%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                                                       %
%  Curve Sketching II                                   %
%                                                       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\problemgrouptitle{Curve Sketching II}

\begin{problemblock}{Apply calculus techniques to identify all important features of the graph of each function and then sketch it.}

\item $f(x)=x^3-3x-2$
\answer{Notice $f'(x)=3x^2-3=3(x+1)(x-1)$ and $f''(x)=6x$.
$D=\mathset{R}=(-\infty,\infty)$;
$x$-int$=-1,\ 2$; $y$-int$=-2$;
No asymptotes;
No symmetry; 
Increasing on: $(-\infty,-1)\cup(1,\infty)$; Decreasing on: $(-1,1)$;
\Relative maxima: $f(-1)=0$; \Relative minimum: $f(1)=-4$;
Concave upward on: $(0,\infty)$; Concave downward on $(-\infty,0)$;
Inflection point: $(0,-2)$; Graph:
\begin{center}
\resizebox{4cm}{!}{%
\includegraphics{graphics/curve-sketching-II1}%
}
\end{center}
}

\item $\displaystyle y=\frac{3x^2}{x^2-1}$ 
\answer{Notice that the first and second derivatives simplify to $\displaystyle y'(x)=-\frac{6x}{(x^2-1)^2}$ and $\displaystyle y''=\frac{6(3x^2+1)}{(x^2-1)^3}$.
$D=\mathset{R}-\{-1,1\}=(-\infty,-1)\cup(-1,1)\cup(1,\infty)$;
$x$-int$=0$; $y$-int$=0$;
Horizontal asymptote: $y=3$; Vertical asymptotes: $x=-1$, $x=1$;
Symmetric about the $y$-axis;  
Increasing on: $(-\infty,-1)\cup(-1,0)$; Decreasing on: $(0,1)\cup(1,\infty)$;
\Relative maxima: $f(0)=0$; No~\relative minimum;
Concave upward on: $(-\infty,-1)\cup(1,\infty)$; Concave downward on $(-1,1)$;
No~inflection points; Graph:
\begin{center}
\resizebox{5cm}{!}{%
\includegraphics{graphics/curve-sketching-II2}%
}
\end{center}
}

\item $f(x)=\displaystyle\frac{x^2}{x^2+2x+1}$
\answer{ Notice that the derivatives simplify to $f'(x)=\displaystyle\frac{2x}{\left(x+1\right)^3}$ and $f''(x)=\displaystyle\frac{2-4x}{\left(x+1\right)^4}$.
$D=\mathset{R}-\{-1\}=(-\infty,-1)\cup(-1,\infty)$;
$x$-int$=0$; $y$-int$=0$;
Horizontal asymptote: $y=1$; Vertical asymptote: $x=-1$;
No symmetry;  
Increasing on: $(-\infty,-1)\cup(0,\infty)$; Decreasing on: $(-1,0)$;
No~\relative maxima; \Relative minimum: $f(0)=0$;
Concave upward on: $(-\infty,-1)\cup(-1,1/2)$; Concave downward on $(1/2,\infty)$;
Inflection point: $(1/2,1/9)$; Graph:
\begin{center}
\resizebox{6cm}{!}{%
\includegraphics{graphics/curve-sketching-II3}%
}
\end{center}
}

\end{problemblock}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: "../master"
%%% End:
