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\begin{document}
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% !!! IMAGES START HERE !!!

{\newpage\clearpage
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$dy/dx$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline347}%
$y = e^{x^3 + 3 \sqrt{x}}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline349}%
$f'(1/3)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline351}%
$\displaystyle f(x) = \frac{2 + \ln(3x)}{3 + x^2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline353}%
$\displaystyle \frac{dy}{dx} = e^{x^3 + 3\sqrt{x}} \cdot \frac{d}{dx}(x^3 + 3x^{1/2})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline355}%
$\displaystyle \frac{dy}{dx} = e^{x^3+3\sqrt{x}}(3x^2 + (3/2) x^{-1/2})=
e^{x^3+3\sqrt{x}}\left(3x^2 + \frac{3}{2\sqrt{x}}\right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline357}%
$\displaystyle f'(x) = \frac{(3+x^2)(2+\ln(3x))' - (3+x^2)'(2+\ln(3x))}
{(3+x^2)^2} = \frac{(3+x^2) \cdot 3 \cdot 1/(3x) - (2x)(2+\ln(3x))}
{(3+x^2)^2} = \frac{(3+x^2) - 2x^2(2+\ln(3x))}{x(3+x^2)^2} =
 \frac{3-3x^2 - 2x^2 \ln(3x)}{x(3+x^2)^2} $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline359}%
$\displaystyle f'(1/3) = \frac{3-3(1/3)^2-2(1/3)^2 \ln(3 \cdot 1/3)}
 {1/3(3+(1/3)^2)^2} = \frac{3 - 1/3 - 0}{1/3(28/9)^2}=
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\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline361}%
$\displaystyle f'(1/3) = 8 \left( \frac{9}{28} \right)^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline363}%
$\displaystyle \lim_{y \rightarrow 3} \frac{y-3}{\sqrt{y+1} \ -2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline365}%
$\displaystyle \lim_{x \rightarrow 0}
\frac{1-\cos(2x)}{x^2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline367}%
$\displaystyle \lim_{y\rightarrow 3} \frac{y-3}{\sqrt{y+1} \ -2} =
 \lim_{y\rightarrow 3} \frac{y-3}{\sqrt{y+1} \ -2} 
 \cdot \frac{\sqrt{y+1}+2}{\sqrt{y+1}+2} =
 \lim_{y\rightarrow 3} \frac{(y-3)(\sqrt{y+1} \ +2)}{(y+1)-4} =
 \lim_{y\rightarrow 3} \sqrt{y+1} +2 = \sqrt{3+1}+2 = 4$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline369}%
$\displaystyle \lim_{x \rightarrow 0} \frac{1-\cos(2x)}{x^2} =
\lim_{x \rightarrow 0} \frac{1-\cos(2x)}{x^2} \cdot
\frac{1+\cos(2x)}{1+\cos(2x)}  = 
\lim_{x \rightarrow 0} \frac{1-\cos^2(2x)}{x^2(1+\cos(2x))} =
=\lim_{x \rightarrow 0} \frac{\mbox{sen} \, ^2(2x)}{x^2(1+\cos(2x))} =
\lim_{x \rightarrow 0} \frac{\mbox{sen} \, (2x)}{x} \frac{\mbox{sen} \, (2x)}{x}
\frac{1}{1+\cos(2x)} =
\lim_{x \rightarrow 0} 4 \frac{\mbox{sen} \, (2x)}{2x} \frac{\mbox{sen} \, {2x}}{2x}\frac{1}
{1+\cos(2x)} =
4\cdot (1)^2 \frac{1}{1+1} = 2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline371}%
$f(x)=5x^2 - 3
\sqrt[3]{x}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline373}%
$\displaystyle f(x) = 5x^2 - 3 x^{1/3}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline375}%
$\displaystyle f'(x) = 5(2)x - 3(1/3)x^{-2/3} = 10x - \frac{1}{x^{2/3}}=
10x - \frac{1}{\sqrt[3]{x^2}} \ , \ x \neq 0$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline377}%
$\displaystyle f(1) = 5(1)^2 - 3\sqrt[3]{1} = 5-3 = 2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline379}%
$\displaystyle f'(1) = 10(1) - \frac{1}{\sqrt[3]{(1)^2}} = 10-1=9$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline381}%
$\displaystyle y-2 = f'(1)(x-1) \Rightarrow y-2 = 9(x-1)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline383}%
$\displaystyle f(x) = |x^2 - 9| - 6$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline385}%
$y = x^2$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline387}%
$f$%
\lthtmlinlinemathZ
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{\newpage\clearpage
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$f(x)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
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$f'(3)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline393}%
$x^2-9 \geq 0$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline395}%
$x^2 \geq 9$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline397}%
$\displaystyle \sqrt{x^2} \geq \sqrt{9} \Leftrightarrow
|x| \geq 3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline399}%
$\displaystyle f(x) = (x^2-9)-6 = x^2 - 15$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline401}%
$|x| < 3$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline403}%
$\displaystyle f(x) = -(x^2-9)-6 = 3-x^2$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline405}%
$\displaystyle f(x) = \left\{ \begin{array}{lr}
x^2 - 15, & |x| \geq 3 \\
3 - x^2, & |x| < 3 \end{array} \right. $%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline407}%
$f(3) = |3^2-9|-6=-6$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline409}%
$\displaystyle  \lim_{x \rightarrow 3^{-}} \frac{f(x)-f(3)}{x-3} =
\lim_{x \rightarrow 3^{-}} \frac{3-x^2-(-6)}{x-3} =
\lim_{x \rightarrow 3^{-}} \frac{9-x^2}{x-3}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline411}%
$9 - x^2 = -(x^2-9)=-(x+3)(x-3)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline413}%
$\displaystyle  \lim_{x \rightarrow 3^{-}} \frac{f(x)-f(3)}{x-3} =
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\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline415}%
$\displaystyle  \lim_{x \rightarrow 3^{+}} \frac{f(x)-f(3)}{x-3} =
\lim_{x \rightarrow 3^{+}} \frac{x^2-15-(-6)}{x-3} =
\lim_{x \rightarrow 3^{+}} \frac{x^2-9}{x-3} =
 \lim_{x \rightarrow 3^{+}} (x+3) = 3 + 3 = 6$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline419}%
$x(t)$%
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{\newpage\clearpage
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$y(t)$%
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{\newpage\clearpage
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$s(t)$%
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{\newpage\clearpage
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$t$%
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{\newpage\clearpage
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$y=8$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline429}%
$ \displaystyle dy/dt = - 50 \ , \ x=6  \  ,  \ ds/dt = 20 .$%
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{\newpage\clearpage
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$dx/dt$%
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{\newpage\clearpage
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$s^2 = x^2 + y^2$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline435}%
$\displaystyle  2 s \frac{ds}{dt} = 2x \frac{dx}{dt} + 2y \frac{dy}{dt}$%
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\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline437}%
$\displaystyle   s \frac{ds}{dt} = x \frac{dx}{dt} + y \frac{dy}{dt}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline439}%
$\displaystyle x = 6, y = 8 \Rightarrow s^2 = 6^2 + 8^2 = 36 + 64 = 100
\Rightarrow s = 10$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline441}%
$\displaystyle   (10) (20) = (6) \frac{dx}{dt} + 8 (-50)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline443}%
$\displaystyle \Rightarrow \frac{dy}{dt} = \frac{(10)(20)+(8)(50)}{6} 
= \frac{600}{6} = 100$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline445}%
$x^2 = y^2 + \mbox{sen} \, (xy)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline449}%
$y = f(x)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline451}%
$f(\sqrt{\pi})=\sqrt{\pi}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline453}%
$f'(\sqrt{\pi})$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline457}%
$\displaystyle f(x) = \frac{a x^n + 3}{3x^5 + 2}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline459}%
$y=3$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline463}%
$a$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline465}%
$n$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline467}%
$\displaystyle \frac{dx^2}{dx} = \frac{d}{dx} \left[y^2 + \mbox{sen} \, (xy) \right]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline469}%
$\displaystyle 2x = 2y \frac{dy}{dx} + \cos(xy) \left(
(1)y + x \frac{dy}{dx} \right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline471}%
$\displaystyle
\Longrightarrow 2x- y \cos(xy) = (2y + x \cos(xy))\frac{dy}{dx} \Rightarrow
\frac{dy}{dx} = \frac{2x-y\cos(xy)}{2y + x \cos(xy)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline473}%
$x=\sqrt{\pi} \Rightarrow y = \sqrt{\pi}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline475}%
$\displaystyle \frac{dy}{dx} = \frac{ 2 \sqrt{\pi} - \sqrt{\pi} \cos(\pi)}
{2\sqrt{\pi} + \sqrt{\pi} \cos(\pi)} =
\frac{2\sqrt{\pi}-\sqrt{\pi}(-1)}{2 \sqrt{\pi} + \sqrt{\pi}(-1)} = 
\frac{3\sqrt{\pi}}{\sqrt{\pi}} = 3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline477}%
$x = \sqrt{\pi}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline481}%
$f(x) \rightarrow 3$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline483}%
$x \rightarrow + \infty$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline485}%
$x \rightarrow -\infty$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline487}%
$a=9$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline489}%
$\displaystyle \lim_{x \rightarrow \pm \infty} \frac{a x^n+3}{3x^5 + 2}
= \lim_{x \rightarrow \pm \infty} \frac{a x^n}{3 x^5}
\Rightarrow 
 \left\{ \begin{array}{l} n = 5 \\a/3 = 3 \Rightarrow
a = 9 \end{array} \right.$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}


\end{document}
