 2 } \right) \mathbf { k }$
B) $4 \mathbf { i } + \left( 36 t ^ { 2 } - 40 t \right) \mathbf { j } + \left( 8 t ^ { 2 } - 45 t \right) \mathbf { k }$
C) $4 t \mathbf { i } + \left( 36 t ^ { 2 } + 40 t \right) \mathbf { j } - \left( 8 t + 45 t ^ { 2 } \right) \mathbf { k }$ 
D) $4 \mathbf { i } + \left( 36 t ^ { 2 } + 40 t \right) \mathbf { j } + \left( 8 t + 45 t ^ { 2 } \right) \mathbf { k }$
E) $4 \mathbf { i } + \left( 36 t ^ { 2 } - 40 t \right) \mathbf { j } + \left( 8 t - 45 t ^ { 2 } \right) \mathbf { k }$

Quizplus · Accepted Answer

First, calculate $2\mathbf{r}(t) - 5\mathbf{u}(t)$, then differentiate the result. For $2\mathbf{r}(t)$, you get $4t\mathbf{i} + 12t^3\mathbf{j} + 4t^2\mathbf{k}$, and for $5\mathbf{u}(t)$, you get $10\mathbf{i} + 20t^2\mathbf{j} + 15t^3\mathbf{k}$. Subtracting these gives $4\mathbf{i} + (12t^3 - 20t^2)\mathbf{j} + (4t^2 - 15t^3)\mathbf{k}$. Differentiating this expression yields $4\mathbf{i} + (36t^2 - 40t)\mathbf{j} + (8t - 45t^2)\mathbf{k}$.

Dt[2r(t)−5u(t)] given the following D _ { t } [ 2 \mathbf { r } ( t ) - 5 \mathbf { u } ( t ) ] \text { given the following }Dt​[2r(t)−5u(t)] given the following

$D _ { t } [ 2 \mathbf { r } ( t ) - 5 \mathbf { u } ( t ) ] \text { given the following }$