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Unformatted text preview: A motivating problem... Suppose the upward velocity of a piston t seconds after it starts to move is given by v ( t ) = sin( t ) m/s. If the piston is 3 meters high when it starts to move, how high will it be 4 seconds later? Where might this lead us? How could we approximate the answer to the question above? We might recall that distance = rate time , but that is only true when the rate is a constant velocity - and clearly the velocity in the problem above is changing over time. (Note: sometimes, the piston even moves backwards!) However, if the time interval was just shorter...the velocity wouldnt be chang- ing by as much, right? Consider the difference in velocity at the following two times: v (1) . = 0 . 84147 v (1 . 0001) . = 0 . 84152 So maybe, for just that brief interval of time, distance = rate time could give a decent approximation of the small change in position during that tiny time interval... distance . = 0 . 8415 (1 . 0001- 1) Note, in the interest of getting an even better approximation, we tried to split the difference here with regard to the rate used something between 0.84147 and 0.84152. There is more to say about this - but for now, recall that we are at least assurred (by intermediate value theorem, in this case) that at some x * we have v ( x * ) = 0 . 8415... Consequently, if we denote the small change in position that happens over this tiny time interval by s , and if we denote the change in time (i.e., time elapsed) from the start to the end of that tiny time interval by t , then we can say s . = v ( x * ) t Now, to approximate the total change in position from t = 0 to t = 4, we could find all of these little-changes-in-position and just add them up! (Keep in mind, some of these could be negative, indicating a distance traveled back- wards.) Consider the following possible way to cut up the time from t = 0 to t = 2 into nine tiny intervals no longer than 0.6 seconds and this is just one of many, many ways to do this......
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This note was uploaded on 12/05/2011 for the course MATH 111 taught by Professor Bang during the Fall '08 term at Emory.
- Fall '08