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October 23rd, 2012, 04:48 PM  #1 
Member Joined: Apr 2011 Posts: 47 Thanks: 0  Upper and Lower Estimates Problem
Hey guys, I've been struggling with this one. I believe I'm overthinking way too much. Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to stop. Jeff’s data follow: Time since start (min) 0 15 30 45 60 75 90 Speed (mph) 16 15 14 14 12 11 0 (a) Assuming that Roger’s speed is never increasing, give upper and lower estimates for the distance Roger ran during the first half hour. Lower estimate = miles Upper estimate = miles (b) Give upper and lower estimates for the distance Roger ran in total during the entire hour and a half. Lower estimate = miles Upper estimate = miles I thought I had to do (16*15)+(15*15)...etc. Multiplying the mph by the change in X. But that equals some insane amount of miles. Can ya guys help me out? Thanks. 
October 23rd, 2012, 07:25 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 464 Math Focus: Calculus/ODEs  Re: Upper and Lower Estimates Problem
For the lower estimate, let's assume that for each 15 minute interval, Roger runs at the speed clocked at the end of that interval and for the upper estimate let's assume Roger runs at the speed clocked at the beginning of that interval. a) For the first 30 minutes: lower estimate: upper estimate: Can you use this same technique now for part b)? 
October 23rd, 2012, 08:02 PM  #3 
Member Joined: Apr 2011 Posts: 47 Thanks: 0  Re: Upper and Lower Estimates Problem
Thanks for the quick response Mark. The formula you suggested did indeed work for part b. Thank you! How did you come to the conclusion to multiply by .25? Also, I thought lower estimates were supposed to be calculated starting with the very first value and missing the [i]last[i] value. I thought the same for the upper estimate. Skip the first value, but include the last one. It was mixed here but still worked. Am I missing something obvious? Thanks again! 
October 23rd, 2012, 08:06 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 464 Math Focus: Calculus/ODEs  Re: Upper and Lower Estimates Problem
15 minutes is 1/4 of an hour, and since the speed is in miles per hour, we want to use hours as our unit of time. The lower estimate is obtained by using the lowest possible speed for the intervals which occurs at the end of each interval since his speed is decreasing. Likewise, the upper estimate uses the highest possible speed for the intervals which occurs at the beginning. 

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