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## Q. 4.1

Let $\mathbf{f}(t)=t \mathbf{i}+t^{3} \mathbf{j}, \mathbf{g}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}$ and $\mathbf{v}=2 \mathbf{i}-3 \mathbf{j}$. Calculate $(\mathbf{a})(\mathbf{f}+\mathbf{g})^{\prime}$,
(b) $(\mathbf{v} \cdot \mathbf{f})^{\prime}$, and (c) $(\mathbf{f} \cdot \mathbf{g})^{\prime}$.

## Verified Solution

(a) $(\mathbf{f}+\mathbf{g})^{\prime}=\mathbf{f}^{\prime}+\mathbf{g}^{\prime}=\left(\mathbf{i}+3 t^{2} \mathbf{j}\right)+[-(\sin t) \mathbf{i}+(\cos t) \mathbf{j}]$
$=(1-\sin t) \mathbf{i}+\left(3 t^{2}+\cos t\right) \mathbf{j}$
(b) $(\mathbf{v} \cdot \mathbf{f})^{\prime}=\mathbf{v} \cdot \mathbf{f}^{\prime}=(2 \mathbf{i}-3 \mathbf{j}) \cdot\left(\mathbf{i}+3 t^{2} \mathbf{j}\right)=2-9 t^{2}$.

(c) $(\mathbf{f} \cdot \mathbf{g})^{\prime}=\mathbf{f} \cdot \mathbf{g}^{\prime}+\mathbf{f}^{\prime} \cdot \mathbf{g}$

$=\left(t \mathbf{i}+t^{3} \mathbf{j}\right) \cdot[-(\sin t) \mathbf{i}+(\cos t) \mathbf{j}]+\left(\mathbf{i}+3 t^{2} \mathbf{j}\right) \cdot[(\cos t) \mathbf{i}+(\sin t) \mathbf{i}]$
$=-t \sin t+t^{3} \cos t+\cos t+3 t^{2} \sin t$
$=(\cos t)\left(t^{3}+1\right)+(\sin t)\left(3 t^{2}-t\right)$