# LaTeX2HTML 2K.1beta (1.62)
# Associate images original text with physical files.


$key = q/displaystyleintfrac{ln(x)dx}{sqrt{x}}=2sqrt{x}ln(x)-2frac{x^{1slash2}}{1slash2}+C=2sqrt{x}ln(x)-4sqrt{x}+C;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="470" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$\displaystyle \int \frac{ \ln(x) dx}{\sqrt{x}} =
2 \sqrt{x} \ln(x) - 2 \frac{x^{1/2}}{1/2} + C =
2 \sqrt{x} \ln(x) - 4 \sqrt{x} + C $">|; 

$key = q/displaystylef'(pi)=cosleft(2piright)cdotfrac{1}{2(2pi)}cdot(2pi)=frac{1}{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$\displaystyle f'(\pi) = \cos \left( 2 \pi \right)
\cdot \frac{1}{2 (2\pi) } \cdot (2\pi) = \frac{1}{2}$">|; 

$key = q/(-infty,2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$(-\infty,2)$">|; 

$key = q/Rightarrowdisplaystyle3x^2+4x+2equivA(x^2+2x+1)+B(x^2+x)+Cx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$ \Rightarrow \displaystyle 3x^2 + 4x + 2 \equiv
A(x^2 + 2x + 1) + B(x^2 + x) + Cx $">|; 

$key = q/x=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$x=0$">|; 

$key = q/displaystylefrac{3x^2+4x+2}{x(x^2+2x+1)}=frac{A(x+1)^2+Bx(x+1)+Cx}{x(x+1)^2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$\displaystyle \frac{3x^2 + 4x + 2}{x(x^2+2x+1)} =
\frac{ A(x+1)^2 + B x(x+1) + C x}{x(x+1)^2} $">|; 

$key = q/displaystyleleft{array{{ccc}A+B&=32A+B+C&=4A&=2array{right.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="81" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$\displaystyle \left\{ \begin{array}{ccc}
A + B &amp;= 3\ 2A + B + C &amp; = 4 \ A &amp;= 2
\end{array} \right. $">|; 

$key = q/f(x)=mbox{sen},left(sqrt{x^2+3pi^2}right);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$f(x) = \mbox{sen}   \left( \sqrt{x^2+3\pi^2} \right)$">|; 

$key = q/X;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$X$">|; 

$key = q/2m;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$2 m$">|; 

$key = q/x;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$x$">|; 

$key = q/displaystylef'(x)<0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$\displaystyle f'(x) &lt; 0$">|; 

$key = q/r=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="$r = 1$">|; 

$key = q/[-1,0];MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$[-1,0]$">|; 

$key = q/(0,sqrt{12});MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$(0,\sqrt{12})$">|; 

$key = q/displaystylelim_{xrightarrow-infty}f(x)=+infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$
\displaystyle \lim_{x \rightarrow -\infty} f(x) = +\infty$">|; 

$key = q/displaystyleintfrac{3x^2+4x+2}{x(x^2+2x+1)}dx=intleft(frac{2}{x}+frac{1}{x+1}+frac{-1}{(x+1)^2}right)dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="415" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$\displaystyle \int \frac{3x^2 + 4x + 2}{x(x^2+2x+1)} dx =
\int \left( \frac{2}{x} + \frac{1}{x+1} + \frac{-1}{(x+1)^2}
\right) dx $">|; 

$key = q/(-infty,-1);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$(-\infty,-1)$">|; 

$key = q/r;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$r$">|; 

$key = q/Rightarrowdisplaystyle3x^2+4x+2equiv(A+B)x^2+(2A+B+C)x+A;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$ \Rightarrow \displaystyle 3x^2 + 4x + 2 \equiv
(A+B) x^2 + (2A + B + C)x + A$">|; 

$key = q/x=3-5slash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$x = 3-5/2$">|; 

$key = q/displaystyleV=piint_0^1(2^2-(x^3+1)^2)dx=4pi(1-0)-piint_0^1(x^3+1)^2dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$\displaystyle V = \pi \int_0^1 (2^2 - (x^3 + 1)^2) dx =
4 \pi (1-0) - \pi \int_0^1(x^3 + 1)^2 dx $">|; 

$key = q/x=-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$x=-1$">|; 

$key = q/y=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$y = 2$">|; 

$key = q/V(3,-2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$V(3,-2)$">|; 

$key = q/displaystylef'(x)=cosleft(sqrt{x^2+3pi^2}right)cdotfrac{1}{2sqrt{x^2+3pi^2}}cdot(2x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$\displaystyle f'(x) = \cos \left( \sqrt{x^2+3\pi^2} \right)
\cdot \frac{1}{2\sqrt{x^2 + 3 \pi^2}} \cdot (2x) $">|; 

$key = q/displaystylefrac{3x^2+4x+2}{x(x^2+2x+1)}=frac{3x^2+4x+2}{x(x+1)^2}=frac{A}{x}+frac{B}{x+1}+frac{C}{(x+1)^2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="434" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$\displaystyle \frac{3x^2 + 4x + 2}{x(x^2+2x+1)} =
\frac{3x^2 + 4x + 2}{x(x+1)^2} =
\frac{A}{x} + \frac{B}{x+1} + \frac{C}{(x+1)^2}$">|; 

$key = q/displaystylefrac{3x^2+4x+2}{x(x^2+2x+1)}=frac{2}{x}+frac{1}{x+1}+frac{-1}{(x+1)^2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$\displaystyle \frac{3x^2 + 4x + 2}{x(x^2+2x+1)} =
\frac{2}{x} + \frac{1}{x+1} + \frac{-1}{(x+1)^2}$">|; 

$key = q/f''(x)>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$f''(x) &gt; 0$">|; 

$key = q/displaystyley-f(pi)=f'(pi)(x-pi);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$\displaystyle y - f(\pi) = f'(\pi)(x-\pi) $">|; 

$key = q/f;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$f$">|; 

$key = q/x>4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$x &gt; 4$">|; 

$key = q/C;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$C$">|; 

$key = q/x=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$x = 2$">|; 

$key = q/displaystyley^2+4y+4=10x-34+4Longrightarrow(y+2)^2=10(x-3);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$\displaystyle y^2 + 4y + 4 = 10x - 34 + 4 \Longrightarrow
(y+2)^2 = 10(x-3)$">|; 

$key = q/displaystyle4=4pi(1)^2frac{dr}{dt}Longrightarrowfrac{dr}{dt}=frac{1}{pi};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$\displaystyle 4 = 4 \pi (1)^2  \frac{dr}{dt} \Longrightarrow
\frac{dr}{dt} = \frac{1}{\pi}$">|; 

$key = q/displaystyleRightarrowy-0=frac{1}{2}(x-pi);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\displaystyle \Rightarrow y - 0 = \frac{1}{2} (x-\pi) $">|; 

$key = q/4m^2slashh;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$4 m^2/h$">|; 

$key = q/displaystyleV'(x)=3-3x^2slash4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$\displaystyle V'(x) = 3 - 3x^2/4 $">|; 

$key = q/displaystylearray{{ll}u=ln(x)&longrightarrowdisplaystyledu=frac{dx}{x}dv=frac{dx}{sqrt{x}}&longrightarrowv=2sqrt{x}array{;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="78" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$\displaystyle \begin{array}{ll}
u = \ln(x) &amp; \longrightarrow \displaystyle du =...
...x}{x} \\\\
dv = \frac{dx}{\sqrt{x}} &amp; \longrightarrow v = 2\sqrt{x}
\end{array} $">|; 

$key = q/displaystyleV(r)=frac{4pir^3}{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$\displaystyle V(r) = \frac{4 \pi r^3}{3}$">|; 

$key = q/f(2)=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$f(2) = 2 $">|; 

$key = q/displaystylefrac{dV}{dt}=frac{dV}{dr}cdotfrac{dr}{dt}=4pir^2frac{dr}{dt};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$\displaystyle \frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}
= 4 \pi r^2  \frac{dr}{dt} $">|; 

$key = q/displaystylef'(pi)=cosleft(sqrt{(pi)^2+3pi^2}right)cdotfrac{1}{2sqrt{(pi)^2+3pi^2}}cdot(2pi);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$\displaystyle f'(\pi) = \cos \left( \sqrt{(\pi)^2+3\pi^2} \right)
\cdot \frac{1}{2\sqrt{(\pi)^2 + 3 \pi^2}} \cdot (2\pi) $">|; 

$key = q/displaystyleintfrac{ln(x)dx}{sqrt{x}}=2sqrt{x}ln(x)-int2sqrt{x}cdotfrac{dx}{x}=2sqrt{x}ln(x)-2intx^{-1slash2}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="504" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$\displaystyle \int \frac{ \ln(x) dx}{\sqrt{x}} =
2\sqrt{x} \ln(x) - \int 2 \sqrt{x} \cdot \frac{dx}{x} =
2 \sqrt{x} \ln(x) - 2 \int x^{-1/2} dx $">|; 

$key = q/displaystylef'(x)>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$\displaystyle f'(x) &gt; 0$">|; 

$key = q/f'(x)<0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$f'(x) &lt; 0$">|; 

$key = q/(y-k)^2=4p(x-h);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$(y-k)^2 = 4p(x-h)$">|; 

$key = q/e^{2x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$e^{2x}$">|; 

$key = q/y^2+4y-10x+34=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$y^2 + 4 y - 10 x
+ 34 = 0$">|; 

$key = q/x=pi;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$x=\pi$">|; 

$key = q/(-infty,-1]bigcup[0,+infty);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$(-\infty,-1] \bigcup [0,+\infty) $">|; 

$key = q/F(3+5slash2,-2)=F(11slash2,-2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$F(3+5/2,-2) = F(11/2,-2)$">|; 

$key = q/displaystylex^2+4xh=12Longrightarrowh=frac{12-x^2}{4x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$\displaystyle x^2 + 4 x h = 12 \Longrightarrow h = \frac{12 - x^2}{4x}$">|; 

$key = q/12-x^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$12 - x^2$">|; 

$key = q/displaystyleA=2RightarrowB=3-A=3-2=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$\displaystyle A =2 \Rightarrow B = 3-A = 3-2 = 1 $">|; 

$key = q/y=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$y=1$">|; 

$key = q/h;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$h$">|; 

$key = q/displaystylek=-2,h=3,4p=10Longrightarrowp=10slash4=5slash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="$\displaystyle k = -2, h = 3, 4p = 10 \Longrightarrow
p = 10/4 = 5/2$">|; 

$key = q/y=x(1+x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$y = x(1+x)$">|; 

$key = q/displaystylef'(x)=2xe^{2x}+x^2(2)e^{2x}=2x(1+x)e^{2x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$\displaystyle f'(x) = 2 x e^{2x} + x^2 (2) e^{2x} = 2 x(1+x)e^{2x}$">|; 

$key = q/displaystylelim_{xrightarrow-infty}f(x)=lim_{xrightarrow-infty}x^2e^{2x}stackrel{inftycdot0}{=}lim_{xrightarrow-infty}frac{x^2}{e^{-2x}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$\displaystyle \lim_{x \rightarrow -\infty} f(x) =
\lim_{x \rightarrow -\infty}...
...\stackrel{\infty \cdot 0}{=}
\lim_{x \rightarrow -\infty} \frac{x^2}{e^{-2x}} $">|; 

$key = q/displaystyleintfrac{3x^2+4x+2}{x(x^2+2x+1)}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$\displaystyle \int \frac{3x^2+4x+2}{x(x^2+2x+1)} dx $">|; 

$key = q/xrightarrow+infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$x \rightarrow +\infty$">|; 

$key = q/1slashpimslashh;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$1/\pi  m/h$">|; 

$key = q/h=12slash2^2=3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$h = 12/2^2 = 3$">|; 

$key = q/displaystyleintfrac{3x^2+4x+2}{x(x^2+2x+1)}dx=2ln|x|+ln|x+1|+frac{1}{(x+1)}+C;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="433" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$\displaystyle \int \frac{3x^2 + 4x + 2}{x(x^2+2x+1)} dx =
2 \ln\vert x\vert + \ln\vert x+1\vert + \frac{1}{(x+1)} + C$">|; 

$key = q/x<2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$x&lt; 2$">|; 

$key = q/displaystylelim_{xrightarrow-infty}f(x)=lim_{xrightarrow-infty}frac{2x}{-2e^{-2x}}stackrel{LH}{=}lim_{xrightarrow-infty}frac{2}{4e^{-2x}}=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$\displaystyle \lim_{x \rightarrow -\infty} f(x) =
\lim_{x \rightarrow -\infty}...
...-2e^{-2x}} \stackrel{LH}{=}
\lim_{x \rightarrow -\infty} \frac{2}{4e^{-2x}} = 0$">|; 

$key = q/r=1m;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$r = 1m$">|; 

$key = q/V'(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$V'(x)$">|; 

$key = q/y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$y$">|; 

$key = q/3m;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$3 m$">|; 

$key = q/x^2slash4=1Rightarrowx=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$x^2/4 = 1 \Rightarrow x = 2$">|; 

$key = q/(0,+infty)Longrightarrowf(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$(0,+\infty)
\Longrightarrow f(x) $">|; 

$key = q/12m^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$12 m^2$">|; 

$key = q/displaystylelim_{xrightarrow+infty}f(x)=lim_{xrightarrow+infty}x^2e^{2x}stackrel{inftycdotinfty}{=}+infty;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$\displaystyle \lim_{x \rightarrow +\infty} f(x) =
\lim_{x \rightarrow +\infty} x^2 e^{2x} \stackrel{\infty \cdot \infty}{=}
+\infty$">|; 

$key = q/y=x^3+1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$y = x^3 + 1$">|; 

$key = q/V(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$V(x)$">|; 

$key = q/V(r);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$V(r)$">|; 

$key = q/(-1,0)Longrightarrowf(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$(-1,0) \Longrightarrow f(x) $">|; 

$key = q/displaystyleintfrac{ln(x)dx}{sqrt{x}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$\displaystyle \int \frac{ \ln(x) dx}{\sqrt{x}} $">|; 

$key = q/(4,+infty);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$(4,+\infty)$">|; 

$key = q/displaystyleV=4pi-piint_0^1(x^6+2x^3+1)dx=4pi-pileft[frac{x^7}{7}-frac{2x^4}{4}+ght)=4pi-frac{2-7+14}{14}pi=frac{(56-9)pi}{14}=frac{47pi}{14};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="940" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$\displaystyle V = 4 \pi - \pi \int_0^1(x^6 + 2x^3+ 1) dx =
4 \pi - \pi \left[ \...
...)
= 4 \pi - \frac{2 - 7 + 14}{14} \pi = \frac{(56-9)\pi}{14}=
\frac{47\pi}{14}$">|; 

$key = q/x=1slash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$x = 1/2$">|; 

$key = q/Rightarrowy-mbox{sen},left(2piright)=frac{1}{2}(x-pi);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$\Rightarrow y - \mbox{sen}   \left( 2 \pi \right) =
\frac{1}{2} (x-\pi) $">|; 

$key = q/f(x)=x^2e^{2x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$f(x) = x^2 e^{2x}$">|; 

$key = q/displaystyleV=x^2hLongrightarrowV(x)=x^2cdotfrac{12-x^2}{4x}=3x-x^3slash4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$\displaystyle V = x^2 h \Longrightarrow V(x) = x^2 \cdot
\frac{12 - x^2}{4x} = 3 x - x^3/4$">|; 

$key = q/displaystyleintfrac{3x^2+4x+2}{x(x^2+2x+1)}dx=2ln|x|+ln|x+1|-frac{(x+1)^{-1}}{-1}+C;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="451" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$\displaystyle \int \frac{3x^2 + 4x + 2}{x(x^2+2x+1)} dx =
2 \ln\vert x\vert + \ln\vert x+1\vert - \frac{(x+1)^{-1}}{-1} + C$">|; 

$key = q/displaystyleV''(x)=-3xslash2<0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$\displaystyle V''(x) = - 3 x /2 &lt; 0 $">|; 

$key = q/y=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$y=0$">|; 

$key = q/f(4)=3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$f(4)=3$">|; 

$key = q/displaystylerightarrowC=4-2A-B=4-4-1=-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$\displaystyle \rightarrow C = 4-2A-B= 4-4-1 = -1$">|; 

$key = q/displaystylelim_{xrightarrow+infty}f(x)=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$\displaystyle
\lim_{x \rightarrow + \infty} f(x) = 1$">|; 

$key = q/Rightarrowy-mbox{sen},left(sqrt{(pi)^2+3pi^2}right)=frac{1}{2}(x-pi);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$\Rightarrow y - \mbox{sen}   \left( \sqrt{(\pi)^2 + 3 \pi^2} \right) =
\frac{1}{2} (x-\pi) $">|; 

1;