1# LaTeX2HTML 2002-2-1 (1.70) 2# Associate images original text with physical files. 3 4 5$key = q/displaystyle{frac{{x^4+x^3-4x^2-4x}}{{x(x-1)(x+1)^2}}};MSF=1.6;AAT/; 6$cached_env_img{$key} = q|<IMG 7 WIDTH="122" HEIGHT="57" ALIGN="MIDDLE" BORDER="0" 8 SRC="|."$dir".q|img62.png" 9 ALT="$\displaystyle {\frac{{x^4+x^3-4x^2-4x}}{{x(x-1)(x+1)^2}}}$">|; 10 11$key = q/array{{ccc}2a-1<col_mark>a<col_mark>2a-1<row_mark>a^2+a-2<col_mark>a^2-1<col_mark>a-1<row_mark>a^2+a-1<col_mark>a^2+a-1<col_mark>aarray{;MSF=1.6;AAT/; 12$cached_env_img{$key} = q|<IMG 13 WIDTH="220" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 14 SRC="|."$dir".q|img39.png" 15 ALT="$ \begin{array}{ccc} 162a-1 & a & 2a-1\\\\ 17a^2+a-2 & a^2-1 & a-1\\\\ 18a^2+a-1 & a^2+a-1 & a 19\end{array}$">|; 20 21$key = q/displaystylearray{{lcl}x(u,v)<col_mark>=<col_mark>u,cos(v)<row_mark>y(u,v)<col_m>=<col_mark>u,sin(v)<row_mark>z(u,v)<col_mark>=<col_mark>1-u;.array{;MSF=1.6;AAT/; 22$cached_env_img{$key} = q|<IMG 23 WIDTH="156" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 24 SRC="|."$dir".q|img97.png" 25 ALT="$\displaystyle \begin{array}{lcl} 26x(u,v)&=& u \cos(v)\\\\ 27y(u,v)&=& u \sin(v)\\\\ 28z(u,v)&=& 1-u\;. 29\end{array}$">|; 30 31$key = q/displaystyle{frac{{t^2}}{{1-t^4}}};MSF=1.6;AAT/; 32$cached_env_img{$key} = q|<IMG 33 WIDTH="43" HEIGHT="57" ALIGN="MIDDLE" BORDER="0" 34 SRC="|."$dir".q|img29.png" 35 ALT="$\displaystyle {\frac{{t^2}}{{1-t^4}}}$">|; 36 37$key = q/displaystyle{frac{{1}}{{t(t^2+1)}}};MSF=1.6;AAT/; 38$cached_env_img{$key} = q|<IMG 39 WIDTH="61" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 40 SRC="|."$dir".q|img33.png" 41 ALT="$\displaystyle {\frac{{1}}{{t(t^2+1)}}}$">|; 42 43$key = q/displaystyle{frac{{ln(cos(x))}}{{exp(x+x^2)}}};MSF=1.6;AAT/; 44$cached_env_img{$key} = q|<IMG 45 WIDTH="80" HEIGHT="54" ALIGN="MIDDLE" BORDER="0" 46 SRC="|."$dir".q|img35.png" 47 ALT="$\displaystyle {\frac{{\ln(\cos(x))}}{{\exp(x+x^2)}}}$">|; 48 49$key = q/left.vphantom{array{{ccc}1<col_mark>11<col_mark>-7<row_mark>-3<col_mark>-9<col_mark>9<row_mark>5<col_mark>-5<col_mark>1array{}right];MSF=1.6;AAT/; 50$cached_env_img{$key} = q|<IMG 51 WIDTH="16" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 52 SRC="|."$dir".q|img49.png" 53 ALT="$ \left.\vphantom{ 54\begin{array}{ccc} 551 & 11 & -7\\\\ 56-3 & -9 & 9\\\\ 575 & -5 & 1 58\end{array}}\right]$">|; 59 60$key = q/lambda;MSF=1.6;AAT/; 61$cached_env_img{$key} = q|<IMG 62 WIDTH="14" HEIGHT="32" ALIGN="MIDDLE" BORDER="0" 63 SRC="|."$dir".q|img54.png" 64 ALT="$ \lambda$">|; 65 66$key = q/displaystylett{left[left[array{{rrr}1<col_mark>-3<col_mark>0<row_mark>1<col_markmark>0<col_mark>0<row_mark>0<col_mark>0<col_mark>0array{right]right];MSF=1.6;AAT/; 67$cached_env_img{$key} = q|<IMG 68 WIDTH="236" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 69 SRC="|."$dir".q|img57.png" 70 ALT="$\displaystyle \tt\left[ \left[ 71\begin{array}{rrr} 721 & -3 & 0\\\\ 731 & 0 & -3 ... 74...n{array}{ccc} 753 & 0 & 0\\\\ 760 & 0 & 0\\\\ 770 & 0 & 0 78\end{array}\right] 79\right]$">|; 80 81$key = q/displaystyle{frac{{acdotb}}{{2}}};MSF=1.6;AAT/; 82$cached_env_img{$key} = q|<IMG 83 WIDTH="32" HEIGHT="52" ALIGN="MIDDLE" BORDER="0" 84 SRC="|."$dir".q|img2.png" 85 ALT="$\displaystyle {\frac{{a\cdot b}}{{2}}}$">|; 86 87$key = q/displaystyle{frac{{x^2-y}}{{x^2-1}}};MSF=1.6;AAT/; 88$cached_env_img{$key} = q|<IMG 89 WIDTH="45" HEIGHT="57" ALIGN="MIDDLE" BORDER="0" 90 SRC="|."$dir".q|img67.png" 91 ALT="$\displaystyle {\frac{{x^2-y}}{{x^2-1}}}$">|; 92 93$key = q/displaystyle{frac{{a}}{{displaystylefrac{2}{b}}}};MSF=1.6;AAT/; 94$cached_env_img{$key} = q|<IMG 95 WIDTH="19" HEIGHT="70" ALIGN="MIDDLE" BORDER="0" 96 SRC="|."$dir".q|img3.png" 97 ALT="$\displaystyle {\frac{{a}}{{\displaystyle \frac{2}{b}}}}$">|; 98 99$key = q/displaystylett{left[array{{ccc}-a+1<col_mark>0<col_mark>0<row_mark>0<col_mark>a+2<col_mark>0<row_mark>0<col_mark>0<col_mark>a-1array{right];MSF=1.6;AAT/; 100$cached_env_img{$key} = q|<IMG 101 WIDTH="184" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 102 SRC="|."$dir".q|img53.png" 103 ALT="$\displaystyle \tt\left[ 104\begin{array}{ccc} 105-a+1 & 0 & 0\\\\ 1060 & a+2 & 0\\\\ 1070 & 0 & a-1 108\end{array}\right]$">|; 109 110$key = q/displaystyle{frac{{1}}{{x}}};MSF=1.6;AAT/; 111$cached_env_img{$key} = q|<IMG 112 WIDTH="15" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 113 SRC="|."$dir".q|img78.png" 114 ALT="$\displaystyle {\frac{{1}}{{x}}}$">|; 115 116$key = q/left[vphantom{array{{ccc}1<col_mark>1<col_mark>a<row_mark>1<col_mark>a<col_mark>1<row_mark>a<col_mark>1<col_mark>1array{}right.;MSF=1.6;AAT/; 117$cached_env_img{$key} = q|<IMG 118 WIDTH="16" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 119 SRC="|."$dir".q|img50.png" 120 ALT="$ \left[\vphantom{ 121\begin{array}{ccc} 1221 & 1 & a\\\\ 1231 & a & 1\\\\ 124a & 1 & 1 125\end{array}}\right.$">|; 126 127$key = q/displaystyle{frac{{sin(x)^2}}{{cos(2x)}}};MSF=1.6;AAT/; 128$cached_env_img{$key} = q|<IMG 129 WIDTH="55" HEIGHT="57" ALIGN="MIDDLE" BORDER="0" 130 SRC="|."$dir".q|img31.png" 131 ALT="$\displaystyle {\frac{{\sin(x)^2}}{{\cos(2x)}}}$">|; 132 133$key = q/array{{ccc}1<col_mark>1<col_mark>a<row_mark>1<col_mark>a<col_mark>1<row_mark>a<col_mark>1<col_mark>1array{;MSF=1.6;AAT/; 134$cached_env_img{$key} = q|<IMG 135 WIDTH="75" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 136 SRC="|."$dir".q|img51.png" 137 ALT="$ \begin{array}{ccc} 1381 & 1 & a\\\\ 1391 & a & 1\\\\ 140a & 1 & 1 141\end{array}$">|; 142 143$key = q/displaystyleleft{vphantom{array{{lcl}x(t)<col_mark>=<col_mark>t,cos(at)<row_mark_mark>t,sin(at)<row_mark>z(t)<col_mark>=<col_mark>1-t;.array{}right.;MSF=1.6;AAT/; 144$cached_env_img{$key} = q|<IMG 145 WIDTH="19" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 146 SRC="|."$dir".q|img98.png" 147 ALT="$\displaystyle \left\{\vphantom{ 148\begin{array}{lcl} 149x(t)&=& t \cos(a t)\\\\ 150y(t)&=& t \sin(a t)\\\\ 151z(t)&=& 1-t\;. 152\end{array}}\right.$">|; 153 154$key = q/displaystylesum_{{j=0}}^{{n-1}};MSF=1.6;AAT/; 155$cached_env_img{$key} = q|<IMG 156 WIDTH="24" HEIGHT="61" ALIGN="MIDDLE" BORDER="0" 157 SRC="|."$dir".q|img82.png" 158 ALT="$\displaystyle \sum_{{j=0}}^{{n-1}}$">|; 159 160$key = q/int_{a}^{b};MSF=1.6;AAT/; 161$cached_env_img{$key} = q|<IMG 162 WIDTH="19" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" 163 SRC="|."$dir".q|img76.png" 164 ALT="$ \int_{a}^{b}$">|; 165 166$key = q/displaystylelim_{{xrightarrow+infty}}^{};MSF=1.6;AAT/; 167$cached_env_img{$key} = q|<IMG 168 WIDTH="38" HEIGHT="40" ALIGN="MIDDLE" BORDER="0" 169 SRC="|."$dir".q|img86.png" 170 ALT="$\displaystyle \lim_{{x\rightarrow +\infty}}^{}$">|; 171 172$key = q/displaystylearray{{lcl}x(t)<col_mark>=<col_mark>sin(t)<row_mark>y(t)<col_mark>=<col_mark>cos^3(t)array{;MSF=1.6;AAT/; 173$cached_env_img{$key} = q|<IMG 174 WIDTH="133" HEIGHT="55" ALIGN="MIDDLE" BORDER="0" 175 SRC="|."$dir".q|img90.png" 176 ALT="$\displaystyle \begin{array}{lcl} 177x(t)&=& \sin(t)\\\\ 178y(t)&=& \cos^3(t) 179\end{array}$">|; 180 181$key = q/displaystyleint_{{-2}}^{{-1}};MSF=1.6;AAT/; 182$cached_env_img{$key} = q|<IMG 183 WIDTH="33" HEIGHT="55" ALIGN="MIDDLE" BORDER="0" 184 SRC="|."$dir".q|img77.png" 185 ALT="$\displaystyle \int_{{-2}}^{{-1}}$">|; 186 187$key = q/displaystyle{frac{{x+1}}{{x-3}}};MSF=1.6;AAT/; 188$cached_env_img{$key} = q|<IMG 189 WIDTH="39" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 190 SRC="|."$dir".q|img19.png" 191 ALT="$\displaystyle {\frac{{x+1}}{{x-3}}}$">|; 192 193$key = q/displaystylearray{{llllllr}x<col_mark>+<col_mark>y<col_mark>+<col_mark>az<col_mamark>+<col_mark>y<col_mark>+<col_mark>z<col_mark>=<col_mark>-2array{;MSF=1.6;AAT/; 194$cached_env_img{$key} = q|<IMG 195 WIDTH="216" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 196 SRC="|."$dir".q|img16.png" 197 ALT="$\displaystyle \begin{array}{llllllr} 198x &+& y &+& az&=&1\\\\ 199x & +& a y&+& z&=&1 \\\\ 200ax & +&y &+& z&=&-2 201\end{array}$">|; 202 203$key = q/left[vphantom{array{{ccc}1<col_mark>11<col_mark>-7<row_mark>-3<col_mark>-9<col_mark>9<row_mark>5<col_mark>-5<col_mark>1array{}right.;MSF=1.6;AAT/; 204$cached_env_img{$key} = q|<IMG 205 WIDTH="16" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 206 SRC="|."$dir".q|img47.png" 207 ALT="$ \left[\vphantom{ 208\begin{array}{ccc} 2091 & 11 & -7\\\\ 210-3 & -9 & 9\\\\ 2115 & -5 & 1 212\end{array}}\right.$">|; 213 214$key = q/displaystyle{frac{{y-1}}{{x^2-1}}};MSF=1.6;AAT/; 215$cached_env_img{$key} = q|<IMG 216 WIDTH="45" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 217 SRC="|."$dir".q|img71.png" 218 ALT="$\displaystyle {\frac{{y-1}}{{x^2-1}}}$">|; 219 220$key = q/Delta;MSF=1.6;AAT/; 221$cached_env_img{$key} = q|<IMG 222 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" 223 SRC="|."$dir".q|img114.png" 224 ALT="$ \Delta$">|; 225 226$key = q/displaystyle{frac{{y-1}}{{x+1}}};MSF=1.6;AAT/; 227$cached_env_img{$key} = q|<IMG 228 WIDTH="39" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 229 SRC="|."$dir".q|img70.png" 230 ALT="$\displaystyle {\frac{{y-1}}{{x+1}}}$">|; 231 232$key = q/square;MSF=1.6;AAT/; 233$cached_env_img{$key} = q|<IMG 234 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" 235 SRC="|."$dir".q|img58.png" 236 ALT="$ \square$">|; 237 238$key = q/bullet;MSF=1.6;AAT/; 239$cached_env_img{$key} = q|<IMG 240 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0" 241 SRC="|."$dir".q|img1.png" 242 ALT="$ \bullet$">|; 243 244$key = q/displaystyle{frac{{x-7}}{{(x-3)^2}}};MSF=1.6;AAT/; 245$cached_env_img{$key} = q|<IMG 246 WIDTH="58" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 247 SRC="|."$dir".q|img20.png" 248 ALT="$\displaystyle {\frac{{x-7}}{{(x-3)^2}}}$">|; 249 250$key = q/left[vphantom{array{{ccc}2a-1<col_mark>a<col_mark>2a-1<row_mark>a^2+a-2<col_markl_mark>a-1<row_mark>a^2+a-1<col_mark>a^2+a-1<col_mark>aarray{}right.;MSF=1.6;AAT/; 251$cached_env_img{$key} = q|<IMG 252 WIDTH="16" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 253 SRC="|."$dir".q|img38.png" 254 ALT="$ \left[\vphantom{ 255\begin{array}{ccc} 2562a-1 & a & 2a-1\\\\ 257a^2+a-2 & a^2-1 & a-1\\\\ 258a^2+a-1 & a^2+a-1 & a 259\end{array}}\right.$">|; 260 261$key = q/simeq;MSF=1.6;AAT/; 262$cached_env_img{$key} = q|<IMG 263 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" 264 SRC="|."$dir".q|img8.png" 265 ALT="$ \simeq$">|; 266 267$key = q/displaystylearray{{ccc}a-1<col_mark>2a^3+3a+1<col_mark>-2a^3+a^2+a-1<row_mark>-a-2a+1<row_mark>a^3-2a+1<col_mark>-a^3+2a-1<col_mark>a^3-2a^2+1array{;MSF=1.6;AAT/; 268$cached_env_img{$key} = q|<IMG 269 WIDTH="363" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 270 SRC="|."$dir".q|img43.png" 271 ALT="$\displaystyle \begin{array}{ccc} 272a-1 & 2a^3+3a+1 & -2a^3+a^2+a-1\\\\ 273-a^2+1 & -2a^3+a^2+2a-1 & 2a^3-a^2-2a+1\\\\ 274a^3-2a+1 & -a^3+2a-1 & a^3-2a^2+1 275\end{array}$">|; 276 277$key = q/boxtimes;MSF=1.6;AAT/; 278$cached_env_img{$key} = q|<IMG 279 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" 280 SRC="|."$dir".q|img59.png" 281 ALT="$ \boxtimes$">|; 282 283$key = q/displaystyleleft[vphantom{array{{ccc}a-1<col_mark>2a^3+3a+1<col_mark>-2a^3+a^2+aow_mark>a^3-2a+1<col_mark>-a^3+2a-1<col_mark>a^3-2a^2+1array{}right.;MSF=1.6;AAT/; 284$cached_env_img{$key} = q|<IMG 285 WIDTH="16" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 286 SRC="|."$dir".q|img42.png" 287 ALT="$\displaystyle \left[\vphantom{ 288\begin{array}{ccc} 289a-1 & 2a^3+3a+1 & -2a^3+a^2+a... 290...+2a-1 & 2a^3-a^2-2a+1\\\\ 291a^3-2a+1 & -a^3+2a-1 & a^3-2a^2+1 292\end{array}}\right.$">|; 293 294$key = q/displaystyleint_{0}^{1};MSF=1.6;AAT/; 295$cached_env_img{$key} = q|<IMG 296 WIDTH="24" HEIGHT="55" ALIGN="MIDDLE" BORDER="0" 297 SRC="|."$dir".q|img79.png" 298 ALT="$\displaystyle \int_{0}^{1}$">|; 299 300$key = q/displaystylePhi;MSF=1.6;AAT/; 301$cached_env_img{$key} = q|<IMG 302 WIDTH="15" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" 303 SRC="|."$dir".q|img109.png" 304 ALT="$\displaystyle \Phi$">|; 305 306$key = q/displaystyle{frac{{1}}{{xsqrt{1+x^2}}}};MSF=1.6;AAT/; 307$cached_env_img{$key} = q|<IMG 308 WIDTH="66" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 309 SRC="|."$dir".q|img73.png" 310 ALT="$\displaystyle {\frac{{1}}{{x\sqrt{1+x^2}}}}$">|; 311 312$key = q/{frac{{1}}{{pi^4}}};MSF=1.6;AAT/; 313$cached_env_img{$key} = q|<IMG 314 WIDTH="20" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" 315 SRC="|."$dir".q|img87.png" 316 ALT="$ {\frac{{1}}{{\pi^4}}}$">|; 317 318$key = q/displaystyleleft{vphantom{array{{lcl}x(t)<col_mark>=<col_mark>sin(132,t)<row_mark>y(t)<col_mark>=<col_mark>cos^3(126,t)array{}right.;MSF=1.6;AAT/; 319$cached_env_img{$key} = q|<IMG 320 WIDTH="17" HEIGHT="55" ALIGN="MIDDLE" BORDER="0" 321 SRC="|."$dir".q|img93.png" 322 ALT="$\displaystyle \left\{\vphantom{ 323\begin{array}{lcl} 324x(t)&=& \sin(132 t)\\\\ 325y(t)&=& \cos^3(126 t) 326\end{array}}\right.$">|; 327 328$key = q/displaystylelim_{{xrightarrow0}}^{};MSF=1.6;AAT/; 329$cached_env_img{$key} = q|<IMG 330 WIDTH="26" HEIGHT="40" ALIGN="MIDDLE" BORDER="0" 331 SRC="|."$dir".q|img83.png" 332 ALT="$\displaystyle \lim_{{x\rightarrow 0}}^{}$">|; 333 334$key = q/sqrt{{2}};MSF=1.6;AAT/; 335$cached_env_img{$key} = q|<IMG 336 WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" 337 SRC="|."$dir".q|img9.png" 338 ALT="$ \sqrt{{2}}$">|; 339 340$key = q/displaystyle{frac{{y-1}}{{x-1}}};MSF=1.6;AAT/; 341$cached_env_img{$key} = q|<IMG 342 WIDTH="39" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 343 SRC="|."$dir".q|img69.png" 344 ALT="$\displaystyle {\frac{{y-1}}{{x-1}}}$">|; 345 346$key = q/mathbb{Z};MSF=1.6;AAT/; 347$cached_env_img{$key} = q|<IMG 348 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" 349 SRC="|."$dir".q|img14.png" 350 ALT="$ \mathbb {Z}$">|; 351 352$key = q/{frac{{1}}{{12}}};MSF=1.6;AAT/; 353$cached_env_img{$key} = q|<IMG 354 WIDTH="19" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" 355 SRC="|."$dir".q|img46.png" 356 ALT="$ {\frac{{1}}{{12}}}$">|; 357 358$key = q/displaystyle{frac{{x^3-yx^2-yx+y^2}}{{x^3-yx^2-x+y}}};MSF=1.6;AAT/; 359$cached_env_img{$key} = q|<IMG 360 WIDTH="120" HEIGHT="57" ALIGN="MIDDLE" BORDER="0" 361 SRC="|."$dir".q|img66.png" 362 ALT="$\displaystyle {\frac{{x^3-yx^2-yx+y^2}}{{x^3-yx^2-x+y}}}$">|; 363 364$key = q/displaystyle{frac{{exp(x)^2-exp(x)+1}}{{exp(x)^3+exp(x)}}};MSF=1.6;AAT/; 365$cached_env_img{$key} = q|<IMG 366 WIDTH="139" HEIGHT="57" ALIGN="MIDDLE" BORDER="0" 367 SRC="|."$dir".q|img23.png" 368 ALT="$\displaystyle {\frac{{\exp(x)^2-\exp(x)+1}}{{\exp(x)^3+\exp(x)}}}$">|; 369 370$key = q/scriptstylealpha;MSF=1.6;AAT/; 371$cached_env_img{$key} = q|<IMG 372 WIDTH="11" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" 373 SRC="|."$dir".q|img37.png" 374 ALT="$\scriptstyle \alpha$">|; 375 376$key = q/displaystyle{frac{{1}}{{(x-1)(x+1)}}};MSF=1.6;AAT/; 377$cached_env_img{$key} = q|<IMG 378 WIDTH="95" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 379 SRC="|."$dir".q|img65.png" 380 ALT="$\displaystyle {\frac{{1}}{{(x-1)(x+1)}}}$">|; 381 382$key = q/displaystylelim_{{x->+infty}}^{};MSF=1.6;AAT/; 383$cached_env_img{$key} = q|<IMG 384 WIDTH="44" HEIGHT="40" ALIGN="MIDDLE" BORDER="0" 385 SRC="|."$dir".q|img25.png" 386 ALT="$\displaystyle \lim_{{x->+\infty}}^{}$">|; 387 388$key = q/displaystyle{frac{{(x+2)(x-2)}}{{(x-1)(x+1)}}};MSF=1.6;AAT/; 389$cached_env_img{$key} = q|<IMG 390 WIDTH="95" HEIGHT="54" ALIGN="MIDDLE" BORDER="0" 391 SRC="|."$dir".q|img63.png" 392 ALT="$\displaystyle {\frac{{(x+2)(x-2)}}{{(x-1)(x+1)}}}$">|; 393 394$key = q/displaystylesum_{{p=1}}^{infty};MSF=1.6;AAT/; 395$cached_env_img{$key} = q|<IMG 396 WIDTH="25" HEIGHT="55" ALIGN="MIDDLE" BORDER="0" 397 SRC="|."$dir".q|img106.png" 398 ALT="$\displaystyle \sum_{{p=1}}^{\infty}$">|; 399 400$key = q/displaystyleleft.vphantom{frac{1+x+y}{(1+x)(1+y)}}right)^{3}_{};MSF=1.6;AAT/; 401$cached_env_img{$key} = q|<IMG 402 WIDTH="23" HEIGHT="61" ALIGN="MIDDLE" BORDER="0" 403 SRC="|."$dir".q|img119.png" 404 ALT="$\displaystyle \left.\vphantom{ \frac{1+x+y}{(1+x)(1+y)} }\right)^{3}_{}$">|; 405 406$key = q/displaystyle{frac{{a}}{{2cdotb}}};MSF=1.6;AAT/; 407$cached_env_img{$key} = q|<IMG 408 WIDTH="32" HEIGHT="44" ALIGN="MIDDLE" BORDER="0" 409 SRC="|."$dir".q|img4.png" 410 ALT="$\displaystyle {\frac{{a}}{{2\cdot b}}}$">|; 411 412$key = q/displaystyleleft.vphantom{frac{x}{1+y},,,frac{y}{1+x}}right);MSF=1.6;AAT/; 413$cached_env_img{$key} = q|<IMG 414 WIDTH="17" HEIGHT="54" ALIGN="MIDDLE" BORDER="0" 415 SRC="|."$dir".q|img113.png" 416 ALT="$\displaystyle \left.\vphantom{\frac{x}{1+y} , \frac{y}{1+x}}\right)$">|; 417 418$key = q/displaystyleleft{vphantom{array{{llllllr}x<col_mark>+<col_mark>y<col_mark>+<col_col_mark>y<col_mark>+<col_mark>z<col_mark>=<col_mark>-2array{}right.;MSF=1.6;AAT/; 419$cached_env_img{$key} = q|<IMG 420 WIDTH="19" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 421 SRC="|."$dir".q|img15.png" 422 ALT="$\displaystyle \left\{\vphantom{ \begin{array}{llllllr} 423x &+& y &+& az&=&1\\\\ 424x & +& a y&+& z&=&1 \\\\ 425ax & +&y &+& z&=&-2 426\end{array}}\right.$">|; 427 428$key = q/displaystylearray{{cccccc}0<col_mark>1<col_mark>0<col_mark>ldots<col_mark><col_mmark>-a_1<col_mark><col_mark>ldots<col_mark><col_mark>-a_{d-1}array{;MSF=1.6;AAT/; 429$cached_env_img{$key} = q|<IMG 430 WIDTH="247" HEIGHT="152" ALIGN="MIDDLE" BORDER="0" 431 SRC="|."$dir".q|img104.png" 432 ALT="$\displaystyle \begin{array}{cccccc} 0&1&0&\ldots&&0 \vdots&\ddots&\ddots&\dd... 433...ots&\ddots&0 0&\ldots&&\ldots&0&1 -a_0&-a_1&&\ldots&&-a_{d-1} \end{array}$">|; 434 435$key = q/mapsto;MSF=1.6;AAT/; 436$cached_env_img{$key} = q|<IMG 437 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0" 438 SRC="|."$dir".q|img12.png" 439 ALT="$ \mapsto$">|; 440 441$key = q/mathbb{R};MSF=1.6;AAT/; 442$cached_env_img{$key} = q|<IMG 443 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" 444 SRC="|."$dir".q|img13.png" 445 ALT="$ \mathbb {R}$">|; 446 447$key = q/displaystyleleft{vphantom{array{{lcl}x(u,v)<col_mark>=<col_mark>u,cos(v)<row_marmark>u,sin(v)<row_mark>z(u,v)<col_mark>=<col_mark>1-u;.array{}right.;MSF=1.6;AAT/; 448$cached_env_img{$key} = q|<IMG 449 WIDTH="19" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 450 SRC="|."$dir".q|img96.png" 451 ALT="$\displaystyle \left\{\vphantom{ 452\begin{array}{lcl} 453x(u,v)&=& u \cos(v)\\\\ 454y(u,v)&=& u \sin(v)\\\\ 455z(u,v)&=& 1-u\;. 456\end{array}}\right.$">|; 457 458$key = q/displaystyle{frac{{x^2}}{{(x-1)(x+1)}}};MSF=1.6;AAT/; 459$cached_env_img{$key} = q|<IMG 460 WIDTH="95" HEIGHT="57" ALIGN="MIDDLE" BORDER="0" 461 SRC="|."$dir".q|img64.png" 462 ALT="$\displaystyle {\frac{{x^2}}{{(x-1)(x+1)}}}$">|; 463 464$key = q/displaystyle{frac{{1}}{{2a^4-2a^3-2a^2+2a}}};MSF=1.6;AAT/; 465$cached_env_img{$key} = q|<IMG 466 WIDTH="141" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 467 SRC="|."$dir".q|img41.png" 468 ALT="$\displaystyle {\frac{{1}}{{2a^4-2a^3-2a^2+2a}}}$">|; 469 470$key = q/displaystylett{left[array{{rrr}-1<col_mark>1<col_mark>1<row_mark>0<col_mark>1<col_mark>-2<row_mark>-1<col_mark>1<col_mark>1array{right];MSF=1.6;AAT/; 471$cached_env_img{$key} = q|<IMG 472 WIDTH="121" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 473 SRC="|."$dir".q|img55.png" 474 ALT="$\displaystyle \tt\left[ 475\begin{array}{rrr} 476-1 & 1 & 1\\\\ 4770 & 1 & -2\\\\ 478-1 & 1 & 1 479\end{array}\right]$">|; 480 481$key = q/array{{ccc}1<col_mark>11<col_mark>-7<row_mark>-3<col_mark>-9<col_mark>9<row_mark>5<col_mark>-5<col_mark>1array{;MSF=1.6;AAT/; 482$cached_env_img{$key} = q|<IMG 483 WIDTH="112" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 484 SRC="|."$dir".q|img48.png" 485 ALT="$ \begin{array}{ccc} 4861 & 11 & -7\\\\ 487-3 & -9 & 9\\\\ 4885 & -5 & 1 489\end{array}$">|; 490 491$key = q/_size;MSF=1.6;AAT/; 492$cached_env_img{$key} = q|<IMG 493 WIDTH="35" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" 494 SRC="|."$dir".q|img36.png" 495 ALT="$ \_size$">|; 496 497$key = q/displaystyle{frac{{x}}{{1+y}}};MSF=1.6;AAT/; 498$cached_env_img{$key} = q|<IMG 499 WIDTH="39" HEIGHT="44" ALIGN="MIDDLE" BORDER="0" 500 SRC="|."$dir".q|img111.png" 501 ALT="$\displaystyle {\frac{{x}}{{1+y}}}$">|; 502 503$key = q/displaystyle{frac{{ln(x)}}{{x(x^2+1)^2}}};MSF=1.6;AAT/; 504$cached_env_img{$key} = q|<IMG 505 WIDTH="72" HEIGHT="54" ALIGN="MIDDLE" BORDER="0" 506 SRC="|."$dir".q|img75.png" 507 ALT="$\displaystyle {\frac{{\ln(x)}}{{x(x^2+1)^2}}}$">|; 508 509$key = q/pi;MSF=1.6;AAT/; 510$cached_env_img{$key} = q|<IMG 511 WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" 512 SRC="|."$dir".q|img7.png" 513 ALT="$ \pi$">|; 514 515$key = q/displaystyle{frac{{1}}{{2}}};MSF=1.6;AAT/; 516$cached_env_img{$key} = q|<IMG 517 WIDTH="15" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 518 SRC="|."$dir".q|img26.png" 519 ALT="$\displaystyle {\frac{{1}}{{2}}}$">|; 520 521$key = q/displaystyle{frac{{1}}{{1+sin(x)+cos(x)}}};MSF=1.6;AAT/; 522$cached_env_img{$key} = q|<IMG 523 WIDTH="127" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 524 SRC="|."$dir".q|img74.png" 525 ALT="$\displaystyle {\frac{{1}}{{1+\sin(x)+\cos(x)}}}$">|; 526 527$key = q/displaystyleleft.vphantom{array{{cccccc}0<col_mark>1<col_mark>0<col_mark>ldots<c_1<col_mark><col_mark>ldots<col_mark><col_mark>-a_{d-1}array{}right);MSF=1.6;AAT/; 528$cached_env_img{$key} = q|<IMG 529 WIDTH="19" HEIGHT="152" ALIGN="MIDDLE" BORDER="0" 530 SRC="|."$dir".q|img105.png" 531 ALT="$\displaystyle \left.\vphantom{ \begin{array}{cccccc} 0&1&0&\ldots&&0 \vdots&... 532...s&0 0&\ldots&&\ldots&0&1 -a_0&-a_1&&\ldots&&-a_{d-1} \end{array} }\right)$">|; 533 534$key = q/displaystylett{left[left[array{{rrr}-1<col_mark>1<col_mark>1<row_mark>0<col_mark>a+2<col_mark>0<row_mark>0<col_mark>0<col_mark>a-1array{right]right];MSF=1.6;AAT/; 535$cached_env_img{$key} = q|<IMG 536 WIDTH="323" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 537 SRC="|."$dir".q|img56.png" 538 ALT="$\displaystyle \tt\left[ \left[ 539\begin{array}{rrr} 540-1 & 1 & 1\\\\ 5410 & 1 & -2 ... 542...}{ccc} 543-a+1 & 0 & 0\\\\ 5440 & a+2 & 0\\\\ 5450 & 0 & a-1 546\end{array}\right] 547\right]$">|; 548 549$key = q/displaystyleint_{1}^{2};MSF=1.6;AAT/; 550$cached_env_img{$key} = q|<IMG 551 WIDTH="24" HEIGHT="55" ALIGN="MIDDLE" BORDER="0" 552 SRC="|."$dir".q|img27.png" 553 ALT="$\displaystyle \int_{1}^{2}$">|; 554 555$key = q/displaystylearray{{lcl}x(t)<col_mark>=<col_mark>sin(132,t)<row_mark>y(t)<col_mark>=<col_mark>cos^3(126,t)array{;MSF=1.6;AAT/; 556$cached_env_img{$key} = q|<IMG 557 WIDTH="159" HEIGHT="55" ALIGN="MIDDLE" BORDER="0" 558 SRC="|."$dir".q|img94.png" 559 ALT="$\displaystyle \begin{array}{lcl} 560x(t)&=& \sin(132 t)\\\\ 561y(t)&=& \cos^3(126 t) 562\end{array}$">|; 563 564$key = q/displaystyleint_{0}^{x};MSF=1.6;AAT/; 565$cached_env_img{$key} = q|<IMG 566 WIDTH="23" HEIGHT="49" ALIGN="MIDDLE" BORDER="0" 567 SRC="|."$dir".q|img24.png" 568 ALT="$\displaystyle \int_{0}^{x}$">|; 569 570$key = q/displaystylesqrt{{cos(x)}};MSF=1.6;AAT/; 571$cached_env_img{$key} = q|<IMG 572 WIDTH="60" HEIGHT="40" ALIGN="MIDDLE" BORDER="0" 573 SRC="|."$dir".q|img81.png" 574 ALT="$\displaystyle \sqrt{{\cos(x)}}$">|; 575 576$key = q/epsilon;MSF=1.6;AAT/; 577$cached_env_img{$key} = q|<IMG 578 WIDTH="11" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" 579 SRC="|."$dir".q|img88.png" 580 ALT="$ \epsilon$">|; 581 582$key = q/displaystyleleft(vphantom{array{{cccccc}0<col_mark>1<col_mark>0<col_mark>ldots<c_1<col_mark><col_mark>ldots<col_mark><col_mark>-a_{d-1}array{}right.;MSF=1.6;AAT/; 583$cached_env_img{$key} = q|<IMG 584 WIDTH="19" HEIGHT="152" ALIGN="MIDDLE" BORDER="0" 585 SRC="|."$dir".q|img103.png" 586 ALT="$\displaystyle \left(\vphantom{ \begin{array}{cccccc} 0&1&0&\ldots&&0 \vdots&... 587...s&0 0&\ldots&&\ldots&0&1 -a_0&-a_1&&\ldots&&-a_{d-1} \end{array} }\right.$">|; 588 589$key = q/geq;MSF=1.6;AAT/; 590$cached_env_img{$key} = q|<IMG 591 WIDTH="15" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" 592 SRC="|."$dir".q|img11.png" 593 ALT="$ \geq$">|; 594 595$key = q/notin;MSF=1.6;AAT/; 596$cached_env_img{$key} = q|<IMG 597 WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" 598 SRC="|."$dir".q|img45.png" 599 ALT="$ \not\in$">|; 600 601$key = q/displaystyle{frac{{sin(x)}}{{x}}};MSF=1.6;AAT/; 602$cached_env_img{$key} = q|<IMG 603 WIDTH="45" HEIGHT="54" ALIGN="MIDDLE" BORDER="0" 604 SRC="|."$dir".q|img84.png" 605 ALT="$\displaystyle {\frac{{\sin(x)}}{{x}}}$">|; 606 607$key = q/displaystyleint;MSF=1.6;AAT/; 608$cached_env_img{$key} = q|<IMG 609 WIDTH="17" HEIGHT="48" ALIGN="MIDDLE" BORDER="0" 610 SRC="|."$dir".q|img30.png" 611 ALT="$\displaystyle \int$">|; 612 613$key = q/displaystylearray{{lcl}x(t)<col_mark>=<col_mark>sin(4,t)<row_mark>y(t)<col_mark>=<col_mark>cos^3(6,t)array{;MSF=1.6;AAT/; 614$cached_env_img{$key} = q|<IMG 615 WIDTH="143" HEIGHT="55" ALIGN="MIDDLE" BORDER="0" 616 SRC="|."$dir".q|img92.png" 617 ALT="$\displaystyle \begin{array}{lcl} 618x(t)&=& \sin(4 t)\\\\ 619y(t)&=& \cos^3(6 t) 620\end{array}$">|; 621 622$key = q/displaystyle{frac{{t^2-t+1}}{{t^4+t^2}}};MSF=1.6;AAT/; 623$cached_env_img{$key} = q|<IMG 624 WIDTH="65" HEIGHT="57" ALIGN="MIDDLE" BORDER="0" 625 SRC="|."$dir".q|img34.png" 626 ALT="$\displaystyle {\frac{{t^2-t+1}}{{t^4+t^2}}}$">|; 627 628$key = q/displaystyleiint_{D}^{};MSF=1.6;AAT/; 629$cached_env_img{$key} = q|<IMG 630 WIDTH="28" HEIGHT="48" ALIGN="MIDDLE" BORDER="0" 631 SRC="|."$dir".q|img116.png" 632 ALT="$\displaystyle \iint_{D}^{}$">|; 633 634$key = q/leq;MSF=1.6;AAT/; 635$cached_env_img{$key} = q|<IMG 636 WIDTH="15" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" 637 SRC="|."$dir".q|img10.png" 638 ALT="$ \leq$">|; 639 640$key = q/Phi^{{-1}}_{};MSF=1.6;AAT/; 641$cached_env_img{$key} = q|<IMG 642 WIDTH="31" HEIGHT="36" ALIGN="MIDDLE" BORDER="0" 643 SRC="|."$dir".q|img115.png" 644 ALT="$ \Phi^{{-1}}_{}$">|; 645 646$key = q/displaystyle{frac{{y}}{{1+x}}};MSF=1.6;AAT/; 647$cached_env_img{$key} = q|<IMG 648 WIDTH="39" HEIGHT="44" ALIGN="MIDDLE" BORDER="0" 649 SRC="|."$dir".q|img112.png" 650 ALT="$\displaystyle {\frac{{y}}{{1+x}}}$">|; 651 652$key = q/displaystyle{frac{{x^4+x^3-4x^2-4x}}{{x^4+x^3-x^2-x}}};MSF=1.6;AAT/; 653$cached_env_img{$key} = q|<IMG 654 WIDTH="122" HEIGHT="57" ALIGN="MIDDLE" BORDER="0" 655 SRC="|."$dir".q|img60.png" 656 ALT="$\displaystyle {\frac{{x^4+x^3-4x^2-4x}}{{x^4+x^3-x^2-x}}}$">|; 657 658$key = q/displaystyleleft(vphantom{frac{1+x+y}{(1+x)(1+y)}}right.;MSF=1.6;AAT/; 659$cached_env_img{$key} = q|<IMG 660 WIDTH="17" HEIGHT="54" ALIGN="MIDDLE" BORDER="0" 661 SRC="|."$dir".q|img117.png" 662 ALT="$\displaystyle \left(\vphantom{ \frac{1+x+y}{(1+x)(1+y)} }\right.$">|; 663 664$key = q/displaystyleleft.vphantom{array{{ccc}a-1<col_mark>2a^3+3a+1<col_mark>-2a^3+a^2+aow_mark>a^3-2a+1<col_mark>-a^3+2a-1<col_mark>a^3-2a^2+1array{}right];MSF=1.6;AAT/; 665$cached_env_img{$key} = q|<IMG 666 WIDTH="16" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 667 SRC="|."$dir".q|img44.png" 668 ALT="$\displaystyle \left.\vphantom{ 669\begin{array}{ccc} 670a-1 & 2a^3+3a+1 & -2a^3+a^2+a... 671...+2a-1 & 2a^3-a^2-2a+1\\\\ 672a^3-2a+1 & -a^3+2a-1 & a^3-2a^2+1 673\end{array}}\right]$">|; 674 675$key = q/neq;MSF=1.6;AAT/; 676$cached_env_img{$key} = q|<IMG 677 WIDTH="15" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" 678 SRC="|."$dir".q|img17.png" 679 ALT="$ \neq$">|; 680 681$key = q/displaystylearray{{lcl}x(t)<col_mark>=<col_mark>a,cos(t)<row_mark>y(t)<col_mark>=<col_mark>a,sin(t)<row_mark>z(t)<col_mark>=<col_mark>1-a;.array{;MSF=1.6;AAT/; 682$cached_env_img{$key} = q|<IMG 683 WIDTH="138" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 684 SRC="|."$dir".q|img101.png" 685 ALT="$\displaystyle \begin{array}{lcl} 686x(t)&=& a \cos(t)\\\\ 687y(t)&=& a \sin(t)\\\\ 688z(t)&=& 1-a\;. 689\end{array}$">|; 690 691$key = q/displaystyle{frac{{t^p}}{{p!}}};MSF=1.6;AAT/; 692$cached_env_img{$key} = q|<IMG 693 WIDTH="21" HEIGHT="52" ALIGN="MIDDLE" BORDER="0" 694 SRC="|."$dir".q|img107.png" 695 ALT="$\displaystyle {\frac{{t^p}}{{p!}}}$">|; 696 697$key = q/sqrt{{x^2+y^2}};MSF=1.6;AAT/; 698$cached_env_img{$key} = q|<IMG 699 WIDTH="63" HEIGHT="39" ALIGN="MIDDLE" BORDER="0" 700 SRC="|."$dir".q|img95.png" 701 ALT="$ \sqrt{{x^2+y^2}}$">|; 702 703$key = q/alpha;MSF=1.6;AAT/; 704$cached_env_img{$key} = q|<IMG 705 WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" 706 SRC="|."$dir".q|img18.png" 707 ALT="$ \alpha$">|; 708 709$key = q/displaystyle{frac{{1}}{{t^2}}};MSF=1.6;AAT/; 710$cached_env_img{$key} = q|<IMG 711 WIDTH="19" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 712 SRC="|."$dir".q|img32.png" 713 ALT="$\displaystyle {\frac{{1}}{{t^2}}}$">|; 714 715$key = q/displaystyle{frac{{1+x+y}}{{(1+x)(1+y)}}};MSF=1.6;AAT/; 716$cached_env_img{$key} = q|<IMG 717 WIDTH="95" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 718 SRC="|."$dir".q|img118.png" 719 ALT="$\displaystyle {\frac{{1+x+y}}{{(1+x)(1+y)}}}$">|; 720 721$key = q/displaystyle{frac{{1}}{{x^3+1}}};MSF=1.6;AAT/; 722$cached_env_img{$key} = q|<IMG 723 WIDTH="45" HEIGHT="51" ALIGN="MIDDLE" BORDER="0" 724 SRC="|."$dir".q|img28.png" 725 ALT="$\displaystyle {\frac{{1}}{{x^3+1}}}$">|; 726 727$key = q/displaystylearray{{lcl}x(t)<col_mark>=<col_mark>t,cos(at)<row_mark>y(t)<col_mark>=<col_mark>t,sin(at)<row_mark>z(t)<col_mark>=<col_mark>1-t;.array{;MSF=1.6;AAT/; 728$cached_env_img{$key} = q|<IMG 729 WIDTH="143" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 730 SRC="|."$dir".q|img99.png" 731 ALT="$\displaystyle \begin{array}{lcl} 732x(t)&=& t \cos(a t)\\\\ 733y(t)&=& t \sin(a t)\\\\ 734z(t)&=& 1-t\;. 735\end{array}$">|; 736 737$key = q/displaystylesqrt{{e^x-1}};MSF=1.6;AAT/; 738$cached_env_img{$key} = q|<IMG 739 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" 740 SRC="|."$dir".q|img72.png" 741 ALT="$\displaystyle \sqrt{{e^x-1}}$">|; 742 743$key = q/infty;MSF=1.6;AAT/; 744$cached_env_img{$key} = q|<IMG 745 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0" 746 SRC="|."$dir".q|img6.png" 747 ALT="$ \infty$">|; 748 749$key = q/Phi;MSF=1.6;AAT/; 750$cached_env_img{$key} = q|<IMG 751 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" 752 SRC="|."$dir".q|img108.png" 753 ALT="$ \Phi$">|; 754 755$key = q/displaystyleleft(vphantom{frac{x}{1+y},,,frac{y}{1+x}}right.;MSF=1.6;AAT/; 756$cached_env_img{$key} = q|<IMG 757 WIDTH="17" HEIGHT="54" ALIGN="MIDDLE" BORDER="0" 758 SRC="|."$dir".q|img110.png" 759 ALT="$\displaystyle \left(\vphantom{\frac{x}{1+y} , \frac{y}{1+x}}\right.$">|; 760 761$key = q/displaystyle{frac{{x^2-y}}{{(x-1)(x+1)}}};MSF=1.6;AAT/; 762$cached_env_img{$key} = q|<IMG 763 WIDTH="95" HEIGHT="57" ALIGN="MIDDLE" BORDER="0" 764 SRC="|."$dir".q|img68.png" 765 ALT="$\displaystyle {\frac{{x^2-y}}{{(x-1)(x+1)}}}$">|; 766 767$key = q/displaystyleleft{vphantom{array{{lcl}x(t)<col_mark>=<col_mark>sin(4,t)<row_mark>y(t)<col_mark>=<col_mark>cos^3(6,t)array{}right.;MSF=1.6;AAT/; 768$cached_env_img{$key} = q|<IMG 769 WIDTH="17" HEIGHT="55" ALIGN="MIDDLE" BORDER="0" 770 SRC="|."$dir".q|img91.png" 771 ALT="$\displaystyle \left\{\vphantom{ 772\begin{array}{lcl} 773x(t)&=& \sin(4 t)\\\\ 774y(t)&=& \cos^3(6 t) 775\end{array}}\right.$">|; 776 777$key = q/displaystyleiint_{Delta}^{};MSF=1.6;AAT/; 778$cached_env_img{$key} = q|<IMG 779 WIDTH="27" HEIGHT="48" ALIGN="MIDDLE" BORDER="0" 780 SRC="|."$dir".q|img120.png" 781 ALT="$\displaystyle \iint_{\Delta}^{}$">|; 782 783$key = q/left.vphantom{array{{ccc}1<col_mark>1<col_mark>a<row_mark>1<col_mark>a<col_mark>1<row_mark>a<col_mark>1<col_mark>1array{}right];MSF=1.6;AAT/; 784$cached_env_img{$key} = q|<IMG 785 WIDTH="16" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 786 SRC="|."$dir".q|img52.png" 787 ALT="$ \left.\vphantom{ 788\begin{array}{ccc} 7891 & 1 & a\\\\ 7901 & a & 1\\\\ 791a & 1 & 1 792\end{array}}\right]$">|; 793 794$key = q/left.vphantom{array{{ccc}2a-1<col_mark>a<col_mark>2a-1<row_mark>a^2+a-2<col_markl_mark>a-1<row_mark>a^2+a-1<col_mark>a^2+a-1<col_mark>aarray{}right];MSF=1.6;AAT/; 795$cached_env_img{$key} = q|<IMG 796 WIDTH="16" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 797 SRC="|."$dir".q|img40.png" 798 ALT="$ \left.\vphantom{ 799\begin{array}{ccc} 8002a-1 & a & 2a-1\\\\ 801a^2+a-2 & a^2-1 & a-1\\\\ 802a^2+a-1 & a^2+a-1 & a 803\end{array}}\right]$">|; 804 805$key = q/sqrt{{-1}};MSF=1.6;AAT/; 806$cached_env_img{$key} = q|<IMG 807 WIDTH="37" HEIGHT="36" ALIGN="MIDDLE" BORDER="0" 808 SRC="|."$dir".q|img5.png" 809 ALT="$ \sqrt{{-1}}$">|; 810 811$key = q/cup;MSF=1.6;AAT/; 812$cached_env_img{$key} = q|<IMG 813 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0" 814 SRC="|."$dir".q|img22.png" 815 ALT="$ \cup$">|; 816 817$key = q/displaystyle{frac{{1}}{{displaystyle{a_1+frac{1}{displaystyle{a_2+frac{1}{ddots+displaystyle{frac{1}{a_n}}}}}}}}};MSF=1.6;AAT/; 818$cached_env_img{$key} = q|<IMG 819 WIDTH="130" HEIGHT="158" ALIGN="MIDDLE" BORDER="0" 820 SRC="|."$dir".q|img102.png" 821 ALT="$\displaystyle {\frac{{1}}{{\displaystyle{a_1+ 822\frac{1}{\displaystyle{a_2+\frac{1}{\ddots+\displaystyle{\frac{1}{a_n}}}}}}}}}$">|; 823 824$key = q/displaystyleleft{vphantom{array{{lcl}x(t)<col_mark>=<col_mark>a,cos(t)<row_mark>l_mark>a,sin(t)<row_mark>z(t)<col_mark>=<col_mark>1-a;.array{}right.;MSF=1.6;AAT/; 825$cached_env_img{$key} = q|<IMG 826 WIDTH="19" HEIGHT="74" ALIGN="MIDDLE" BORDER="0" 827 SRC="|."$dir".q|img100.png" 828 ALT="$\displaystyle \left\{\vphantom{ 829\begin{array}{lcl} 830x(t)&=& a \cos(t)\\\\ 831y(t)&=& a \sin(t)\\\\ 832z(t)&=& 1-a\;. 833\end{array}}\right.$">|; 834 835$key = q/displaystyleleft{vphantom{array{{lcl}x(t)<col_mark>=<col_mark>sin(t)<row_mark>y(t)<col_mark>=<col_mark>cos^3(t)array{}right.;MSF=1.6;AAT/; 836$cached_env_img{$key} = q|<IMG 837 WIDTH="17" HEIGHT="55" ALIGN="MIDDLE" BORDER="0" 838 SRC="|."$dir".q|img89.png" 839 ALT="$\displaystyle \left\{\vphantom{ 840\begin{array}{lcl} 841x(t)&=& \sin(t)\\\\ 842y(t)&=& \cos^3(t) 843\end{array}}\right.$">|; 844 845$key = q/displaystyle{frac{{(x+2)(x+1)(x-2)}}{{x^3+x^2-x-1}}};MSF=1.6;AAT/; 846$cached_env_img{$key} = q|<IMG 847 WIDTH="139" HEIGHT="54" ALIGN="MIDDLE" BORDER="0" 848 SRC="|."$dir".q|img61.png" 849 ALT="$\displaystyle {\frac{{(x+2)(x+1)(x-2)}}{{x^3+x^2-x-1}}}$">|; 850 851$key = q/displaystyleint_{0}^{{pislash2}};MSF=1.6;AAT/; 852$cached_env_img{$key} = q|<IMG 853 WIDTH="37" HEIGHT="56" ALIGN="MIDDLE" BORDER="0" 854 SRC="|."$dir".q|img80.png" 855 ALT="$\displaystyle \int_{0}^{{\pi/2}}$">|; 856 857$key = q/displaystylelim_{{xrightarrow0^+}}^{};MSF=1.6;AAT/; 858$cached_env_img{$key} = q|<IMG 859 WIDTH="34" HEIGHT="40" ALIGN="MIDDLE" BORDER="0" 860 SRC="|."$dir".q|img85.png" 861 ALT="$\displaystyle \lim_{{x\rightarrow 0^+}}^{}$">|; 862 863$key = q/in;MSF=1.6;AAT/; 864$cached_env_img{$key} = q|<IMG 865 WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0" 866 SRC="|."$dir".q|img21.png" 867 ALT="$ \in$">|; 868 8691; 870 871