Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every 15 minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to stop. Jeff's data follows.
Time since start (min) 0 15 30 45 60 75 90
Speed (mph) 10 9 7 7 6 5 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.
4 miles (lower estimate) (correct)
4.75 miles (upper estimate) (correct)
(b) Give upper and lower estimates for the distance Roger ran in total during the entire hour and a half.
8.5 miles (lower estimate) (correct)
11 miles (upper estimate) (correct)
(c) How often would Jeff have needed to measure Roger's speed in order to find lower and upper estimates within 0.1 mile of the actual distance he ran?
But I am stuck here, my answer was 0.5 but it was wrong
Let $s_1 = 10+9+7+7+6+5 = 44$, and $s_2 = 9+7+7+6+5 = 34$, and the time interval is $\delta$ (measured in hours).
The upper bound is $s_1\delta$, and the lower bound is $s_2\delta$
And you are estimating within $s_1\delta - s_2\delta = 10\delta$.
So to get that to equal 0.1 miles you need $\delta = 1/100$.