133 lines
3.7 KiB
TeX
133 lines
3.7 KiB
TeX
\documentclass{report}
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\input{preamble}
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\input{macros}
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\input{letterfonts}
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\title{\Huge{Calculus III}}
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\author{\huge{Krishna Ayyalasomayajula}}
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\date{}
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\begin{document}
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\maketitle
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\newpage% or \cleardoublepage
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% \pdfbookmark[<level>]{<title>}{<dest>}
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\pdfbookmark[section]{\contentsname}{toc}
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\tableofcontents
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\pagebreak
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\chapter{Lab 7}
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\section{Work}
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\qs{}{
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\begin{align*}
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\vec{r} \in \mathbb{R} \therefore t \ge 0 \because e^{\sqrt{t}}\notin\mathbb{R} \; t|t<0 \\
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\text{This corresponds with answer choice D}
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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\forall t>4, \quad \vec{r}\cdot\hat{j}\notin\mathbb{R} \\
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\ln(t-1) \text{ is not defined } \forall t\le1 \\
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\text{This corresponds with answer choice D} \\
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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\vec{r}\cdot\hat{j}\in[-4,4] \quad \land \quad \vec{r}\cdot\hat{i}\in[-3,3] \\
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\frac{\mathrm{d}\vec{r}}{\mathrm{d}t}=\langle -3\sin t ,4\cos t \rangle \\
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\frac{\mathrm{d}\vec{r}}{\mathrm{d}t} \right |_{t=\tfrac{\pi}{2}}= \langle -3,0 \rangle \quad \vec{r}(0)=\langle 3,0 \rangle \\
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\text{This corresponds with answer choice B}
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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\text{This can be directly evaluated to: } \\
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\langle 28,-49 \rangle \\
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\text{This corresponds with answer choice D}
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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\vec{r}\cdot\hat{j} \text{ is not defined for } t=1 \text{ but the limit as $t\to6$ can be evaluated without affecting the proceess} \\
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\vec{r}\cdot\hat{i} \text{ is not defined for } t=1 \text{ but the limit as $t\to6$ can be evaluated without affecting the proceess} \\
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\text{Using directevaluation: } \\
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\langle \frac{5}{35},-\frac{36+12-3}{5} \rangle = \langle \tfrac{1}{7},-9 \rangle \\
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\text{This corresponds with answer choice B}
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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\implies \langle 0,-6 \rangle \\
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\text{This corresponds with answer choice B}
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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e^{-\ln 6}=6^{-1}=\tfrac{1}{6} \\
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\implies \lim_{t\to\ln 6}{\langle 6e^{-t}, 3e^{-t} \rangle}=\langle 1,\tfrac{1}{2} \rangle \\
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\text{This corresponds witha nswer choice B}
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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\frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle -14t,\tfrac{1}{3}t^2 \rangle \\
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\text{This corresponds with answer choice C}
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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\frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle -\csc^2 t, -\cot t\csc t \rangle \\
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\text{This corresponds with answer choice A}
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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\frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle 8te^{t^2}, -3, 2t \rangle \\
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\text{This corresponds with answer choice B}
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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\frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle 18\frac{1}{6t},6t^2 \rangle \\
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\frac{\mathrm{d}^2\vec{r}}{\mathrm{d}t^2} = \langle \frac{-108}{36t^2}, 12t \rangle \langle \frac{-3}{t^2}, 12t \rangle \\
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\text{This corresponds with answer choice C}
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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\frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle 15t^4,-60t^4,20t^4 \rangle \\
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\| \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} \| \\
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&= \sqrt{15^2\cdot t^8 +60^2\cdot t^8 +20^2\cdot t^8} \\
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&= t^4 \sqrt{15^2+60^2+20^2} = 65t^4 \\
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\hat{T}=\langle \tfrac{15}{65},\tfrac{-60}{65},\tfrac{20}{65} \rangle \\
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\text{This corresponds with answer choice B}
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\end{align*}
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}
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\qs{}{
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\begin{align*}
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\frac{\mathrm{d}\vec{r}}{\mathrm{d}t} = \langle 36\sin^2(2t)\cos(2t), -36\cos^2(2t)\sin(2t) \rangle \\
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\| \frac{\mathrm{d}\vec{r}}{\mathrm{d}t} \| \\
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&= \sqrt{36^2 \sin^4(2t) \cos^2(2t)+36^2\cos^4(2t)\sin^2(2t)} \\
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&=\sqrt{(36^2 \sin^2(2t) \cos^2(2t))\cdot (\sin^2(2t)+cos^2(2t))} \\
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&=36\sin(2t)\cos(2t) \\
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&=18\sin(4t)
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\end{align*}
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}
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\end{document}
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