# LaTeX2HTML 2002-2-1 (1.70)
# Associate images original text with physical files.


$key = q/[-3,-1];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.gif"
 ALT="$[-3,-1]$">|; 

$key = q/x=3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6.gif"
 ALT="$x=3$">|; 

$key = q/(0,+infty);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.gif"
 ALT="$(0,+\infty)$">|; 

$key = q/pislash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.gif"
 ALT="$\pi/2$">|; 

$key = q/x=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img30.gif"
 ALT="$x=0$">|; 

$key = q/xrightarrow-infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.gif"
 ALT="$x \rightarrow -\infty$">|; 

$key = q/displaystylembox{sen},(x)=-1Rightarrowx=3pislash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.gif"
 ALT="$\displaystyle \mbox{sen}   (x)=-1 \Rightarrow x = 3\pi /2$">|; 

$key = q/displaystyleint(x+1)e^{x}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.gif"
 ALT="$\displaystyle \int (x+1)e^{x} dx $">|; 

$key = q/displaystylelim_{xrightarrow0^+}frac{ln(1+x)}{4x}array{{c}0slash0=mbox{LH}array{sh(1+x)}{4}=lim_{xrightarrow0^+}frac{1slash4}{1+x}=frac{1}{4};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.gif"
 ALT="$\displaystyle \lim_{x \rightarrow 0^+} \frac{\ln(1+x)}{4x}
\begin{array}{c} 0/...
...^+} \frac{1/(1+x)}{4} =
\lim_{x \rightarrow 0^+} \frac{1/4}{1+x} = \frac{1}{4}$">|; 

$key = q/x;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img43.gif"
 ALT="$x$">|; 

$key = q/displaystylef'(x)=2cos(x)-2cos(x)(-sen(x))=2cos(x)(1+mbox{sen},(x));MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="457" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.gif"
 ALT="$\displaystyle f'(x) = 2 \cos(x) - 2\cos(x)(-sen(x)) =
2 \cos(x) (1+\mbox{sen}   (x))$">|; 

$key = q/f(3pislash2)=2(-1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.gif"
 ALT="$f(3\pi/2)=2(-1) $">|; 

$key = q/displaystylelim_{xrightarrow0}mbox{cotg},(x)(mbox{sen},(5x)-mbox{sen},(2x))stackrow0}frac{5cos(5x)-2cos(2x)}{sec^2(x)}=frac{5(1)-2(1)}{1^2}=3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="831" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.gif"
 ALT="$\displaystyle \lim_{x \rightarrow 0} \mbox{cotg}   (x)(\mbox{sen}   (5x)-\mbo...
...rightarrow 0} \frac{5\cos(5x)-2\cos(2x)}{\sec^2(x)} =
\frac{5(1)-2(1)}{1^2} = 3$">|; 

$key = q/displaystyleint2x(4-x^2)^{1slash3}dx=intu^{1slash3}(-)du=-intu^{1slash3}du=-fracash3}=-frac{3u^{4slash3}}{4}+C=-frac{3(4-x^2)^{4slash3}}{4}+C;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="727" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.gif"
 ALT="$\displaystyle \int 2x (4-x^2)^{1/3} dx =
\int u^{1/3}(-)du = - \int u^{1/3} du...
...frac{u^{4/3}}{4/3} = - \frac{3 u^{4/3}}{4} + C =
- \frac{3(4-x^2)^{4/3}}{4} + C$">|; 

$key = q/displaystyleh=frac{2000}{x^2}RightarrowC(x)=2x^2+16xfrac{2000}{x^2}=2x^2+frac{16(2000)}{x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="418" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.gif"
 ALT="$\displaystyle h = \frac{2000}{x^2} \Rightarrow
C(x) = 2 x^2 + 16x \frac{2000}{x^2} = 2x^2 + \frac{16(2000)}{x}$">|; 

$key = q/displaystylearray{{rl}u=x+1,&du=dxdv=e^xdx,&v=e^xarray{;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.gif"
 ALT="$\displaystyle \begin{array}{rl}
u = x+1, &amp; du = dx \\\\
dv = e^x dx, &amp; v = e^x \end{array} $">|; 

$key = q/inftyslashinfty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.gif"
 ALT="$\infty/ \infty$">|; 

$key = q/C'(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.gif"
 ALT="$C'(x)$">|; 

$key = q/f(2pi)=-(1)^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.gif"
 ALT="$f(2\pi) = -(1)^2 $">|; 

$key = q/xin{{mathbb{R}{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.gif"
 ALT="$x \in {\mathbb{R}}$">|; 

$key = q/displaystyleRightarrowx=3pislash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.gif"
 ALT="$\displaystyle \Rightarrow x=3\pi/2$">|; 

$key = q/'[-2,0];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.gif"
 ALT="$'[-2,0]$">|; 

$key = q/displaystylelim_{xrightarrow-infty}x^4e^{2x}=displaystylelim_{xrightarrow-infty}elim_{xrightarrow-infty}frac{24}{16e^{-2x}}=frac{1}{+infty}=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="634" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.gif"
 ALT="$\displaystyle \lim_{x \rightarrow -\infty} x^4 e^{2x} =
\displaystyle \lim_{x \...
...ystyle \lim_{x \rightarrow -\infty} \frac{24}{16e^{-2x}} =
\frac{1}{+\infty}=0$">|; 

$key = q/I_1Longrightarrow;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.gif"
 ALT="$I_1
\Longrightarrow $">|; 

$key = q/displaystyle4x=frac{16(2000)}{x^2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.gif"
 ALT="$\displaystyle 4x = \frac{16(2000)}{x^2}$">|; 

$key = q/f;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.gif"
 ALT="$f$">|; 

$key = q/displaystyleC'(x)=4x-frac{16(2000)}{x^2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.gif"
 ALT="$\displaystyle C'(x) = 4x - \frac{16(2000)}{x^2} $">|; 

$key = q/C;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img63.gif"
 ALT="$C$">|; 

$key = q/(-infty,-2]bigcup[0,+infty);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.gif"
 ALT="$(-\infty,-2] \bigcup [0,+\infty)$">|; 

$key = q/x=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img18.gif"
 ALT="$x=2$">|; 

$key = q/f(0)=-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.gif"
 ALT="$f(0)= -1$">|; 

$key = q/displaystyleint(x+1)e^xdx=(x+1)e^x-inte^xdx=(x+1)e^x-e^x+C;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.gif"
 ALT="$\displaystyle \int (x+1) e^x dx =
(x+1)e^x - \int e^x dx = (x+1)e^x - e^x + C$">|; 

$key = q/displaystylelim_{xrightarrow+infty}x^4e^{2x}stackrel{inftycdotinfty}{=}+infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.gif"
 ALT="$\displaystyle \lim_{x \rightarrow +\infty} x^4 e^{2x}
\stackrel{\infty \cdot \infty}{=} + \infty$">|; 

$key = q/displaystyleC'(x)=frac{4(x^3-8000)}{x^2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.gif"
 ALT="$\displaystyle C'(x) = \frac{4(x^3-8000)}{x^2} $">|; 

$key = q/displaystylef'(x)=4x^3e^{2x}+x^4(2)e^{2x}=2x^3(2+x)e^{2x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.gif"
 ALT="$\displaystyle f'(x) = 4x^3 e^{2x} + x^4 (2) e^{2x} =
2x^3(2 + x) e^{2x}$">|; 

$key = q/displaystyleLongrightarrowf''(x)geq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.gif"
 ALT="$\displaystyle \Longrightarrow f''(x) \geq 0 $">|; 

$key = q/x=3.5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img19.gif"
 ALT="$x=3.5$">|; 

$key = q/displaystylelim_{xrightarrow-infty}x^4e^{2x}stackrel{inftycdot0}{=}displaystylelrray{displaystylelim_{xrightarrow-infty}frac{4x^3}{-2e^{-2x}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.gif"
 ALT="$\displaystyle \lim_{x \rightarrow -\infty} x^4 e^{2x}
\stackrel{\infty \cdot 0...
...\end{array}
\displaystyle \lim_{x \rightarrow -\infty} \frac{4x^3}{-2e^{-2x}} $">|; 

$key = q/xrightarrowinfty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img35.gif"
 ALT="$x \rightarrow \infty$">|; 

$key = q/displaystyleint2x(4-x^2)^{1slash3}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.gif"
 ALT="$\displaystyle \int 2x (4-x^2)^{1/3} dx$">|; 

$key = q/e^{2x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img26.gif"
 ALT="$e^{2x}$">|; 

$key = q/f(x)=2mbox{sen},(x)-cos^2(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.gif"
 ALT="$f(x) = 2\mbox{sen}   (x)-\cos^2(x)$">|; 

$key = q/displaystyleRightarrowx=pislash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.gif"
 ALT="$\displaystyle \Rightarrow x=\pi/2$">|; 

$key = q/x^3(2+x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.gif"
 ALT="$x^3(2+x)$">|; 

$key = q/2000cm^3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img41.gif"
 ALT="$2000 cm^3$">|; 

$key = q/I_2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.gif"
 ALT="$I_2$">|; 

$key = q/displaystyleLongrightarrowf''(x)leq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.gif"
 ALT="$\displaystyle \Longrightarrow f''(x) \leq 0 $">|; 

$key = q/h;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img44.gif"
 ALT="$h$">|; 

$key = q/displaystylecos(x)=0Rightarrowx=pislash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.gif"
 ALT="$\displaystyle \cos(x) = 0 \Rightarrow x = \pi /2 $">|; 

$key = q/xrightarrow+infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.gif"
 ALT="$x \rightarrow +\infty$">|; 

$key = q/x=4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7.gif"
 ALT="$x=4$">|; 

$key = q/displaystylelim_{xrightarrow0^+}frac{ln(1+x)}{4x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.gif"
 ALT="$\displaystyle \lim_{x \rightarrow 0^+} \frac{\ln(1+x)}{4x}$">|; 

$key = q/C(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.gif"
 ALT="$C(x)$">|; 

$key = q/f(pislash2)=2(1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.gif"
 ALT="$f(\pi/2) = 2(1)$">|; 

$key = q/I_2=(-infty,1]bigcup[2,3.5];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.gif"
 ALT="$I_2 = (-\infty,1] \bigcup
[2, 3.5]$">|; 

$key = q/3pislash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.gif"
 ALT="$3\pi/2$">|; 

$key = q/x=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img17.gif"
 ALT="$x=1$">|; 

$key = q/cm^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img42.gif"
 ALT="$cm^2$">|; 

$key = q/C(x)rightarrow+infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.gif"
 ALT="$C(x) \rightarrow +\infty$">|; 

$key = q/displaystyleC=2x^2(1)+4xh(4)=2x^2+16xh;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.gif"
 ALT="$\displaystyle C = 2x^2 (1) + 4xh (4) = 2x^2 + 16 xh$">|; 

$key = q/f(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.gif"
 ALT="$f(x)$">|; 

$key = q/displaystyleC''(x)=4+frac{32(2000)}{x^3}>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.gif"
 ALT="$\displaystyle C''(x) = 4 + \frac{32(2000)}{x^3} &gt; 0$">|; 

$key = q/bullet;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img8.gif"
 ALT="$\bullet$">|; 

$key = q/f'(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.gif"
 ALT="$f'(x)$">|; 

$key = q/2pi;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img69.gif"
 ALT="$2\pi$">|; 

$key = q/I_2Longrightarrow;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.gif"
 ALT="$I_2
\Longrightarrow $">|; 

$key = q/(-infty,-3]bigcup[-1,+infty);;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.gif"
 ALT="$ (-\infty,-3] \bigcup [
-1,+\infty);$">|; 

$key = q/[0,2pi];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.gif"
 ALT="$[0,2\pi]$">|; 

$key = q/displaystylelim_{xrightarrow0}mbox{cotg},(x)(mbox{sen},(5x)-mbox{sen},(2x));MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.gif"
 ALT="$\displaystyle \lim_{x \rightarrow 0} \mbox{cotg}   (x)(\mbox{sen}   (5x)-\mbox{sen}   (2x))$">|; 

$key = q/f(x)=x^4e^{2x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.gif"
 ALT="$f(x) =
x^4 e^{2x}$">|; 

$key = q/0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img68.gif"
 ALT="$0$">|; 

$key = q/displaystyleu=4-x^2Rightarrowdu=-2xdx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.gif"
 ALT="$\displaystyle u = 4 - x^2 \Rightarrow du = -2x dx $">|; 

$key = q/I_1=[1,2]bigcup[3.5,+infty);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.gif"
 ALT="$I_1 = [1,2] \bigcup
[3.5, +\infty)$">|; 

$key = q/f(-1)=0.13,quadf(-2)=0.29,quadf(-3)=0.20;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.gif"
 ALT="$f(-1) = 0.13, \quad f(-2) = 0.29, \quad
f(-3)=0.20$">|; 

$key = q/x=-2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.gif"
 ALT="$x=-2$">|; 

$key = q/x=20;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img54.gif"
 ALT="$x=20$">|; 

$key = q/displaystylex^3=4(2000)=8000Rightarrowx=sqrt[3]{8000}=20;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.gif"
 ALT="$\displaystyle x^3 = 4(2000) = 8000 \Rightarrow x = \sqrt[3]{8000} = 20$">|; 

$key = q/I_1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.gif"
 ALT="$I_1$">|; 

$key = q/y=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.gif"
 ALT="$y = 0$">|; 

$key = q/displaystyleV=x^2h=2000;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img46.gif"
 ALT="$\displaystyle V = x^2 h = 2000$">|; 

1;