October 21, 2005 12:18masterSheet number 88 Page number 7272Lecture Sixit must rest for half a day. Then, the maximal distance it cantravel in 1 day is 1 km but it can travel this distance in onlyhalf a day.b. For (2) to imply (1) we need to assume the turtle cannot “jump”a positive distance in zero time. Contrast this with the case inwhich the turtle’s speed is 1 km/day but after a day of travelingit can “jump” 1 km. Thus, it needs 1 day to travel 1 km butwithin 1 day it can travel 2 km.The assumptions that in any positive interval of time the turtlecan travel a positive distance and that the turtle cannot “jump” aresufﬁcient for the equivalence of (1) and (2). LetM(t)be the maximaldistance the turtle can travel in timet. Assume that the functionMis strictly increasing and continuous. Then, the statement, “Themaximal distance a turtle can travel int∗isx∗” is equivalent to thestatement, “The minimal time it takes a turtle to travel
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