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-119,13 +119,48 @@ } \qs{}{ + \begin{align*} - \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle 36\sin^2(2t)\cos(2t), -36\cos^2(2t)\sin(2t) \rangle \\ - \| \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} \| \\ - &= \sqrt{36^2 \sin^4(2t) \cos^2(2t)+36^2\cos^4(2t)\sin^2(2t)} \\ - &=\sqrt{(36^2 \sin^2(2t) \cos^2(2t))\cdot (\sin^2(2t)+cos^2(2t))} \\ - &=36\sin(2t)\cos(2t) \\ - &=18\sin(4t) + \vec{r}(t) &= 6\sin^3(2t)\,\hat{i} + 6\cos^3(2t)\,\hat{j} \\[6pt] + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} + &= \langle 36\sin^2(2t)\cos(2t),\; -36\cos^2(2t)\sin(2t) \rangle \\[6pt] + &= 36\sin(2t)\cos(2t)\,(\sin(2t)\,\hat{i} - \cos(2t)\,\hat{j}) + \end{align*} + + \begin{align*} + \left\|\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}\right\| + &= \sqrt{(36\sin^2(2t)\cos(2t))^2 + (-36\cos^2(2t)\sin(2t))^2} \\[6pt] + &= 36\sqrt{\sin^4(2t)\cos^2(2t) + \cos^4(2t)\sin^2(2t)} \\[6pt] + &= 36\sqrt{\sin^2(2t)\cos^2(2t)} \\[6pt] + &= 36\,|\sin(2t)\cos(2t)| \\[6pt] + &= 18\,|\sin(4t)| + \end{align*} + + \begin{align*} + \hat{T}(t) + &= \frac{\dfrac{d\vec r}{dt}}{\left\|\dfrac{d\vec r}{dt}\right\|} + = \frac{36\sin(2t)\cos(2t)(\sin(2t)\,\hat{i} - \cos(2t)\,\hat{j})}{36|\sin(2t)\cos(2t)|} \\[6pt] + &= \sin(2t)\,\hat{i} - \cos(2t)\,\hat{j} + \end{align*} + + \text{This corresponds with answer choice B.} +} + +\qs{}{ + Let $\vec{C}$ encode the components of the constant of integration such that $\vec{C}\in\mathbb{R}^3$ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle t^4+5t^2,4t \rangle \\ + \int\mathrm{d}\vec{r}= \int \langle t^4+5t^2,4t \rangle \mathrm{d}t \\ + \vec{r}(t)&=\langle \tfrac{1}{5}t^5+\tfrac{5}{3}t^3,2t^2 \rangle + \vec{C}\\ + \vec{C}&=\vec{r}(0)=\langle 5,-6 \rangle + \end{align*} + This corresponds with answer choice C. +} + +\qs{}{ + \begin{align*} + \implies \int_0^3{\left \langle \frac{2}{\sqrt{1+t}},-9t^2,\frac{6}{(1+t^2)^2}\right \rangle \mathrm{d}t }\\ + \implies \int_0^3{\left \langle 2(1+t)^{-\tfrac{1}{2}},-9t^2,\frac{6}{(1+t^2)^2}\right \rangle \mathrm{d}t } \end{align*} }