\hfill \thepage} %} \input{tcilatex} \begin{document} \subsection*{Ma116 \hspace{2.0in} Project 1 \hfill 2/11/00} \subsection*{{\protect\small Name: \protect\underline{\hspace{2.5in}} \hfill ID: \protect\underline{\hspace{2.0in}}}} \subsection*{{\protect\small E-Mail: \protect\underline{\hspace{2.5in}} \hfill T.A./Recitation: \protect\underline{\hspace{2.0in}}}} \hfill {\small \textit{I pledge my honor that I have abided by the Stevens Honor System.}\hfill\underline{\hspace{2.3in}}} \hfill \textbf{Please enter your final answers within the underlined regions along the right-hand side of the page. Details leading to the final answers are to be included in the space below the question.} \hfill \subsubsection*{Introduction} The object of this project is to study tangent lines to curves as well as area between two curves. In each instance, having found a tangent line to a certain cubic curve, you will be asked to find the area between that tangent and the cubic. The first six questions perform the calculations using the simple cubic $y=x^3$. You will be asked to generate a plot within this SNB document similar to the plot shown in the figure below. The figure is provided as a reference to verify that you are generating the correct geometry. The main result in this project has to do with the ratio of the areas of the regions labeled $A_1$ and $A_2$ in the figure. \[ \FRAME{itbpF}{2.87in}{4.04in}{0in}{\Qcb{{\protect\small \mathbf{Figure 1.% \hspace{5pt} Plot of $y=x^3$ with tangent lines.}}}}{}{lens.gif} {\special% {language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width 2.87in;height 4.04in;depth 0in;original-width 2.87in;original-height 4.04in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename '/document/graphics/lens.gif';file-properties "XNPEU";}} \] \pagebreak \subsubsection*{The cubic curve {\protect\large $\,y=x^3$}} \vspace{0pt} \begin{description} \item[1.] In the space below, use Scientific Notebook to plot $y=x^{3}$. \end{description} \hfill \hfill \hfill \hfill \hfill \begin{description} \item[2.] As shown in Figure 1, use $L_1$ to denote the the line tangent to $% y=x^3$ through the point $(-1,-1)$. After answering the following two questions, add the graph of this tangent line to your preceding plot. \end{description} \hfill \begin{description} \item[a)] What is the slope of the tangent line to $\ x^{3}$ at $x=-1$\/? \hfill\left.\underline{\hspace{1.5in}}\right. \end{description} \hfill \begin{description} \item[b)] What is the equation of the tangent line at $x=-1$\/? \hfill\left.% \underline{\hspace{1.5in}}\right. \end{description} \hfill \hfill \begin{description} \item[3.] In addition to the intersection at $(-1,-1)$, the tangent line $% L_{1}$ in part \textbf{1} intersects the cubic $y=x^{3}$ at a second point denoted by the letter $P$\/? Set up an appropriate polynomial equation for determining this intersection point.\\[2pt] \underline{\textit{Note}}: \hspace{1pt} You can use SNB to solve the cubic equation that gives the point of intersection.\\[5pt] What are the coordinates of $P$\/? \hfill \left. \underline{\hspace{1.5in}}% \right. \end{description} \hfill \hfill \begin{description} \item[4.] We use $L_2$ to denote the the line tangent to $y=x^3$ through the point $P$ (see Figure 1). After answering the following two questions, add the graph of this second tangent line to your plot in part \textbf{1}. \end{description} \hfill \begin{description} \item[a)] What is the slope of the tangent line at the point $P$? \hfill\left.\underline{\hspace{1.5in}}\right. \end{description} \hfill \begin{description} \item[b)] What is the equation of the tangent line through the point $P$? \hfill\left.\underline{\hspace{1.5in}}\right. \end{description} \hfill \hfill \begin{description} \item[5.] This second tangent line $L_2$ intersects the cubic at a new point $Q$. Set up an appropriate polynomial equation for determining the intersection point $Q$.\\[5pt] What are the coordinates of $Q$? \hfill\left.\underline{\hspace{1.5in}}% \right. \end{description} \pagebreak \begin{description} \item[6.] You should now have a plot of $y=x^{3}$ on which is superimposed two tangent lines. Between the tangent line $L_1$ and the cubic $y=x^3$ there is a lens-shaped region, denoted $A_1$. Between the second tangent line $L_2$ and the cubic there is a second lens-shaped region, denoted $A_2$% . Refer to Figure 1 to verify that you have the correct geometry in your plot.\\[2pt] \underline{\textit{Note}}: \hspace{1pt} After setting up the definite integrals for computing areas, use SNB to perform the evaluation. \end{description} \hfill \begin{description} \item[a)] What is the exact area of region $A_1$\/? \hfill\left.\underline{% \hspace{1.5in}}\right. \end{description} \hfill \begin{description} \item[b)] What is the exact area of region $A_2$\/? \hfill\left.\underline{% \hspace{1.5in}}\right. \end{description} \hfill \hfill \begin{description} \item[7.] What is the ratio of the larger area to the smaller? \hfill\left.% \underline{\hspace{1.5in}}\right. \end{description} \hfill \hfill \subsubsection*{Another cubic curve} \vspace{0pt} \begin{description} \item[8.] Select $y=x(x+4)(x-2)$ in place of $y=x^{3}$. Repeat steps 1 through 7 starting with the point determined by setting $x=-2$. \end{description} \hfill \begin{description} \item[a)] The equation of the first tangent line $L_1$\/: \hfill\left.% \underline{\hspace{1.5in}}\right. \end{description} \hfill \begin{description} \item[b)] The coordinates of the second point $P$\/: \hfill\left.\underline{% \hspace{1.5in}}\right. \end{description} \hfill \begin{description} \item[c)] The equation of the second tangent line $L_2$\/: \hfill\left.% \underline{\hspace{1.5in}}\right. \end{description} \hfill \begin{description} \item[d)] The coordinates of the third point $Q$\/: \hfill\left.\underline{% \hspace{1.5in}}\right. \end{description} \hfill \begin{description} \item[e)] The first area $A_1$\/: \hfill\left.\underline{\hspace{1.5in}}% \right. \end{description} \hfill \begin{description} \item[f)] The second area $A_2$\/: \hfill\left.\underline{\hspace{1.5in}}% \right. \end{description} \hfill \begin{description} \item[g)] The ratio of the larger area to the smaller area: \hfill\left.% \underline{\hspace{1.5in}}\right. \end{description} \hfill \end{document} %%%%%%%%%%%%%%%%%%%%%% End /document/proj1_V3.tex %%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% Start /document/graphics/lens.gif %%%%%%%%%%%%%%%%%% GedQx\SXUDPaADO@@@@@@HklrV^yesB@@@@@UDPaA@`@~sycin\{OLJgtj}bsM@ppu`abcd efghI\]CliopqrSnsvjluw{|}Zz~`mjEp`FOblHrR_Zm|It^Jc~{sTjXs~S{HkWyBN|FDC}tTlb UgQsnmJrgK{vlsyv[WSn\wj^Ryut^|ua_@bXPBfsZxQXbPbUYMzHeQ^yS^]hdObigGuw_]~I i~hv]gVjjv@thbyJM[nJimZykvFkls~iNprUjVx{n{zBpsI\ZHO|lEc[XJ_@tKs|eNccnSS\dB[ \v^s@vZYmt_o\x\jDyeO}yTQYwkGOm{in|vsVhDu~}|\_go~IpcEP@VoAbi}ihEmu\PDnmBE `@dOB}PHfHErJ~{pDgXmzeuphjthPxXG}xGaSHEImDG@NyImdAWIKye{LIL{XQfYMWGIEqJrYNe IM{Iu@uKEzPSWQMzHWXRUJLquE]Jd~IEBjTUjmjjUJQvzViZSAkYugXG{U|zYQhVSkTY[[]z[a [\E@Vgkgrk~bj]yJ^uKFY_hh_[LKk`MGHpIGqIbuI@B|b]KFjkIa\dy@ek|@qV]eLJcldqK_} |UcU\cjfm{dEPgQ]|ShgJ@w|kcmlywTjmPa\kUKlM}nAdb]pNOlZmMs]nuMR[VqSf\}`Jq]sK lmGNlO]cGtqO[x_`oIMWUNgy~xSaffNxN]pMjwNw{^VBh{qQWbo|~_~h|yCgwm_BX|@PpQHT^} gX}XiW}A]PA|d_DQULTJH\uGNhzyRQh{]\egrYaKE@z`DxwqHRxAfU|gGEbgwHfaeXUtLqxlIKk XLubIVCfauGM^YKBJjb`X\EbXHO^d~XPrCIxXaXstQBe]WRJC|TCYPjdAZeYiY\eZICyexwLmH] Y_}OzhXbfiyNJfjWUre[YYFfnY\ZLbilUPhYTvczI_vgdhRVHii_rcox`ZhmI`VIsaBeEJcZhS YbZbjqcViVj^FiRZFXdWZgziAhfzO{xgRjWZfFjybcVjkj_^jhJDPilZcJkHykBa\JmBcyJLZkQ WlnkDJpja}jihIB{l^~sn^D[pDGczqngI{rNCuis~ljWtJmeTmXmQij~iJ[v~^Igwrmd[gzjJIG nB\KfBlhKJBKkK{vm{z{vaad|NozKqEn[Jrdo{{|ZohFt@pA\yvoLKhUmGlpBnhZ}XpMLIJkC\[ }PL|DG`lzCChUUFkq]wjvqp{CCraL^MrI|BHWgLJ[nmjEsRUg~`rt{wJsPDMsev|M{iLRN[O{\C us~liZe}zPCGEmQWiM[PgMJ}uptH{QseajTgLTm{jRIZVsrp|VhNDIKR_mxJvyJHAu|RYkbiMm nvw{@{r[]XSbr}]zvWP][wNMuQp{M_wwT}]chWsb]vU@aWxLz{HxfCdKcp~wM]_Nn_y[rvQwoXx L]gsT[Ys^OJ^UPym{`F^bnOIyLQg{yPJPqz[hcXzgneG{ZnvZGLnXGvJKigI]el{ou^ijQ@oob\ C`{zEOoVEQ~a`|oms_nG_nSBE|rkpUyb`xWulx[oxw}UMRb}YtxGKeoyCch_zg{|Ne{moObL} R`|OMjD^QP~CwNJugwhQ^Z^@pZES@BkL`imBU}C~qIX_vnCW~YNuPvHRpJwCZwLg@_lGFxA lGuov^orgB\feddXYpnUB[yUB{^gPsvTzEz|barpiEHTY_Zaip\XHtFNXzakP~tdAHFDqaCqv uCldeD{VKqDXStInrnAtpbw~OqJhO\~]EoWQqswVlxTEM]FbiHFlK~|HcHq_VVlb}EgNaQUXHkM zFsa\fpHGSNrA}auquHira`FKIVOiTQkW|G_AUqtc]lOfFJdh`CYnIQjFWdLRS@Mt`XsvISrVW` L@qGmZRrLYlsTA`ZPjrUz@Tn_De}hWCblSRDGChrocuBR~JaeqRbWrrUJEmewrAWnT`uK`Rv eQ\DjKWeBL|HWt]]LYaAsFYsdSBGkXxogIysLRSTf`EZyKcFXMW_XslrwDEMwWMEmiUa[jq@Mc sEXx\quNK`]QECzthhN[XgczYOy^Fp|PqsMPydXjO{PssIeL`bpLhZrdXqDMrO~GXnEThDMFJB RbNQWa`kAZPdbVQmhOtDZGUb~Q_V]LMj[Aazpr^VtVpPdoXREHcTGfKucbRkWTRQZMUmmROhYpF JcLfZSwiYo\JuThNTe`PnWzhchZ|JQJUaFSUQZHMh@UvhP`jFUcdWuYUUeEC^jEHSxtbkfUEec uxYWUjReHkUORHZmlzt@kLqZhQUXZEeksUqH^EJZEeBmstjY]I}WakPJtu^MrU{^WGVbAaEuiXG g@vqf}ZqrLZlwNc@e]UDYOT@qL[iXsVXSB_SPKjKPLZubuuoje]WfZMdkv}WkEPv{|l|oFHTcvf P^mDJwqsdtbiX[xvjan]UJIzmCwa~W|v^iomQEBfKWBEvKx^UNn{p{csezzSdhrgK^CLH]m_X wXKU|xf]YkWroK_}RJ^Go}lrKxxVr]eRxCxkn}Jb^QEjW|PlL~z^QN{wW{W|~RPbcAx{zlMJ`i ]kSTqvurv_aUMxIvCVan\SpiKIlENoZNspZX}EzdC{]qgifIIi]B{MIqpU}IHvetOUqYrMkIV\R b{gIDT|FnFScjqXxYNvC[SzqBtu[``inciiPTa\R`hnb}ZytbL_CHSRulErnjS~IGkjUYklCm fnWWrwJRZW^JcPsrWfkLJ]qzeGryRFsVVfrNLsiTdldHNKgcB_Tr|N}NWgc`ztoQZFih~[Cw~gq lMs`{`nXEh}Idq{I^nKlhOf_wCmTPWvh]tSQnYevRsCKjrCI}N@Ss[nt]rLmQuUziqZcz}lhij j}n|qL|iNMjUgMkluHi~DDkUmjUXzf^VskMf~z_]xzMZvzuVJo\nVHX}fY@{c]ZNY[W[mZ|AC| VK^KaV{kms^ZSmcktea]sK[oVsVOq}WE\{x|tP[\}g{TonidNNsr{~hFZwfd[CgyfU\oa]bg}] L{byoIbs[bEc[`Sp}]F|dFLK_cd{[L|Omtjn}paxKKsuDgbkpkeInJNyzbcqmUgLO~GUEroS\\ fkIg`crSy`\UUBOZ\pO_\[aKt]fSs]_v|~wFoHQn\^gr_Cm|sh[LcS\k`D]NI|Ji}gRzd|QyOg eEtQzAigNxlzrtaXORjNToiK_}X`\i^NOwvuC{b]^~t^OR]H`d]UcpNyYtzJOKB}pl_VeTriqu [WSaybPj}}BjVOH|Lxp]@J`Gtewu{dm^SB_pKnzZ`mEo^_Batn`zCBVdOqceh{u}tUewqfV{~C I_fGNAzmsghGPhoWzmeZ~^^JbdOuEDImingGb]uiCs^lh~IwwaRFD\cnogwCFA[~aN`SxGxH~ E_}v{Yuo{@hN_~`rKzSmg~TWQAlbsdY|~Of\Dw{k}}X\nox{W~CikqODN}k~IOg_t_I{zg| tU~C^[Q\{}_Xg~uq^~_|g\oA@Vr`y_}I@f@MBuyUYT@@pN %%%%%%%%%%%%%%%%%%% End /document/graphics/lens.gif %%%%%%%%%%%%%%%%%%%