\begin{homework}\
\label{homework:4}

\medskip
\exercise
Find the equation of an ellipse centered at the origin if 

(a) it passes through (2,0) and also through (0,1/2);

(b) it passes through (10,0) and has foci at ($\pm 6$,0);

(c) it has foci at ($\pm 7$,0) and eccentricity 0.28; 

(d) it passes through (1,0) and also through (0,3); 

(e) it has foci at (0,$\pm 3$) and passes through (4,0).

\medskip
\exercise
Find the foci and eccentricity of the ellipses 

(a) $x^2\!/25+y^2\!/16=1$; (b) $x^2\!/49+y^2\!/625=1$; (c) $4x^2+9y^2=1$.

\medskip
\exercise
A particle travels at constant velocity in a straight line from the
point (1,3) to the point (7,11) in 2 seconds.  

(a) Find parametric equations $x(t)$, $y(t)$ for the motion of the
particle, and find the distance the particle travels using the formula for
the arclength of a parameterized curve.

(b) Find the equation of the line the particle travels along in the form
$y=mx+b$, and then find the distance the particle travels using the formula
for the arclength of an explicit function $y=f(x)$.

(c) Find the distance the particle travels by simply using the distance
formula between its starting point and stopping point.  Your answers to
parts (a), (b) and (c) should be the same.

\medskip
\exercise
Let $n$ be greater than 1.  

(a) Write a definite integral for the arclength of $y=x^n$ from $x=0$ to
$x=b$. 

(b) There is one (and only one) case when this integral can be easily
evaluated by the anti-derivative method, namely, when $n=3/2$.  In that
case use a suitable substitution to evaluate the integral and find a
formula in terms of $b$ for the arclength in part (a).

\medskip
\exercise
Write a definite integral for 

(a) the total distance traveled by a ball that is thrown from a 38.8-meter
high window at 14 m/sec at an angle of 45${}^\circ$ above the horizontal;


(b) the distance one-quarter the way around an ellipse having semi-major
axis $a$ and semi-minor axis $b$, i.e. the part of the circumference
that is in the first quadrant (use parametric equations for the
ellipse); 

(c) the distance traveled by the pebble in Example 4.4 if the pebble is
caught in a spoke half-way from the center to the rim.

\medskip
\exercise
A bug is crawling up the arch of the sine curve $y=\sin x$ at speed 0.5.
When the bug reaches the point $(\pi/6,0.5)$, find

(a) the rate at which its $x$-coordinate is increasing;

(b) the velocity of its shadow on the $x$-axis, if the sun is shining from
the left from an angle of $21.8^\circ$ above the horizontal;

(c) the same as part (b), but no longer assume that the angle from which
the sun is shining remains constant.  Suppose that the units of time are
hours, and the sun is setting at the rate of 15${}^\circ$ per hour.

\end{homework}
 
