Sherlock Holmes (1859-1930), perhaps the greatest detective of all time, had to confront many difficult situations his opponents put him through on this way to solving mysteries. In the case of Sherlock Holmes' Train Adventure, you will use mathematical analysis to help get Sherlock out of a bind...

And what a bind it is. Our hero finds himself bound and gagged in a train car headed hundreds of miles away from the case he's been working on, and through enemy territory. Fortunately, Holmes' captors do not kill him, for there is a reward for him at their destination. Fortunately also, Holmes knows a trick or two from his good friend Harry Houdini (1874-1926), and could release himself at any time.

Unfortunately, though he knows how to jump off a train without seriously injuring himself, doing so in enemy territory or in the middle of the desert would be extremely dangerous. Holmes does know of one safe place for jumping off the train -- a small town some 75 miles from where he was abducted. Of course, his captors do not put him in a car with a large window through which he could notice when to make his escape, when the train passes by the town. Through a small crack, all Holmes can see are shadows of each telegraph pole the train whizzes by. These are sometimes obstructed for long stretches by trees, so counting the number of poles they pass won't work. Furthermore, if he broke his binds to take a closer look, however, he'd have to jump out quickly -- before his captors check on him again -- which would be a big mistake unless he broke his bonds very close to 75 miles since the train's departure. In a soundproof car and no odometer, what's Sherlock to do?

Then an idea strikes in Sherlock's mind like a bolt of lightning. "That's it!" he exclaims, gags removed, to Watson tied up next to him, "Numerical Integration!" Watson is perplexed, but Holmes just smiles, adding, "It's a good thing my mathematical schooling went far beyond Elementary, my dear Watson!"

Here is Holmes' idea: although the train's speed varies throughout the journey as it passes from wilderness to more inhabited areas and back, by counting the number of telegraph poles the train passes during a given 10 second period, they can get a good estimate of the train's speed at any given time. "And any old fool knows that the integral of velocity will give us the total distance," remarks Watson, "but we don't have a function to integrate, only a handful of velocity readings." Show Watson how it's done. Sherlock has committed the following chart to memory (over, please):

(Watson has converted "telegraph poles per ten seconds" into miles per hour (mph) readings)

Using the left-hand endpoint approximation method (L_n), approximately how far has the train traveled? ________________________ miles.

Sherlock and Watson decide they must make more frequent readings; this is what they find:

Using the same approximation method, approximately how far has the train traveled in total during the first hour and a half? ____________________. Should Sherlock and Watson make their escape at this time?

But wait! Before you answer, keep in mind that it's only safe to jump if they are absolutely sure where they are. Having sized up the train's engine type when they were abducted, Sherlock and Watson are confident that its maximum possible speed is 90 mph and its maximum rate of acceleration or deceleration is 180 mph/hour. Holmes and Watson must be sure they are within 25 miles of the town or jumping train isn't safe -- should they make a run for it?