$3$ Identifying an appropriate domain some graphs and from these decide upon an appropriate domain.

Example $1$:

$y=y_0+v_{0}t-\frac{1}{2}a{t}^2$

Suppose a ball is thrown with an initial velocity of $v_0=10$ meters per second upward from an initial height of $y_0=2$ meters. On what realistic domain will $y(t)0?$ seconds should be long enough, if the model is realistic.

$>t=$ linspace $(0,10)$;

$>$ plot$($t$,$y$)$

If you make this graph, you'll see that $10$ seconds was too long. $($Why$).$ Let's replot using an interval of $0,2.$

$>t=$linspace$(0,2)$;

$>$plot$(t,2+{10}^{**}t-(\frac{1}{2}{)}^{**}{10}^{**}t{*}^{{?}^{?}}2)$

$($We combined the definition of $y$ directly into the plot function.$)$

Now our interval is too small.

Exercise $2$:

Replot the ball problem until you can find a good estimate for the value of $b$ so that $y(t)0$ on the interval $0,b,$ and is negative for $t>b.$ What is the value of $b$ that you found?