\begin{homework}\
\label{homework:10}

\medskip
\exercise
(a) Let $l_1$ be the tangent line to the reciprocal function at the
point $(2,0.5)$.  Find the area under $l_1$ between $x=1$ and $x=3$.  

(b) Let $l_2$ be the chord joining $(1,1)$ to $(2.4,\,1/2.4)$ on the graph
of the reciprocal function.  Find the area under $l_2$.  

(c) Explain how your answers to parts (a) and (b) imply that the number $e$
must be between 2.4 and 3.

\smallskip
\exercise
Simplify the following expressions: 

\begin{tabular}{lll}
(a) $e^{\ln 3+\ln x}$;
        &(c) $e^{\ln x-2\ln y}$; 
             &(e) $\ln\left(\sqrt{e^{x^2}}\right).$\\
(b) $\ln\left(1/e^{x^2}\right)$;
          &(d) $e^{-\ln(1/x)}$;
             &
\end{tabular}

\smallskip
\exercise
Solve for $y$ in terms of $x$:

\begin{tabular}{lll}
(a) $e^{0.1y}=x^2$;
          &(c) $x=\ln\sin y$; 
               &(e) $\ln(1-y)-\ln y=0.01x$.\\
(b) $\ln(y-x)=2\ln x$;
          &(d) $e^{y^2}=x^2$;
          &
\end{tabular}



\smallskip
\exercise
Find the derivative of $$\text{(a)}~e^{2x}\text{Arctan}\,x;\qquad
\text{(b)}~\frac{x}{ e^{\sin(\pi x)}};\qquad\text{(c)}~e^{e^x};\qquad\text{(d)}
~x^2e^{-\cos x};\qquad\text{(e)}~e^{\text{Arcsin}\,x}.$$

\smallskip
\exercise
The function obtained by averaging the function $e^x$ and the function
$e^{-x}$, i.e., $(e^x+e^{-x})/2$, comes up often enough in applications to
have earned the right to have its own name: ``hyperbolic cosine.''  The
function $(e^x-e^{-x})/2$ is called the ``hyperbolic sine.''  These are
abbreviated ``cosh'' and ``sinh'' (``sinh'' is pronounced ``sinch''):
$$\cosh x=\frac{e^x+e^{-x}}{  2};\qquad\qquad\sinh x= \frac{e^x-e^{-x}}{2}.$$
(a) Find derivative formulas for ~cosh~ and ~sinh~ similar to the
derivative formulas for ~cos~ and ~sin. (b) Simplify $\cosh^2x-\sinh^2x$.
(c) Sketch the hyperbolic sine and cosine.

\smallskip
\exercise
Evaluate: $$\text{(a)}~\int_{\ln 0.04}^{\ln
4}e^{x/2}dx,\qquad\text{(b)}~\int_0^1xe^{-x^2}dx,
\qquad\text{(c)}~\int_0^\pi e^{\cos x}\sin x\,dx,\qquad
\text{(d)}~\int_0^\infty\frac{dt}{ 10^t}.$$

\smallskip
\exercise
Find the volume of the solid obtained by rotating the shape in Example~\ref{eg:10.1}
in the text around the $x$-axis.

\smallskip
\exercise
Find the $x$- and $y$-coordinates of the maxima/minima and points of
inflection, and sketch: $$\text{(a)}~y=e^{\frac{2}{ 3}\sin(3x)},~~0\le x\le
\pi/3;\qquad\text{(b)}~ y=x^2\ln(1/x),~~0<x<1.$$

\end{homework} 

