practice_derivatives.tex

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\begin{document}
\begin{center}
\Large
\rm{Math 111:  Practice with derivatives}
\\
\end{center}
\vspace{0.2in}
\fboxsep0.5cm

\begin{enumerate}
\item{In each case find $\frac{dy}{dx}$
\begin{enumerate}
\item{$y = \frac{x}{\sqrt{x^2+1}}$}
\vspace{0.2in}
\item{$y = \sec^2{(3x)}$}
\vspace{0.2in}
\item{$y = x^2e^{\sqrt{x}}$}
\vspace{0.2in}
\item{$y = (4^x+1)^{\pi}$}
\vspace{0.2in}
\item{$\sqrt{x+y} = 1+x^2y^2$}
\vspace{0.2in}
\item{$e^{2y}\tan{x} = \cos{xy}$}
\vspace{0.2in}
\item{$y = \log_{10}{(x-10^{-x})}$}
\vspace{0.2in}
\item{$y = \ln{\left( \frac{x}{4e^{2x}+3}\right)^3}$}
\vspace{0.2in}
\item{$y = \tan^{-1}{\left( \sqrt{\frac{1-x}{1+x}} \right)}$}
\end{enumerate}
}
\item{The graph of $y=(x^2)^x$ has two horizontal tangent lines.   Find equations for both of them.}
\item{The lateral surface area of a cone with radius $r$ and height $h$ is given by $A=\pi r\sqrt{r^2+h^2}$.  Find $\frac{dr}{dh}$ in terms of
$r$ and $h$ when $A=1500\pi$}
\item{The values of $f(x)$ and $f'(x)$ are given in the table.}
\vspace{.15in}

\begin{tabular}{|r|r|r|r|r|r|}
\hline
$x$& -2 & -1 & 0 & 1 & 2   \\
\hline
$f(x)$& -8 & -5 & -2 & -2 & 0   \\
\hline
$f'(x)$& 2 & 4 & 2 & 3 & -3  \\
\hline
\end{tabular}
\vspace{.15in}
\\
If $H(x) = f(f(x)+x)$, what is the value of $H'(1)$?
\item{Suppose $f$ with a graph that passes through the point $(1,4)$ and that the line tangent to the graph at $(1,4)$ is $y=3x+1$.
  If $g(x) = f(x^2)$, find $g(1)$ and $g'(1)$.
}
  
\end{enumerate}
\end{document}