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


$key = q/displaystyleg(x)=x^2-1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.gif"
 ALT="$\displaystyle g(x) = x^2-1$">|; 

$key = q/a^2=b^2+c^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.gif"
 ALT="$a^2 = b^2 + c^2$">|; 

$key = q/Longrightarrowdisplaystyleintfrac{dx}{x^2sqrt{4-x^2}}=intfrac{du}{4sen^2(u)}=frac{1}{4}intcossec^2(u)du=-frac{1}{4}cotg(u)+C;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="551" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.gif"
 ALT="$\Longrightarrow \displaystyle \int \frac{dx}{x^2 \sqrt{4- x^2}} =
\int \frac{du}{4 sen^2(u)} = \frac{1}{4} \int cossec^2(u) du =
-\frac{1}{4} cotg(u) + C$">|; 

$key = q/displaystyleintfrac{3x^2+x-2}{x^2(x-2)}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.gif"
 ALT="$\displaystyle \int \frac{3x^2+x-2}{x^2(x-2)} dx $">|; 

$key = q/displaystyleleft{array{{ccc}A+C&=3B-2A&=1-2B&=-2array{right.;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="82" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.gif"
 ALT="$\displaystyle \left\{ \begin{array}{ccc}
A + C &amp;= 3\ B - 2A &amp; = 1 \ - 2B &amp;= -2
\end{array} \right. $">|; 

$key = q/displaystyle3x^2+x-2=(A+C)x^2+(B-2A)x-2B;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="340" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.gif"
 ALT="$\displaystyle 3x^2 + x -2 = (A + C)x^2 + (B-2A)x - 2B$">|; 

$key = q/a,b,c;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.gif"
 ALT="$a,b,c$">|; 

$key = q/(1,2)bigcup(3,4);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.gif"
 ALT="$(1,2) \bigcup (3,4)$">|; 

$key = q/X;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img62.gif"
 ALT="$X$">|; 

$key = q/displaystyleintfrac{dx}{x^2sqrt{4-x^2}}=intfrac{2cos(u)du}{4sen^2(u)sqrt{4-4sen^2(u)}}=intfrac{2cos(u)du}{4sen^2(u)2cos(u)};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="508" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.gif"
 ALT="$\displaystyle \int \frac{dx}{x^2 \sqrt{4- x^2}} =
\int \frac{2 \cos(u) du}{4 sen^2(u) \sqrt{4 - 4 sen^2(u)}} =
\int \frac{2 \cos(u) du}{4 sen^2(u) 2 \cos(u)} $">|; 

$key = q/displaystyleF_1=left(-5-frac{5sqrt{3}}{2},0right),F_2=left(-5+frac{5sqrt{3}}{2},0right);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.gif"
 ALT="$\displaystyle F_1 = \left( -5 - \frac{5 \sqrt{3}}{2},0 \right)  , \
F_2 = \left( -5 + \frac{5 \sqrt{3}}{2},0 \right) $">|; 

$key = q/x;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img30.gif"
 ALT="$x$">|; 

$key = q/displaystyleA=int_{0}^{2}left[(4x-x^2)-(x^2)right]dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.gif"
 ALT="$\displaystyle A = \int_{0}^{2} \left[
(4x-x^2) - (x^2) \right] dx $">|; 

$key = q/displaystyleint_1^{infty}frac{3}{xsqrt{x}}dx=lim_{ellrightarrowinfty}6left(1-frac{1}{sqrt{ell}}right)=6;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.gif"
 ALT="$\displaystyle \int_1^{\infty} \frac{3}{x \sqrt{x}} dx =
\lim_{\ell \rightarrow \infty} 6 \left( 1 -
\frac{1}{\sqrt{\ell}} \right) = 6$">|; 

$key = q/displaystyle3x^2+x-2=A(x^2-2x)+B(x-2)+Cx^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.gif"
 ALT="$\displaystyle 3x^2 + x -2 = A(x^2 - 2x) + B(x-2) + Cx^2$">|; 

$key = q/C(-5,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.gif"
 ALT="$C(-5,0)$">|; 

$key = q/displaystyley_1=x_1^2=0,y_2=x_2^2=4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.gif"
 ALT="$\displaystyle y_1 = x_1^2 = 0  ,  y_2 = x_2^2 = 4$">|; 

$key = q/displaystylefrac{(x+5)^2}{25}+frac{4y^2}{25}=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.gif"
 ALT="$\displaystyle \frac{(x+5)^2}{25} + \frac{4y^2}{25} = 1$">|; 

$key = q/F'(x)=f(x)>0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.gif"
 ALT="$F'(x) = f(x) &gt; 0 $">|; 

$key = q/displaystylefrac{3x^2+x-2}{x^2(x-2)}=frac{1}{x^2}+frac{3}{x-2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.gif"
 ALT="$\displaystyle \frac{3x^2 + x -2}{x^2(x-2)} = \frac{1}{x^2} +
\frac{3}{x-2} $">|; 

$key = q/displaystyleintfrac{dx}{x^2sqrt{4-x^2}}=-frac{sqrt{4-x^2}}{4x}+C,Cin{{mathbb{R}{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.gif"
 ALT="$\displaystyle \int \frac{dx}{x^2 \sqrt{4- x^2}} =
-\frac{\sqrt{4-x^2}}{4x} + C  ,  C \in {\mathbb{R}}$">|; 

$key = q/y=x^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.gif"
 ALT="$y=x^2$">|; 

$key = q/A_2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.gif"
 ALT="$A_2$">|; 

$key = q/displaystyleV_1(-10,0),V_2(0,0)Longrightarrow2a=10Longrightarrowa=5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.gif"
 ALT="$\displaystyle V_1(-10,0)  ,  V_2(0,0) \Longrightarrow 2a = 10
\Longrightarrow a = 5 $">|; 

$key = q/y=4x-x^2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.gif"
 ALT="$y = 4x-x^2$">|; 

$key = q/x^2=1,x>0Rightarrowx=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.gif"
 ALT="$x^2 = 1 , x &gt; 0 \Rightarrow x=1$">|; 

$key = q/F;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img46.gif"
 ALT="$F$">|; 

$key = q/f;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.gif"
 ALT="$f$">|; 

$key = q/displaystyleA=int_0^2(4x-2x^2)dx=left[2x^2-frac{2x^3}{3}right]_0^2=2(2)^2-frac{2(2)^3}{3}-0+0=;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="495" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.gif"
 ALT="$\displaystyle A = \int_0^2 ( 4x-2x^2 ) dx =
\left[ 2x^2 - \frac{2x^3}{3} \right]_0^2 =
2 (2)^2 - \frac{2 (2)^3}{3} - 0 + 0 = $">|; 

$key = q/displaystyleF(3)=F_2+2=-2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.gif"
 ALT="$\displaystyle F(3) = F_2 + 2 = -2$">|; 

$key = q/displaystyleintfrac{dx}{x^2sqrt{4-x^2}};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.gif"
 ALT="$\displaystyle \int \frac{dx}{x^2 \sqrt{4- x^2}} $">|; 

$key = q/x=2sen(u)Rightarrowdx=2cos(u)du,-pilequleqpi;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.gif"
 ALT="$x = 2 sen(u) \Rightarrow dx = 2 cos(u) du  ,
 -\pi \leq u \leq \pi$">|; 

$key = q/displaystyleF(x)=int_{1}^xf(t)dt,1leqxleq4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.gif"
 ALT="$\displaystyle F(x) = \int_{1}^x f(t) dt  ,  1 \leq x \leq 4$">|; 

$key = q/displaystylefrac{(x-(-5))^2}{a^2}+frac{(y-0)^2}{b^2}=1displaystylefrac{(x+5)^2}{5^2}+frac{y^2}{b^2}=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.gif"
 ALT="$\displaystyle \frac{(x-(-5))^2}{a^2} + \frac{(y-0)^2}{b^2} = 1
\displaystyle \frac{(x+5)^2}{5^2} + \frac{y^2}{b^2} = 1$">|; 

$key = q/displaystylefrac{3x^2+x-2}{x^2(x-2)}=frac{A}{x}+frac{B}{x^2}+frac{C}{x-2}=frac{Ax(x-2)+B(x-2)+Cx^2}{x^2(x-2)};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="489" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.gif"
 ALT="$\displaystyle \frac{3x^2 + x -2}{x^2(x-2)} = \frac{A}{x} +
\frac{B}{x^2} + \frac{C}{x-2} = \frac{Ax(x-2) + B(x-2) + Cx^2}
{x^2(x-2)}$">|; 

$key = q/(-10,0);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.gif"
 ALT="$(-10,0)$">|; 

$key = q/displaystyleC=3-A=3-0=3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.gif"
 ALT="$\displaystyle C = 3 - A = 3 - 0 = 3$">|; 

$key = q/displaystyleA=int_0^8(x_g-x_f)dy=int_0^8left(sqrt{y+1}-frac{y^2slash4-1}{5}right)dy;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.gif"
 ALT="$\displaystyle A = \int_0^8 (x_g - x_f) dy=
\int_0^8 \left( \sqrt{y+1} - \frac{y^2/4-1}{5} \right) dy$">|; 

$key = q/(1,2)bigcup(3,4)LongrightarrowF(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.gif"
 ALT="$(1,2) \bigcup (3,4)
\Longrightarrow F(x) $">|; 

$key = q/A_1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.gif"
 ALT="$A_1$">|; 

$key = q/displaystyleF(4)=F(3)-1=-3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.gif"
 ALT="$\displaystyle F(4) = F(3) -1 = -3$">|; 

$key = q/displaystyleint_1^{infty}frac{3}{xsqrt{x}}dx=lim_{ellrightarrowinfty}int_1^{ell}3x^{-3slash2}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.gif"
 ALT="$\displaystyle \int_1^{\infty} \frac{3}{x \sqrt{x}} dx =
\lim_{\ell \rightarrow
\infty} \int_1^{\ell} 3 x^{-3/2} dx$">|; 

$key = q/displaystyleRightarrowx_1=0,x_2=2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.gif"
 ALT="$\displaystyle \Rightarrow
x_1 = 0  ,  x_2 = 2$">|; 

$key = q/displaystyleB=1Rightarrow2A=B-1=0RightarrowA=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="271" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.gif"
 ALT="$\displaystyle B = 1 \Rightarrow 2A = B-1 = 0 \Rightarrow A = 0 $">|; 

$key = q/displaystyleintfrac{3x^2+x-2}{x^2(x-2)}dx=intleft(x^{-2}+frac{3}{x-2}right)dx=3ln|x-2|-frac{1}{x}+C,Cin{{mathbb{R}{};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="558" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.gif"
 ALT="$\displaystyle \int \frac{3x^2+x-2}{x^2(x-2)} dx =
\int \left( x^{-2} + \frac{3}{x-2} \right) dx =
3 \ln \vert x-2\vert - \frac{1}{x} + C  ,  C \in {\mathbb{R}}$">|; 

$key = q/displaystyleF(2)=F_1-4=-4;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.gif"
 ALT="$\displaystyle F(2) = F_1 - 4 = -4$">|; 

$key = q/(x,y)=(-2,2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.gif"
 ALT="$(x,y) = (-2,2)$">|; 

$key = q/(2,3);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.gif"
 ALT="$(2,3)$">|; 

$key = q/displaystylecotg(u)=frac{cos(u)}{sen(u)}=frac{sqrt{1-(xslash2)^2}}{xslash2}=frac{sqrt{4-x^2}}{x};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.gif"
 ALT="$\displaystyle cotg(u) = \frac{\cos(u)}{sen(u)} =
\frac{\sqrt{1 - (x/2)^2}}{x/2} =
\frac{\sqrt{4 - x^2}}{x} $">|; 

$key = q/f(x)=2sqrt{5x+1};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.gif"
 ALT="$f(x) = 2 \sqrt{5x+1}$">|; 

$key = q/(2,3)LongrightarrowF(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.gif"
 ALT="$(2,3) \Longrightarrow F(x) $">|; 

$key = q/y;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.gif"
 ALT="$y$">|; 

$key = q/displaystyley=2sqrt{5x+1}Rightarrow(yslash2)^2=5x+1Rightarrowx=frac{y^2slash4-1}{5}=x_f;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.gif"
 ALT="$\displaystyle
y = 2\sqrt{5x+1} \Rightarrow (y/2)^2 = 5x + 1 \Rightarrow
x = \frac{y^2/4-1}{5} = x_f$">|; 

$key = q/displaystylex^2=4x-x^2Leftrightarrow2x^2-4x=0Leftrightarrow2x(x-2)=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.gif"
 ALT="$\displaystyle x^2 = 4x-x^2 \Leftrightarrow 2x^2 - 4x = 0
\Leftrightarrow 2x(x-2)=0 $">|; 

$key = q/displaystyleint_1^{+infty}frac{3}{xsqrt{x}}dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.gif"
 ALT="$\displaystyle
\int_1^{+\infty} \frac{3}{x\sqrt{x}} dx$">|; 

$key = q/displaystyleLongrightarrowfrac{4}{b^2}=1-frac{9}{25}=frac{16}{25}Rightarrowfrac{1}{b^2}=frac{4}{25};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.gif"
 ALT="$\displaystyle
\Longrightarrow \frac{4}{b^2} = 1 - \frac{9}{25} = \frac{16}{25}
\Rightarrow \frac{1}{b^2} = \frac{4}{25}$">|; 

$key = q/(-2,2);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.gif"
 ALT="$(-2,2)$">|; 

$key = q/displaystyleint_1^{ell}3x^{-3slash2}dx=3left[frac{x^{-1slash2}}{-1slash2}right]_ell}}-frac{1}{sqrt{1}}right)=6left(1-frac{1}{sqrt{ell}}right);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="569" HEIGHT="77" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.gif"
 ALT="$\displaystyle \int_1^{\ell} 3 x^{-3/2} dx = 3 \left[
\frac{x^{-1/2}}{-1/2} \rig...
...ell}} - \frac{1}{\sqrt{1}} \right)
= 6 \left( 1 - \frac{1}{\sqrt{\ell}} \right)$">|; 

$key = q/displaystyleA=8-frac{2(8)}{3}=left(1-frac{2}{3}right)8=frac{8}{3};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.gif"
 ALT="$\displaystyle A = 8 - \frac{2(8)}{3} = \left( 1 - \frac{2}{3}
\right) 8 = \frac{8}{3} $">|; 

$key = q/y=f(x);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.gif"
 ALT="$y = f(x)$">|; 

$key = q/F(1),F(2),F(3),F(4);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.gif"
 ALT="$F(1), F(2), F(3), F(4)$">|; 

$key = q/F'(x)=f(x)<0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.gif"
 ALT="$F'(x)= f(x) &lt; 0 $">|; 

$key = q/A_3;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.gif"
 ALT="$A_3$">|; 

$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="|."$dir".q|img59.gif"
 ALT="$(0,0)$">|; 

$key = q/b=5slash2;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.gif"
 ALT="$b = 5/2$">|; 

$key = q/5x+1=0Rightarrowx=-1slash5;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.gif"
 ALT="$5x + 1 = 0 \Rightarrow x = -1/5$">|; 

$key = q/displaystylefrac{(-2+5)^2}{5^2}+frac{2^2}{b^2}=1Rightarrowfrac{9}{25}+frac{4}{b^2}=1;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.gif"
 ALT="$\displaystyle \frac{(-2+5)^2}{5^2} + \frac{2^2}{b^2} = 1 \Rightarrow
\frac{9}{25} + \frac{4}{b^2} = 1$">|; 

$key = q/y=x^2-1Rightarrowx=sqrt{y+1}=x_g;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.gif"
 ALT="$ y = x^2 -1 \Rightarrow x = \sqrt{y+1} = x_g$">|; 

$key = q/displaystyleF'(x)=frac{d}{dx}left[int_{1}^xf(t)dtright]=f(x),xin(1,4);MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.gif"
 ALT="$\displaystyle F'(x) = \frac{d}{dx} \left[ \int_{1}^x f(t) dt \right] =
f(x)  ,  x \in (1,4)$">|; 

$key = q/displaystyle5^2=left(frac{5}{2}right)^2+c^2Rightarrowc^2=frac{25(3)}{2}Rightarrowc=frac{5sqrt{3}}{2};MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.gif"
 ALT="$\displaystyle 5^2 = \left(\frac{5}{2} \right)^2 + c^2 \Rightarrow
c^2 = \frac{25(3)}{2} \Rightarrow c = \frac{5 \sqrt{3}}{2}$">|; 

$key = q/displaystyleF(1)=int_{1}^{1}f(t)dt=0;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.gif"
 ALT="$\displaystyle F(1) = \int_{1}^{1} f(t)dt = 0$">|; 

$key = q/g;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.gif"
 ALT="$g$">|; 

$key = q/displaystyleA=int_{-1slash5}^12sqrt{5x+1}dx+int_1^3left[2sqrt{5x+1}-(x^2-1)right]dx;MSF=1.6;LFS=11;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="416" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.gif"
 ALT="$\displaystyle A = \int_{-1/5}^1 2\sqrt{5x+1}dx +
\int_1^3 \left[ 2 \sqrt{5x+1} - (x^2-1) \right] dx $">|; 

1;