%\centerline{\bf Homework on the Natural Logarithm}

\begin{homework}\
\label{homework:8}

\medskip
\exercise
Find the derivative of:

\smallskip

\begin{tabular}{ll}
(a) $\ln(kx)$ ($k$ is a constant);  & (d) $\ln(\tan x)$; 
\\
(b) $\ln^4x$;              &(e) $\ln\left((x^2+16)^2\right)-x\arctan(x/4)$.
\\
(c) $\ln^2(\ln x)$; &
\end{tabular}



\smallskip
\exercise
(a) Find the derivative of the functions $\ln(ax)-\ln x-\ln a$ (where
$a$ is any constant) and $\ln(x^n)-n\ln x$ (where $n$ is any constant).

(b) Use part (a) to give another derivation of Properties 3(a) and 3(d) of
$\ln$.

\smallskip

\exercise
Simplify: 

\smallskip

\begin{tabular}{ll}
(a) $\ln 30 -\ln 2 -\ln 5$; 
         &(d) $\ln\sqrt{6.75}$ (in terms of $\ln 2$ and $\ln 3$); \\
(b) $\ln\sqrt{8}$ (in terms of $\ln 2$);
         &(e) $\ln(\pi r^2)-2\ln r$;\\
(c) $\ln{\root 3\of 9}$ (in terms of $\ln 3$); 
         &(f) $\ln\sqrt{ r^2-x^2}-\frac{1}{ 2}\ln(r-x)$.
\end{tabular}

\smallskip

\exercise
Find: 

\begin{tabular}{ll}
(a) $\disp\int_{-1}^{-0.5}\frac{dx}{ x}$; 
       &(d) $\disp\int_2^3\frac{dx}{ x\,\ln^2x}$;\\
(b) $\disp\int\cot x\, dx$; 
       &(e) $\disp\int_0^{2\pi}\frac{\sin(\theta/2)d\theta}{ 2-\cos(\theta/2)}$.\\
(c) $\disp\int\frac{dx}{ x\,\ln x}$ (hint: substitute $u=\ln x$);&
\end{tabular}


\smallskip
\exercise
Find the center of mass of the uniform plate contained 

(a) between the graphs of $y=1/x$, $y=-1/x$, $x=1$ and $x=2$;

(b) between the graphs of $y=1/x^2$, $y=-1/x^2$, $x=1$ and $x=2$;

(c) between the graphs of $y=1/(1+x^2)$, $y=-1/(1+x^2)$, $x=0$ and $x=1$.

\smallskip

\exercise
Find the area 

(a) between the curves $y=5/(1+x^2)$ and $y=2/x$; 

(b) between the curves $y=1/\sqrt{1-x^2}$ and $y=2/(2-x)$.

\smallskip
\exercise
The region bounded by $y=-1/x^2$, $y=-1/4$ and $y=-1$ is revolved around
the $y$-axis to form a container.  Find its volume.

\smallskip
\exercise
A bug slides down the curve $y=\ln x$, moving leftward (i.e., its
$x$-coordinate is decreasing) at 10 units/min.  Find its speed along the
curve when $x=2$, when $x=1$, and when $x=1/2$.

\smallskip
\exercise
By ``experiment'' (for example, substituting $x=1/10^{1000}$), find the
limit as $x\longrightarrow 0$ (where $x$ is positive) of: 

\smallskip

\begin{tabular}{llll}
(a) $x^3\ln x$,
&
(b) $x\ln x$, 
&
(c) $\sqrt{x}\ln x$, 
&
(d) ${\root 10 \of x} \ln x$.
\end{tabular}

\smallskip
\exercise
Find the derivative of: 

\smallskip

\begin{tabular}{ll}
(a) $y = x^x$; 
          &(d) $y=x^{\ln x}$;  \\
(b) $y=(\sin x)^{\sqrt{x}}$; 
          &(e) $y=(\cos x)^{\text{Arctan}\,x}$.\\
(c) $y=(\ln x)^x$; &

\end{tabular}

\smallskip
\exercise
One morning in January it began to snow and kept on snowing at a constant
rate.  A snow plow began to plow at noon.  It plowed clean and at a constant
rate (volume per unit time).  From one o'clock until two o'clock it went
only half as far as it had gone between noon and one o'clock.  When did
the snow begin falling?


\end{homework} 
 
