\hfill \thepage} %} \input{tcilatex} \begin{document} \subsection*{Ma116 \hspace{2.25in} Exam I \hfill 2/8/00} \subsection*{{\protect\normalsize Name: \protect\underline{\hspace{2.5in}} \hfill ID: \protect\underline{\hspace{2.0in}}}} \subsection*{{\protect\normalsize E-Mail: \protect\underline{\hspace{2.5in}} \hfill Lecturer: \protect\underline{\hspace{2.0in}}}} \hfill {\small \textit{I pledge my honor that I have abided by the Stevens Honor System.}\hspace{2pt}\hfill\underline{\hspace{2.0in}}} \hfill {\small \textbf{SHOW ALL WORK! You may not use a calculator on this exam.}} \hfill \begin{description} \item[1. ] Consider a particle moving along a circle with position vector $% \,\vec{r}(t)=(R\sin (\pi t/2),R\cos (\pi t/2))$\/. \end{description} \begin{description} \item[a) \lbrack 7 pts.\rbrack ] Calculate the unit tangent vector for all $% t$ and using the figure provided, sketch the tangent vectors at $t=0$ and $% t=1$. \end{description} \begin{description} \item \hspace{2pt} \[ \hspace{3.75in}\FRAME{itbpF}{1.62in}{1.62in}{0in}{}{}{circle.gif}{\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width 1.62in;height 1.62in;depth 0in;original-width 117.4375pt;original-height 117.4375pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'q1/circle.gif';file-properties "XNPEU";}} \] \end{description} \begin{description} \item[b) \lbrack 7 pts.\rbrack ] Determine the angle between the velocity vector and the acceleration vector. \end{description} \vspace{1.25in} \hfill \begin{description} \item[{c) [6 pts.]}] Suppose that a particle's position vector in 3-space satisfies $|\,\vec{r}(t)\,| = c$ ($c$ is a constant). Show that $\vec{r}(t)$ and $\vec{r}\,^\prime(t)$ are orthogonal to one another.\\[5pt] \underline{\textit{Hint}}\/: Recall that $|\,\vec{r}(t)\,|^{2} = \vec{r}% (t)\cdot \vec{r}(t)$. \end{description} \newpage \begin{description} \item[{2. [20 pts.]}] Set up, \emph{but do not evaluate}\/, a definite integral which determines the area in the \emph{first quadrant} that lies between $0 \leq x \leq 2$\/ and between the two curves, \[ y = \frac{3x^2}{4} - 1 \hspace{.25in}\text{and}\hspace{.25in} y = \sqrt{2x}% \,. \] \underline{\textit{Note}}\/:\hspace{2pt} You must include a sketch of the region using the grid provided. \end{description} \hfill \hspace{3.50in} \FRAME{itbpF}{2.36in}{2.36in}{0in}{}{}{grid.gif} {\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width 2.36in;height 2.36in;depth 0in;original-width 2.36in;original-height 2.36in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'q1/grid.gif';file-properties "XNPEU";}} \newpage \begin{description} \item[{3. [20 pts.]}] If the area between $y=\sqrt{x}$ and $y = x^2$ for $0 \leq x \leq 1$ is rotated around the $y$-axis, what is the volume of the resulting solid?\\[5pt] \underline{\textit{Note}}\/:\hspace{2pt} You must \emph{evaluate} the integral and include a sketch of the region to be rotated. \end{description} \hfill \hspace{3.50in} \FRAME{itbpF}{2.36in}{2.36in}{0in}{}{}{grid.gif} {\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width 2.36in;height 2.36in;depth 0in;original-width 2.36in;original-height 2.36in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'q1/grid.gif';file-properties "XNPEU";}} \newpage \begin{description} \item[4. ] The figure below shows a portion of the curve described by the parametric equations, \[ \begin{array}[b]{lcl} x & = & \alpha\,t^{2} \hspace{0.65in} (\alpha > 0) \\ y & = & 3t-\dfrac{t^{3}}{3} \\ \rule{0in}{1.25in} & & \end{array} \hspace{0.5in} \FRAME{itbpF}{2.51in}{1.67in}{0in}{}{}{cubic.gif} {\special {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width 2.51in;height 1.67in;depth 0in;original-width 2.51in;original-height 1.67in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename 'q1/cubic.gif';file-properties "XNPEU";}} \] \end{description} \begin{description} \item[{a) [12 pts.]}] Set up, \emph{but do not evaluate}\/, a definite integral with respect to the variable $t$ which expresses the exact arc length of the \emph{closed loop}. \end{description} \vspace{2.5in} \hfill \begin{description} \item[{b) [8 pts.]}] Determine the value of $\alpha$ which makes the curvature at the origin equal to 1. \end{description} \pagebreak \begin{description} \item[5. ] \hspace{1pt} \end{description} \begin{description} \item[{a) [7 pts.]}] Evaluate the improper integral $\,\dint_e^\infty \dfrac{1}{x(\ln x)^2}\,dx$\/. \end{description} \vspace{1.50in} \hfill \begin{description} \item[{b) [7 pts.]}] If the curve $\,y = 1/x$, for $1 \leq x < \infty$, is rotated around the $x$-axis, what is the resulting volume? \end{description} \vspace{2.25in} \hfill \begin{description} \item[{c) [6 pts.]}] Given that the surface area of the solid in part (b) is bigger than $\int_1^\infty 2\pi y\,dx$\/, determine which is larger, the volume of the solid in (b) or the surface area of this solid? \end{description} \end{document} %%%%%%%%%%%%%%%%%%%%%% End /document/quiz1_v3.tex %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% Start /document/q1/circle.gif %%%%%%%%%%%%%%%%%%%% GedQx\SX\B@g@@O@@@@@@xo~sB@@@@@\B@g@@`@~sxcin\{OD@gtj}bszg\{`abS]cfghijkl mnopqrF@eusxy_wtez@SxK@O^AOF\XqaH[JSiDcNgFCJQfTsjbYMlZZbkWpdAYmLzpHdfnmG[ c]KgR|yvocOo~}|{{w_]eW`B^GaENwaHzfbKfVcNzTWSRydQfUeYjIeW~sfK@ef]BSfQx`hcfR egTZjbpIKkzzFp~blsVQmshKnMXyNo}[oLtukfU|pBa\HPRlpClYcDmkKO]XVgZvFcM[WnMwlz }b`GnxMgxxWoez}U~zj~n^rH}vY_}m_o~ic~W|oOQE|YnXAprPpBnbD~OAjAG^AFNpCjfjPqI BqLdQ~o`D[rEkhEMtFqpHsuGQRGYFNNyQXiB]QJuQLevJKILmY~rInfySpY[\YMSFOmEP_IQ}ix RbgPJvTjLYzt`J]VjTwQQMzQiC\nYKABEWBQ{zRsuUHUUIaWUtPyxYMuXuZ[{aX_jVIRHSvANHX CK^{Q]LrUMY\IS`YW_WcMT^Ul_[UcouUPucuuf|QeQldclcuLdW{bQYRFL[EMqB]Cg[N|\|P]P OmSikk}ZZASaiIluRmAv[qMiAul[XoeTnML^GNqEZpG]qSn`e]jM~qS}mILtyxti]uI[MY^DY^p sN{R~wyNIEvecjcXkQwwExMXoI}{[q|QY}Wx}M^ROFF_^|wcyG}E`vE_qGgodUnsAz[BHy] NgUUm`aV}IURXKYaDXtaHTC|`BXh}A^HEGnaKhOMbdsAbS_HoM`ZTKNE|B@z`SxLnXoHp]cJPLv WPFIQKgxamcSUymb{QObVcVNn_BYiQBLyaEA[VSzKGITzbYV^icXiUrLUfW~d\Yw|euXXRfaiN^ fiiZnflY[zfIr[BZqykMgrT]z\witegzy^nK}yJ}g~Y`NhDZaZhGj{`hqbbnPLzVxhjBdzaRJkP i`deZ[~QA@@pN %%%%%%%%%%%%%%%%%%%%% End /document/q1/circle.gif %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% Start /document/q1/grid.gif %%%%%%%%%%%%%%%%%%%%% GedQx\SXcCpx@@O@@@@@@xo~sB@@@@@cCpx@@`@~sxcin\{OLJgtj}bm`Mo{`abcdefghijkl mhF@YJqrHtupxGwx|u~r@fafPLEo\Tcn\KSVqIEhSHnljukXsjuHS]Re|MRXHc_TvqyFiYP]k{w pcKW]wyvocOtzz{{_\mAx`DVxO|ETUkmpbshxcMAYcFRYexIheYjiaXnig_^Ib`NJiYuYbQfj WPnjj}HZiqJ{PpNkmwZVmxnKoA\Zo@c{o\yzaEcZkIjoAwLoC{\t_BmtU[IuVgMaXGEkKklq`c |rSjmy[q}yj[Wzk{~[DmQ]|PoMvos}sgOsxwo~~soIp@RpzAPCNPDVp}ipFrP{qA]}DoEnoYqJ bQTUn~_GGIvF{HHwCACIIuR{JIJUrHGWFiuq\ix`IyRIMmrIkINisJsIOAcD{IPSpMsuKIVLMzE YiFCJSIsO[JPGJTiiTgJJkJ[hhReYKQZWQrSmJNqjXAIYKKGOKZ]gZwwWUyQA[Qe[k~xZuV[oky rk^sF_{[u~k]GlVyzV}zRA\Taa|VxA_xcc\Ba|delckLfolf]hey\fcQDwibSy_G]sD\aQlaWmj [m\mkiCVic]ofMmwemo}mP}\a[RYM]mGuMUCKqqkqOnurmrKEocko_{D{}tk[sIIsoNhrnv}Dw {nf^npCnt{FuMouA{~nzkD{[OgRN|qm|QSjIaqOpc^b~_~{m~~wl|^Ax_L`DX_DOXbgqO`Fem YwpBbPMvgAQTXgy`FTC^aGDDbaPTcA[Gh{A`bxhX`ex]\bhx\H^egHA]Uwxq_nTHnbCbJZcpANf cnqNrcfDIrHwwLzZDi~UdSWM~czaOndWQSzdRRTR_bvQZeFIVFcnhOeJJyTrbAyWBHPiXv_eisE fBuPjeyWVb^rXR^frBYJgjdWVg@YUbgch^rf{g_bdoyVJFRy^n]iiaJGIhErh[fGnaMZD~hQzdz hPzBEZwYbVhph`vf@zUBjniHii[ZzYjYB]bjzTinjtxZziYlFhoH\Z^mjk~ZyJ_bk{zmzkqW_N kazpNj\YgV~oZL}JpRhBKsjhHzs^jRKBFmVks^lvZr~iE[\Rm~aj~laKsNn@[Krm[klfntZvjl_ Kx^mok]JoLY|RodJlnnDk~ZlMKrreYZeJiWz@SiDl@cpGlBSaA|}fbvkC{TX[Dwey[{bnWlzfql [snlT\yzk`|nJryzyjq~kvnq{kJOoTLtNqnl{FszHDOsNBLSTKJOhRlNwssdMwcu\PwO^\DSrn jIsqLOgsi|N[fD}SNtRMRojVMjjtkLT_rLMFsuCgT[JTMQcufzbZiFlZgvl]AwvI|Z{vRZC{sN ]S[_gGJj^kXOux\_wq@n_[tNlVwo^]W[w_MbGvB^E{xONdCtRj{r\]cwVnNayYU^CpT~d{yMNh kkb^h_wY~Ssx[]j_xDEa_of}fB{Ijl[hF~^wmxn^g{lL_Sz}noco@oK|RLaqy`mjkxI~eyD_q Cbp}[opR_\G}rmuc}OF]{|EOwwyGj_{z~xs{{^`{}]Ozwzjzs~ljEc|lNs[z[nfKdMpS{{i~ cg|_]Z{fopW@TymtH`cNww}suQo|_rIFxYB@R~b_borWDT@F|P`wLEh|rAVfx`bIHHELDJAG`so EW{CCjfp_~NPxHlBZ@e_gNzWBD~YAEaupVhM|Efd\A|p^hO|GBDCbBQaHJ|AT@@@pN %%%%%%%%%%%%%%%%%%%%%% End /document/q1/grid.gif %%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%% Start /document/q1/cubic.gif %%%%%%%%%%%%%%%%%%%% GedQx\SXrCPh@@O@@@@@@xo~sB@@@@@rCPh@@`@~sxcin|AMLJgtj}bszMoG^^abcdefgZ`hkl mnohjpstuvprF@j{w~@CzPQO]yDbLirPGKV\IthRgtDtsiXsjuqck[BNYOoaK[nvncgWsNfzv pcWij|zvidc|}}|eigO`u}V`DZw`EbX[GfHcemXcPjucQRYTSVIfH]Yf\vig_BZhbNJieNzf^`j iHjZGmnJ`oralpJHm[\[mJzSnz^Voxz[d@GlpLR\qFWHrIk\lZz\lLG}iT[MUWg]CSkmg\sTwf} gP_G^LYUnyednHlk^jnC~{RE_{sCuxV^oRuSR{GaMBABPbPP{Up[\pDBgEFft@qHBQmeoJIqK bqLjL~_H~rXZpcGyhdLtlbhBIhjhcRqG]aGydHOAjdxOpxp^FUhiftIaTyZftDgrEqY`VTPwrP ecOwuvVZGyixNFCNjbLyJejAciIlj[xzE\Z\jbVWaWGh_ZuXkIXYSYaGB@{ZevphZIS`SSKPZy\ uKnNKHSY\MD]JTa@^UhQNlBG\NqP`wcUTlZy[cmP|^rcYQWr\OY|Bn[}\bdpGF}NilAgCMvlP KmnZKWj|LPMh]i`M^VliK}nmdg[zbImPy\f~MQxUdG^vLnkCnQTNEuKOz]mMgFwHuASRCkMyRw S|PCyU^iwwNoMOyAOTSyWoxQYy]w[oWYOneYvu~}WEq~OTWwX}gwLXAxqd_tcqQ`KSBBI{w[} ZKXbt`iFY[dbDnM|WsAKWX@Y`sfF^[LwWYa|VHVaue}\~gHfJVHD~KlX{xantKj_gHY]JuHzD cnahhcGxM~aXR^|cuVPbFsh||VEYPpaHvMeU}AOfWIYsldLyKTXHUc\HGioieZGVuTXiLzd{Hkd SdIKYfaYOARlIYAfwaZbFVF\ZVrYQQTDFYr^TRUjYW_ygEG`vUoGbJhdXmdhpgpXhLdZEVQzju PTZerVWzjAhPJc~KZjMdd]zC~P`JotffZhf]czFMRiZM}jLJkFkgCmZJsjmnNxSnvdKAn^MtY@q kPCyLlD[^YWq^lgZrVlpJsvlOysvkQktReaZuJFXKLimEGwzm_{UAnbixbfd[DZngKznLj{zrnz i{ngpKIIot[}ZowK~foBi~fi|[zo_J@GpB\xRphjAEZ}jp@TA@pN %%%%%%%%%%%%%%%%%%%%%% End /document/q1/cubic.gif %%%%%%%%%%%%%%%%%%%%%