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


$key = q/displaystylef(0)=frac{5(0)}{0+3}=0,f(pm1)=frac{5(1)}{1+3}=frac{5}{4};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.gif"
 ALT="$\displaystyle f(0) = \frac{5(0)}{0+3} = 0  , \
f(\pm 1) = \frac{5(1)}{1+3} = \frac{5}{4}$">|; 

$key = q/displaystyleintxe^{kx}dx=intx(1)dx=frac{x^2}{2}+C,Cin{{mathbb{R}{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.gif"
 ALT="$\displaystyle \int x e^{kx} dx = \int x (1) dx = \frac{x^2}{2} + C, C \in {\mathbb{R}}$">|; 

$key = q/f''(x)>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.gif"
 ALT="$f''(x) &gt; 0$">|; 

$key = q/includegraphics{P2D_03-2f3.eps};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="700" HEIGHT="77" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img27.gif"
 ALT="\includegraphics{P2D_03-2f3.eps}">|; 

$key = q/resizebox{9cm}{5cm}{includegraphics{P2D_03-2f2.eps}};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="224" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img17.gif"
 ALT="\resizebox{9cm}{5cm}{\includegraphics{P2D_03-2f2.eps}}">|; 

$key = q/y=16-x^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.gif"
 ALT="$y = 16-x^2$">|; 

$key = q/+infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.gif"
 ALT="$+\infty$">|; 

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

$key = q/displaystyleln(L)=lim_{xrightarrow+infty}frac{displaystylefrac{20x+3}{10x^2+3x-4}}{1slashx}=lim_{xrightarrow+infty}frac{x(20x+3)}{10x^2+3x-4};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="95" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.gif"
 ALT="$\displaystyle \ln(L) = \lim_{x\rightarrow +\infty}
\frac{\displaystyle \frac{20...
...}{10x^2+3x-4} }{
1/x} = \lim_{x\rightarrow +\infty} \frac{x(20x+3)}{10x^2+3x-4}$">|; 

$key = q/kneq0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.gif"
 ALT="$k \neq 0$">|; 

$key = q/includegraphics{P2D_03-2f4.eps};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="610" HEIGHT="395" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img40.gif"
 ALT="\includegraphics{P2D_03-2f4.eps}">|; 

$key = q/displaystylef(x)geqf(1slash5)>0Rightarrowx-frac{ln(x)}{5}>0Rightarrowx>frac{ln(x)}{5};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="378" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.gif"
 ALT="$\displaystyle f(x) \geq f(1/5) &gt; 0 \Rightarrow
x - \frac{\ln(x)}{5} &gt; 0 \Rightarrow x &gt; \frac{\ln(x)}{5}$">|; 

$key = q/displaystyleL=lim_{xrightarrow+infty}left(10x^2+3x-4right)^{frac{1}{ln(x)}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.gif"
 ALT="$\displaystyle L = \lim_{x \rightarrow +\infty}
\left( 10x^2+3x-4\right)^{\frac{1}{\ln(x)}}$">|; 

$key = q/(0,4);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.gif"
 ALT="$(0,4)$">|; 

$key = q/x=4slashsqrt{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.gif"
 ALT="$x = 4/\sqrt{3}$">|; 

$key = q/displaystyleln(L)=lim_{xrightarrow+infty}frac{1}{ln(x)}lnleft(10x^2+3x-4right)=lim_{xrightarrow+infty}frac{lnleft(10x^2+3x-4right)}{ln(x)};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="512" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.gif"
 ALT="$\displaystyle \ln(L) = \lim_{x \rightarrow +\infty}
\frac{1}{\ln(x)} \ln \left(...
...) =
\lim_{x \rightarrow +\infty}
\frac{\ln \left( 10x^2+3x-4\right) }{\ln(x)} $">|; 

$key = q/displaystylev=frac{e^{kx}}{k};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.gif"
 ALT="$\displaystyle v = \frac{e^{kx}}{k}$">|; 

$key = q/displaystyleintfrac{tdt}{3t^2+6}=intfrac{duslash6}{u}=frac{1}{6}intfrac{du}{u}=frac{ln|u|}{6}+C=frac{ln|3t^2+6|}{6}+C,Cin{{mathbb{R}{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="542" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.gif"
 ALT="$\displaystyle \int \frac{t dt}{3t^2 + 6 } = \int \frac{du/6}{u} =
\frac{1}{6} \...
...vert u\vert}{6} + C =
\frac{\ln \vert 3t^2+ 6\vert}{6} + C, C \in {\mathbb{R}}$">|; 

$key = q/f(x)geqf(1slash5);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.gif"
 ALT="$f(x) \geq f(1/5)$">|; 

$key = q/f;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.gif"
 ALT="$f$">|; 

$key = q/0slash0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.gif"
 ALT="$0/0$">|; 

$key = q/k;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img66.gif"
 ALT="$k$">|; 

$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="|."$dir".q|img37.gif"
 ALT="$f'(x)$">|; 

$key = q/u=3t^2+6Rightarrowduslashdt=6tRightarrowdu=6tdt;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.gif"
 ALT="$u = 3t^2 + 6 \Rightarrow du/dt = 6t \Rightarrow du = 6 t  dt$">|; 

$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="|."$dir".q|img19.gif"
 ALT="$(0,+\infty)$">|; 

$key = q/displaystyleintxe^{kx}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.gif"
 ALT="$\displaystyle \int x e^{kx}dx$">|; 

$key = q/displaystyleln(L)=lim_{xrightarrow+infty}frac{20x^2}{10x^2}=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.gif"
 ALT="$\displaystyle \ln(L) = \lim_{x\rightarrow +\infty}
\frac{20x^2}{10x^2} = 2$">|; 

$key = q/(-infty,-1)bigcup(1,+infty);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.gif"
 ALT="$(-\infty,-1) \bigcup (1,+\infty)$">|; 

$key = q/displaystyleA'(x)=(1)(16-x^2)+x(-2x)=16-3x^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.gif"
 ALT="$\displaystyle A'(x) = (1)(16-x^2) + x(-2x) = 16 - 3x^2$">|; 

$key = q/0<x<4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.gif"
 ALT="$0 &lt; x &lt; 4$">|; 

$key = q/f'(x)=0Leftrightarrow5x=1Leftrightarrowx=1slash5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.gif"
 ALT="$f'(x) = 0 \Leftrightarrow
5x=1 \Leftrightarrow x = 1/5$">|; 

$key = q/displaystylelim_{xrightarrow+infty}left(10x^2+3x-4right)^{frac{1}{ln(x)}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.gif"
 ALT="$\displaystyle \lim_{x \rightarrow +\infty} \left( 10x^2+3x-4\right)^{\frac{1}{
\ln(x)}}$">|; 

$key = q/Rightarrow;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img72.gif"
 ALT="$\Rightarrow$">|; 

$key = q/k=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img68.gif"
 ALT="$k=0$">|; 

$key = q/D(f')=(-infty,+infty);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.gif"
 ALT="$D(f') = (-\infty,+\infty)$">|; 

$key = q/displaystylef'(x)=1-frac{1}{5x},x>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.gif"
 ALT="$\displaystyle f'(x) = 1 - \frac{1}{5x}  , x &gt; 0$">|; 

$key = q/x;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img41.gif"
 ALT="$x$">|; 

$key = q/dv=e^{kx}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img74.gif"
 ALT="$dv = e^{kx} dx$">|; 

$key = q/f''(1slash5)>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.gif"
 ALT="$f''(1/5) &gt; 0$">|; 

$key = q/displaystylelim_{xrightarrow+infty}frac{5x^2}{x^2+3}=lim_{xrightarrow+infty}frac{5x^2}{x^2}=5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.gif"
 ALT="$\displaystyle \lim_{x \rightarrow +\infty}
\frac{5x^2}{x^2+3} = \lim_{x \rightarrow +\infty} \frac{5x^2}{x^2} = 5$">|; 

$key = q/displaystylef(x)=frac{5x^2}{x^2+3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.gif"
 ALT="$\displaystyle f(x) = \frac{5x^2}{x^2+3}$">|; 

$key = q/displaystylex>frac{ln(x)}{5};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.gif"
 ALT="$\displaystyle x &gt; \frac{\ln(x)}{5}$">|; 

$key = q/(-infty,0];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.gif"
 ALT="$(-\infty,0]$">|; 

$key = q/y=5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.gif"
 ALT="$y=5$">|; 

$key = q/f''(x)<0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.gif"
 ALT="$f''(x) &lt; 0$">|; 

$key = q/ln(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.gif"
 ALT="$ln(x)$">|; 

$key = q/displaystylelim_{xrightarrow+infty}left(10x^2+3x-4right)^{frac{1}{ln(x)}}=L=e^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.gif"
 ALT="$ \displaystyle \lim_{x \rightarrow +\infty}
\left( 10x^2+3x-4\right)^{\frac{1}{\ln(x)}}= L = e^2$">|; 

$key = q/inftyslashinftyRightarrow;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.gif"
 ALT="$\infty/\infty \Rightarrow$">|; 

$key = q/displaystylelim_{xrightarrow-infty}frac{5x^2}{x^2+3}=lim_{xrightarrow-infty}frac{5x^2}{x^2}=5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.gif"
 ALT="$\displaystyle \lim_{x \rightarrow -\infty}
\frac{5x^2}{x^2+3} = \lim_{x \rightarrow -\infty} \frac{5x^2}{x^2} = 5$">|; 

$key = q/(-infty,+infty);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.gif"
 ALT="$(-\infty,+\infty)$">|; 

$key = q/(-infty,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.gif"
 ALT="$(-\infty,0)$">|; 

$key = q/resizebox{9cm}{5cm}{includegraphics{P2D_03-2f1.eps}};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="406" HEIGHT="224" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img16.gif"
 ALT="\resizebox{9cm}{5cm}{\includegraphics{P2D_03-2f1.eps}}">|; 

$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="|."$dir".q|img28.gif"
 ALT="$f(x)$">|; 

$key = q/f''(x)>0forallxin(0,+infty);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.gif"
 ALT="$f''(x) &gt; 0   \forall x \in (0,+\infty)$">|; 

$key = q/displaystylefleft(frac{1}{5}right)=frac{1}{5}-frac{ln(1slash5)}{5}=frac{1}{5}+frac{ln(5)}{5}>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.gif"
 ALT="$\displaystyle f \left( \frac{1}{5} \right) = \frac{1}{5} - \frac{\ln(1/5)}{5}
= \frac{1}{5} + \frac{\ln(5)}{5} &gt; 0$">|; 

$key = q/x=1slash5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.gif"
 ALT="$x=1/5$">|; 

$key = q/-infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.gif"
 ALT="$-\infty$">|; 

$key = q/16-3x^2=0Rightarrowx^2=16slash3Rightarrowx=4slashsqrt{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.gif"
 ALT="$16-3x^2 = 0 \Rightarrow x^2 = 16/3 \Rightarrow
x = 4/\sqrt{3}$">|; 

$key = q/du=dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img73.gif"
 ALT="$ du = dx $">|; 

$key = q/resizebox{11cm}{2.4cm}{includegraphics{P2D_03-2f5.eps}};LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="493" HEIGHT="78" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img47.gif"
 ALT="\resizebox{11cm}{2.4cm}{\includegraphics{P2D_03-2f5.eps}}">|; 

$key = q/inftyslashinfty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.gif"
 ALT="$\infty/\infty$">|; 

$key = q/displaystyleAleft(frac{4}{sqrt{3}}right)=frac{4}{sqrt{3}}cdotleft(16-frac{16}{3}right)=frac{4(2)(16)}{3sqrt{3}}=frac{128sqrt{3}}{9};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.gif"
 ALT="$\displaystyle
A \left(\frac{4}{\sqrt{3}} \right) = \frac{4}{\sqrt{3}} \cdot
\left(16 - \frac{16}{3} \right) = \frac{4(2)(16)}{3 \sqrt{3}}=
\frac{128\sqrt{3}}{9}$">|; 

$key = q/displaystyleA(x)=frac{(2x)y}{2}=frac{2x(16-x^2)}{2}=x(16-x^2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.gif"
 ALT="$\displaystyle A(x) = \frac{(2x) y}{2} = \frac{2x (16-x^2)}{2}=
x (16-x^2)$">|; 

$key = q/displaystylef''(x)=frac{1}{5x^2},x>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.gif"
 ALT="$\displaystyle f''(x) = \frac{1}{5x^2}  ,  x &gt; 0$">|; 

$key = q/x=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img26.gif"
 ALT="$x=0$">|; 

$key = q/displaystyleintxe^{kx}dx=frac{xe^{kx}}{k}-intfrac{e^{kx}dx}{k}=frac{xe^{kx}}{k}-frac{e^{kx}}{k^2}+C,Cin{{mathbb{R}{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.gif"
 ALT="$\displaystyle \int x e^{kx}dx = \frac{x e^{kx}}{k} - \int \frac{e^{kx}dx}{k}
= \frac{x e^{kx}}{k} - \frac{e^{kx}}{k^2} + C, C \in {\mathbb{R}}$">|; 

$key = q/displaystylelim_{xrightarrow0}frac{arcsen(5x)}{3x^2-4x}=lim_{xrightarrow0}frac{d)^2}}}{6x-4}=lim_{xrightarrow0}frac{5slash1}{0-4}=frac{-5}{4};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="418" HEIGHT="108" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.gif"
 ALT="$\displaystyle \lim_{x\rightarrow 0} \frac{ arcsen(5x)}{3x^2-4x} =
\lim_{x \righ...
...\sqrt{1-(5x)^2}}}{6x-4} =
\lim_{x \rightarrow 0} \frac{5/1}{0-4} = \frac{-5}{4}$">|; 

$key = q/displaystylelim_{xrightarrow0^{-}}f(x)=lim_{xrightarrow0^{-}}x-frac{ln(x)}{5}=+infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.gif"
 ALT="$\displaystyle \lim_{x \rightarrow 0^{-}} f(x) = \lim_{x \rightarrow 0^{-}}
x - \frac{\ln(x)}{5} = + \infty$">|; 

$key = q/(0,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.gif"
 ALT="$(0,0)$">|; 

$key = q/displaystyleln(L)=lnleft(lim_{xrightarrow+infty}left(10x^2+3x-4right)^{frac{1}{lrow+infty}lnleft(left(10x^2+3x-4right)^{frac{1}{ln(x)}}right);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="565" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.gif"
 ALT="$\displaystyle \ln(L) = \ln \left( \lim_{x \rightarrow +\infty}
\left( 10x^2+3x-...
...tarrow +\infty}
\ln \left( \left( 10x^2+3x-4\right)^{\frac{1}{\ln(x)}} \right) $">|; 

$key = q/u=x;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img71.gif"
 ALT="$u = x$">|; 

$key = q/(-1,1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.gif"
 ALT="$(-1,1)$">|; 

$key = q/displaystylef(x)=x-frac{ln(x)}{5};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.gif"
 ALT="$\displaystyle f(x) = x - \frac{\ln(x)}{5}$">|; 

$key = q/displaystylef'(x)=frac{(x^2+3)(10x)-5x^2(2x)}{(x^2+3)^2}=frac{30x}{(x^2+3)^2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.gif"
 ALT="$\displaystyle f'(x) = \frac{(x^2+3)(10x) - 5x^2 (2x)}{(x^2+3)^2} =
\frac{30x}{(x^2+3)^2}$">|; 

$key = q/displaystyleintfrac{t}{3t^2+6}dt;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.gif"
 ALT="$\displaystyle \int \frac{t}{3t^2+6} dt$">|; 

$key = q/x>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.gif"
 ALT="$x &gt; 0$">|; 

$key = q/[0,+infty);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.gif"
 ALT="$[0,+\infty)$">|; 

$key = q/displaystylelim_{xrightarrow0}frac{arcsen(5x)}{3x^2-4x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.gif"
 ALT="$\displaystyle \lim_{x \rightarrow 0} \frac{ arcsen(5x)}{3x^2-4x}$">|; 

1;