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% !!! IMAGES START HERE !!!

{\newpage\clearpage
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$\displaystyle \lim_{x \rightarrow 0^+} \frac{\ln(1+x)}{4x}$%
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{\newpage\clearpage
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$\displaystyle \lim_{x \rightarrow 0} \mbox{cotg} \, (x)(\mbox{sen} \, (5x)-\mbox{sen} \, (2x))$%
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$\displaystyle \lim_{x \rightarrow 0^+} \frac{\ln(1+x)}{4x} 
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$\displaystyle \lim_{x \rightarrow 0} \mbox{cotg} \, (x)(\mbox{sen} \, (5x)-\mbox{sen} \, (2x)) 
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$f$%
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$\bullet$%
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$I_1 = [1,2] \bigcup
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{\newpage\clearpage
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$\displaystyle \Longrightarrow  f''(x) \geq 0 $%
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{\newpage\clearpage
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$I_1
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$I_1$%
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$I_2 = (-\infty,1] \bigcup
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{\newpage\clearpage
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$\displaystyle \Longrightarrow  f''(x) \leq 0 $%
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{\newpage\clearpage
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$I_2
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$I_2$%
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$x=3.5$%
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{\newpage\clearpage
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$f(x) =
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{\newpage\clearpage
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$[-3,-1]$%
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{\newpage\clearpage
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$f(-1) = 0.13, \quad f(-2) = 0.29, \quad
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline349}%
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{\newpage\clearpage
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$e^{2x}$%
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$f'(x)$%
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$x^3(2+x)$%
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$x=0$%
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$x \rightarrow -\infty$%
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{\newpage\clearpage
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$\displaystyle \lim_{x \rightarrow -\infty} x^4 e^{2x} 
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$\infty/ \infty$%
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{\newpage\clearpage
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$\displaystyle \lim_{x \rightarrow -\infty} x^4 e^{2x}  =
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\displaystyle \lim_{x \rightarrow -\infty} \frac{24x}{-8e^{-2x}}
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\displaystyle \lim_{x \rightarrow -\infty} \frac{24}{16e^{-2x}} = 
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{\newpage\clearpage
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$x \rightarrow \infty$%
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{\newpage\clearpage
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$\displaystyle \lim_{x \rightarrow +\infty} x^4 e^{2x} 
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{\newpage\clearpage
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$y = 0$%
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{\newpage\clearpage
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$f(x)$%
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{\newpage\clearpage
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$(-\infty,-2] \bigcup [0,+\infty)$%
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{\newpage\clearpage
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$'[-2,0]$%
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{\newpage\clearpage
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$2000 cm^3$%
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{\newpage\clearpage
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$cm^2$%
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{\newpage\clearpage
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$x$%
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{\newpage\clearpage
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$h$%
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{\newpage\clearpage
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$\displaystyle C = 2x^2  (1) +  4xh (4) = 2x^2 + 16 xh$%
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{\newpage\clearpage
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$\displaystyle V = x^2 h  = 2000$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline403}%
$\displaystyle h = \frac{2000}{x^2} \Rightarrow
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{\newpage\clearpage
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$(0,+\infty)$%
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{\newpage\clearpage
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$\displaystyle C'(x) = 4x - \frac{16(2000)}{x^2} $%
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{\newpage\clearpage
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$\displaystyle 4x = \frac{16(2000)}{x^2}$%
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{\newpage\clearpage
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$\displaystyle x^3 = 4(2000) = 8000 \Rightarrow x = \sqrt[3]{8000} = 20$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline417}%
$\displaystyle C'(x) = \frac{4(x^3-8000)}{x^2} $%
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{\newpage\clearpage
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$C'(x)$%
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{\newpage\clearpage
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$x=20$%
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{\newpage\clearpage
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$C(x) \rightarrow +\infty$%
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{\newpage\clearpage
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$x \rightarrow +\infty$%
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{\newpage\clearpage
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$\displaystyle C''(x) = 4 + \frac{32(2000)}{x^3} > 0$%
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{\newpage\clearpage
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$C(x)$%
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{\newpage\clearpage
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$\displaystyle  \int (x+1)e^{x} dx $%
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{\newpage\clearpage
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$\displaystyle \int 2x (4-x^2)^{1/3} dx$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline447}%
$\displaystyle \begin{array}{rl} 
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline449}%
$\displaystyle \int (x+1) e^x dx =
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\lthtmlinlinemathZ
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{\newpage\clearpage
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$C$%
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{\newpage\clearpage
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$\displaystyle u = 4 - x^2 \Rightarrow du = -2x dx $%
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{\newpage\clearpage
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$\displaystyle \int 2x (4-x^2)^{1/3} dx = 
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- \frac{u^{4/3}}{4/3} = - \frac{3 u^{4/3}}{4} + C =
- \frac{3(4-x^2)^{4/3}}{4} + C$%
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{\newpage\clearpage
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$f(x) = 2\mbox{sen} \, (x)-\cos^2(x)$%
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{\newpage\clearpage
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$[0,2\pi]$%
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$0$%
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$2\pi$%
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{\newpage\clearpage
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$\displaystyle f'(x) = 2 \cos(x) - 2\cos(x)(-sen(x)) =
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{\newpage\clearpage
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$\displaystyle \cos(x) = 0 \Rightarrow x = \pi /2 $%
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{\newpage\clearpage
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$\displaystyle \mbox{sen} \, (x)=-1 \Rightarrow x = 3\pi /2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline483}%
$f(0)= -1$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline485}%
$\pi/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline487}%
$f(\pi/2) = 2(1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline489}%
$3\pi/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline491}%
$f(3\pi/2)=2(-1) $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline495}%
$f(2\pi) = -(1)^2 $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline497}%
$\displaystyle \Rightarrow  x=\pi/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline501}%
$\displaystyle \Rightarrow  x=3\pi/2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}


\end{document}
