\begin{homework}\
\label{homework:7}

\medskip


\exercise % 1:
In each case use the tangent line approximation to estimate the
$y$-coordinate of the point on the curve whose horizontal distance is 0.01
(to the right) from the given point $P$: 
\begin{align*}
\text{(a) }&x^2+xy+y^2=7,\;P=(1,2);\\ 
\text{(b) }&\frac{x}{ y}+\frac{y}{ x}+\frac{xy^2}{ 100}=\text{const },P=(2,5);\\
\text{(c) }&\sqrt{x}+\frac{3}{ x}=y-1+\frac{1}{\sqrt{y}+1},\,P=(9,4).
\end{align*}


\medskip

\exercise %2.
Find a formula for the derivative $y'$ at the point $(x,y)$:
\[
\text{(a)}~y^2=1+x^2;\qquad
\text{(b)}~x^2+xy+y^2=7;\qquad
\text{(c)}~
x^3+xy^2=y^3+yx^2.
\]

\medskip

\exercise %3
A \index{ hyperbola} hyperbola passing through $(8,6)$ consists of all points whose distance
from the origin is a constant more than its distance from the point (5,2).
Find the slope of the tangent line to the hyperbola at $(8,6)$.

\medskip

\exercise %4.
In Example 7.5(a), suppose that you are starting from the point $(1,1)$
rather than $(0,1)$.  Find the equation of the line from this point which
is the closest approximation to the curve of constant total light
intensity.  After you've walked a horizontal distance of $0.1$ (to the
right), what is your $y$-coordinate?

\medskip

\exercise %5.
In the situation of Example 7.5, suppose that you don't know the relative
strengths of the sources A and B, but you know that the line $y=0.6x+1$ is
the line that you should start walking along from the point (0,1) if you
want the total \index{ light intensity} light intensity to be as constant as possible.  Find how
much stronger B is than A.

\medskip

\exercise %6.
Suppose that a light source A is at the origin, and a source B twice as
strong is located at $(3,0)$.  Find the slope of the direction you move in
from (1,2) in order for the total light intensity to remain as constant as
possible.

\medskip

\exercise %7.
When \index{ crude oil} crude oil flows from a well, water is frequently mixed with it in an
emulsion.  To remove the water the crude is piped to a device called a
heater--treater, which is simply a large tank in which the oil is warmed
and the water is allowed to settle out.  Operating experience in a
particular oil field indicates that the concentration $C$ of water in the
treater's output can be modeled by the following equation in the
neighborhood of the usual operating point of 135${}^\circ$F and a 2-hour
holding time: 
\[
     C=0.04-0.0032\,h^2-10^{-6}\,T^2+4.1\cdot 10^{-5}h\,T,
\]
where $h$ is the holding time in hours and $T$ is the operating temperature
in degrees F.  (a) Due to random fluctuations in the well's flow rate, the
holding time actually varies slightly around 2 hours.  Suppose you are
given a simple control device that can change the \index{ tank} tank temperature
proportionally to the measured change in holding time.  What constant of
proportionality would best compensate for small holding time fluctuations
and keep the water concentration as constant as possible?  (b) Now as field
equipment ages, its maximum operating settings are generally decreased.
Find the equation of the line that best approximates the way in which the
holding time would have to be increased as the maximum temperature rating
falls slightly below the usual operating temperature.

\medskip

\exercise %8.
One end of a string (the point A) is attached to the $y$-axis at a distance
20 meters from the origin; the string is threaded through a hole (the point
B) at the end of a pole situated along the line $x=10$; and the other end
of the string (the point C) is 
\begin{rightindent}{2.5in}
\noindent
attached to a hook that is free to slide
along the $x$-axis and is being pulled by a force to
the right that is sufficient to keep the string taut.  See the diagram at
the right.  At time
$t=0$, when the top of the pole is at the point (10,10), the pole starts
lowering at the steady rate of 30 cm/sec (=0.30 m/sec).  The sliding end of
the string reaches the point $x=20$ at time $t=10$.
\end{rightindent}
\hfill
%-----------------------------------------------------
\begin{psfigure}{2.5in}{1.9in}{figHA.ps}
\end{psfigure}
%-----------------------------------------------------

\noindent
(a) Write an equation relating $t$ and $x$ (the time is an implicit function
of the location of the sliding end of the
string).  (b) Use the tangent line approximation to
find the time when the sliding end of the string has moved another 20 cm to
the left from the point $x=20$.


\end{homework}


