\def\wsn{29}
\input worksheet.tex

\item{1.}  Let $f(x)=1/x$ be defined on $[1,2]$.  Break the interval
into five equal intervals.  Use these to find an upper sum for 
$$
\int_1^2{1\over x}\,dx.
$$

\bigskip\bigskip
\itemitem{2.\hskip12pt a)}  Suppose that on some interval, the
function $f$ satisfies $f'(x)=f(x)$.  Show that $f(x)=Ke^{x}$ for some
$K$ by considering the function $g(x)={f(x)\over e^{x}}$.

\medskip
\itemitem{b)}  Find the derivative of $f(x)=e^{x-5}$.  Does this
contradict your work from part a)?

\medskip
\itemitem{c)}  Suppose that on some interval, the function
$f$ satisfies $f'(x)=cf(x)$ for some number $c$.  Show that
$f(x)=Ke^{cx}$ for some $K$ by considering the function
$g(x)={f(x)\over e^{cx}}$.


\itemitem{d)}  Suppose that $f'(x)=f(x)g'(x)$ for some $g$.  Show that
$f(x)=Ke^{g(x)}$ for some $K$.
 
\bigskip\bigskip
\item{3.}  Compute the following integrals
$$
\pt a \int{e^{2x}+e^x\over e^x}\,dx\qquad
\pt b \int x(x+1)^2\,dx\qquad
\pt c \int\cos x\sqrt{\sin x}\,dx\qquad
\pt d \int{x^2-1\over x+1}\,dx
$$

\bigskip\bigskip
\item{4.}  {\bf Newton's Law of Cooling} states that the rate of
change of the temperature $T$ of an object is proportional to the
difference between $T$ and the temperature $\tau$ of the surrounding
medium.  A cup of coffee at $200^\circ$ in a room of temperature
$70^\circ$ is stirred continually and reaches $100^\circ$ after ten
minutes.  At what time was it at $120^\circ$?

\smallskip
\item{} Hint:  Consider Problem 2a).

\bigskip\bigskip
\item{5.} Show that
$$
\int_{-\pi}^\pi{7x^3-22x\over x^{50}+\cos x}\,dx=0.
$$

\bigskip\bigskip
\item{6.}  Find the following integrals:
$$
\pt a \int_0^{\pi/4}\sec x\tan^2x\,dx\qquad
\pt b \int _1^e{1\over x\sqrt{\ln x}}\,dx\qquad
\pt c \int_0^\pi{\sin x\over\cos^2 x}\,dx
$$


\bigskip\bigskip
\item{7.}  Let $f(x)=x(\ln x)^2$ on the interval $[e^{-4},e]$.

\medskip
\itemitem{a)}  Differentiate and simplify.

\medskip
\itemitem{b)}  Find all points $c$ where $f'(c)=0$ or $f'(c)$ does not
exist.

\medskip
\itemitem{c)}  Use the first derivative test to classify the above
points.

\medskip
\itemitem{d)}  Find all local and global extrema.

\bigskip\bigskip
\item{8.}  Solve for $x$ if
$$
{e^x\over\sqrt{e^x}}=2.
$$



\bye