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z;N!h;;vU~6u8gouy`H^d8Leg+>G|rNkC(}6#ja1vS6$O)x7M;l>_ep5d!DYdt!o)MM?c^m*=C!F%g%f_A@}oj);bMIXQWs|zmjE>SB_ z{{LV6`pS}L#Y;kGed%+JT6^-ucaLln%fckROHsb(@&$bL?v&}J&GDXlZtCT8SJ}_? z&Of*I{<*nP&)+R5e!u?nid&VRE>%W7wCR2)#cTX;WAC#FllHcoh6_xdExo8!|LAVZ y1QDSVjWH)2xGE>K*7)A*`;{!e>xuk}nL9G4zyHT1%VS||Xkfyns_N?R#svT>g?EYo diff --git a/labs-set-2/template.tex b/labs-set-2/template.tex index 4bbb8b1..cae2a0b 100644 --- a/labs-set-2/template.tex +++ b/labs-set-2/template.tex @@ -426,7 +426,123 @@ This corresponds with answer choice B. \section{Work} \qs{}{ - + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle 1,0, \frac{-\sin{t}}{\cos{t}} \rangle = \langle 1,0,-\tan{t} \rangle \\ + \|\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}\| = \sqrt{1+\tan^2{t}}=\sec{t} \\ + \vec{T}(t)=\cos{t}\cdot \langle 1,0,-\tan{t} \rangle = \langle \cos{t},0,-\sin{t} \rangle \\ + \implies \vec{T}\prime(t) = \langle -\sin{t},0,-\cos{t} \rangle \\ + \implies \|\vec{T}^\prime(t)\|=\sqrt{\sin^2{t}+\cos^2{t}}=1 \\ + \vec{N}(t)=\langle -\sin{t},0,-\cos{t} \rangle + \end{align*} + Corresponds with answer choice D. } +\qs{}{ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} &= \langle 0,2t,2 \rangle \\ + \Big\|\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}\Big\| &= 2\sqrt{t^2+1} \\ + \vec{T}(t) &= \frac{\langle 0,t,1 \rangle}{\sqrt{t^2+1}} \\[6pt] + \frac{\mathrm{d}\vec{T}}{\mathrm{d}t} &= + \left\langle + 0,\, + \frac{\mathrm{d}}{\mathrm{d}t}\Big(\frac{t}{\sqrt{t^2+1}}\Big),\, + \frac{\mathrm{d}}{\mathrm{d}t}\Big(\frac{1}{\sqrt{t^2+1}}\Big) + \right\rangle \\[6pt] + &= + \left\langle + 0,\, + (t^2+1)^{-1/2} - \frac{t^2}{(t^2+1)^{3/2}},\, + -\frac{t}{(t^2+1)^{3/2}} + \right\rangle \\[6pt] + &= + (t^2+1)^{-3/2}\langle 0,1,-t\rangle \\[6pt] + \Big\|\frac{\mathrm{d}\vec{T}}{\mathrm{d}t}\Big\| &= (t^2+1)^{-1} \\[6pt] + \vec{N}(t) &= + \frac{\frac{\mathrm{d}\vec{T}}{\mathrm{d}t}} + {\Big\|\frac{\mathrm{d}\vec{T}}{\mathrm{d}t}\Big\|} + = \frac{(t^2+1)^{-3/2}\langle 0,1,-t\rangle} + {(t^2+1)^{-1}} + = \frac{\langle 0,1,-t\rangle}{\sqrt{t^2+1}}. + \end{align*} + This corresponds with answer choice B. +} + +\qs{}{ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} &= \langle 2t\cos{t}+\cancel{2\sin{t}-2\sin{t}},0,\cancel{2\cos{t}}-2t\sin{t}-\cancel{2\cos{t}} \rangle \\ + \Big\|\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}\Big\| &= 2t\sqrt{\cos^2{t}+\sin^2{t}}=2t \\ + \vec{T}(t) &= \langle \cos{t},0,-\sin{t} \rangle \\ + \frac{\mathrm{d}\vec{T}}{\mathrm{d}t} &= \langle -\sin{t},0,-\cos{t} \rangle \\ + \Big\|\frac{\mathrm{d}\vec{T}}{\mathrm{d}t}\Big\| &= 1 \\ + \vec{N}(t) &= \langle -\sin{t},0,-\cos{t} \rangle + \end{align*} + This corresponds with answer choice C. +} +\newpage +\qs{}{ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} &= \langle 4\cos{2t},-4\sin{2t},3 \rangle \\ + \Big\|\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}\Big\| &= \sqrt{4^2(\cos^2{2t}+\sin^2{2t})+3^2}=5 \\ + \vec{T}(t) &= \langle \tfrac{4}{5}\cos{2t},-\tfrac{4}{5}\sin{2t},\tfrac{3}{5} \rangle \\ + \frac{\mathrm{d}\vec{T}}{\mathrm{d}t} &= \langle -\tfrac{8}{5}\sin{2t},-\tfrac{8}{5}\cos{2t},0\rangle \\ + \Big\|\frac{\mathrm{d}\vec{T}}{\mathrm{d}t}\Big\| &= \frac{8}{5} \\ + \vec{N}(t) &= \langle -\sin{2t},-\cos{2t},0 \rangle + \end{align*} + This corresponds with answer choice C. +} + +\qs{}{ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} &= \langle 6\cos{t},-6\sin{t},-1 \rangle \\ + \Big\|\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}\Big\| &= \sqrt{6^2+1}=\sqrt{37} \\ + \vec{T}(t) &=\frac{1}{\sqrt{37}} \langle 6\cos{t},-6\sin{t},-1 \rangle \\ + \frac{\mathrm{d}\vec{T}}{\mathrm{d}t} &= \frac{1}{\sqrt{37}}\langle -6\sin{t},-6\cos{t},0 \rangle \\ + \Big\|\frac{\mathrm{d}\vec{T}}{\mathrm{d}t}\Big\| &= \sqrt{\frac{6^2}{\sqrt{37}^2}(\sin^2{t}+\cos^2{t})} + = \frac{6}{\sqrt{37}} \\ + \vec{N}(t) &= \frac{1}{\sqrt{37}}\langle -\sin{t},-\cos{t},0 \rangle \\ + \vec{B}(t)&=\vec{T}(t)\times\vec{N}(t) = \begin{vmatrix} + \hat{i} & \hat{j} & \hat{k} \\ + \tfrac{6}{\sqrt{37}}\cos{t} & \tfrac{-6}{\sqrt{37}}\sin{t} & \tfrac{-1}{\sqrt{37}} \\ + \tfrac{-1}{\sqrt{37}}\sin{t} & \tfrac{-1}{\sqrt{37}}\cos{t} & 0 + \end{vmatrix} \\[6pt] + \vec{B}(t)&=\frac{-\langle \cos{t},-\sin{t},6 \rangle }{\sqrt{37}} + \end{align*} + This corresponds with answer choice D. +} + +\qs{}{ + \begin{align*} + \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} &= \langle t,-7,0 \rangle \\ + \Big\|\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}\Big\| &= \sqrt{t^2+49} \\ + \vec{T}(t) &= \langle \frac{t}{\sqrt{t^2+49}},\frac{-7}{\sqrt{t^2+49}},0 \rangle \\ + \frac{\mathrm{d}\vec{T}}{\mathrm{d}t} &= + \left\langle + \frac{\sqrt{t^2+49}-t\cdot \frac{t}{\sqrt{t^2+49}}}{t^2+49},\, + \frac{0 - (-7)\cdot \frac{t}{\sqrt{t^2+49}}}{t^2+49},\, + 0 + \right\rangle \\ + &= \left\langle + \frac{49}{(t^2+49)^{3/2}},\, + \frac{7t}{(t^2+49)^{3/2}},\, + 0 + \right\rangle \\ + \Big\|\frac{\mathrm{d}\vec{T}}{\mathrm{d}t}\Big\| &= \frac{\sqrt{49^2 + (7t)^2}}{(t^2+49)^{3/2}} + = \frac{7\sqrt{t^2+49}}{(t^2+49)^{3/2}} + = \frac{7}{t^2+49} \\ + \vec{N}(t) &= \frac{\frac{\mathrm{d}\vec{T}}{\mathrm{d}t}}{\Big\|\frac{\mathrm{d}\vec{T}}{\mathrm{d}t}\Big\|} + = \frac{\langle 49,7t,0\rangle/(t^2+49)^{3/2}}{7/(t^2+49)} + = \frac{\langle 7, t, 0 \rangle}{\sqrt{t^2+49}} \\ + \vec{B}(t) &= \vec{T}(t) \times \vec{N}(t) + = \begin{vmatrix} + \hat{i} & \hat{j} & \hat{k} \\ + t/\sqrt{t^2+49} & -7/\sqrt{t^2+49} & 0 \\ + 7/\sqrt{t^2+49} & t/\sqrt{t^2+49} & 0 + \end{vmatrix} \\ + &= \frac{1}{t^2+49} \langle 0,0, t^2+49 \rangle + = \langle 0,0,1 \rangle + \end{align*} + This corresponds with answer choice D. +} + + \end{document}