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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 39\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 39} 
\centerline
{\bf a.k.a. Practice Exam 3}

\bigskip

$$
\hbox{1. Compute }\int_2^\infty{x^3-x^2-2\over x^4-x^3+x^2-x}\,dx
\hskip4in
$$

\item{2.}  Graph $r=\cos4\theta$ in polar coordinates.  Find the area
of the region bounded by this graph.  

\medskip
\item{3.}  An object is moving along the path $x=50t$, $y=16-16t^2$.
\itemitem{a)}  What is its speed at $t=1$?
\itemitem{b)}  At what time is the object moving fastest?
\itemitem{c)}  How far does the object travel in the upper half plane
(i.e., where $y>0$)?

$$
\hbox{4. For what values of $p$ does the series
}\sum_{n=1}^\infty{1\over n^p} \hbox{ converge?  Justify your answer.}
\hskip1.58in
$$

\item{5.}  Show that the Taylor series for $\sin x$ converges for all
real numbers and is equal to $\sin x$.  

\medskip
\item{6.}  Consider the plane parallel to the vectors $(1,1,1)$ and
$(2,0,1)$ containing the point $(0,1,0)$.  Find the point on this
plane closest to the origin.

\medskip
\item{7.}  Prove that if the speed of a particle is constant, {\bf a}
is perpendicular to {\bf v}.

\medskip
\item{8.}  Let $f(x,y)$ be a differentiable function of two
variables and {\bf u} a unit vector in the $(x,y)$-plane.  Give the
definitions of $\partial f/\partial x$, $\partial f/\partial y$, and
$D_{\bf u}f$, and then describe how you would actually compute each of
these. 

$$
\hbox{9.  Let $w=f(x-y,y-z,z-x)$.  Show that }
{\partial w\over\partial x}+{\partial w\over\partial y}+{\partial
w\over\partial z}=0.
\hskip2.26in
$$

\item{10.}  Let $T(x,y,z)$ be the temperature at the point $(x,y,z)$.
Assume that $\nabla T$ at $(1,1,1)$ is $\langle2,3,4\rangle$.
\itemitem{a)}  Find $D_{\bf u}T$ at $(1,1,1)$ if {\bf u} is in the
direction of the vector $\langle1,-1,2\rangle$.
\itemitem{b)}  Estimate the change in temperature as you move from the
point $(1,1,1)$ a distance $0.2$ in the direction of the vector
$\langle1,-1,2\rangle$. 
\itemitem{c)}  Find three unit vectors {\bf u} such that $D_{\bf
u}T=0$ at $(1,1,1)$.

$$
\hbox{11.  Evaluate }\int_0^1\int_x^1\sin y^2\,dy\,dx \hbox{ and }
\int_0^1\int_{\sqrt x}^1{dy\over\sqrt{1+y^3}}\,dx.
\hskip2.89in
$$

\item{12.}  Find the center of mass of the region bounded by $y=x$ and
$y=x^2$ in the first quadrant with density at $(x,y)$ given by
$e^{-x}$.

\medskip
\item{13.}  Find the moment of inertia about the origin of the region
in Problem 12. 

\medskip
\item{14.}  Find the average of the function $f(P)$ over the region
within the cardioid $r=1+\cos\theta$ where $f(P)$ is the distance from
the point $P$ to the $x$ axis.


\bye