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    $\begingroup$ You're plotting two different functions and each has a reasonable choice of Exclusions independent of the other. Why don't you use to Plot command and combine the results with Show? You might also try the TrackedSymbols -> {t0,s} in your Manipulate command. Finally, with 9 questions now asked, I think it would be reasonable for you to properly format your code - it's just a matter of indenting your code block four spaces. $\endgroup$ Commented Aug 10, 2014 at 12:59
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    $\begingroup$ Your epilog contains Point[{-4, y1[s]}], but y1 is not defined. $\endgroup$ Commented Aug 10, 2014 at 13:53
  • $\begingroup$ I used Mark's suggestion and made the changes suggested by m_goldberg. All worked well. BTW, the "y1" was a typo; it should have been "y0" $\endgroup$ Commented Aug 11, 2014 at 19:51