1. ## Integral, The Definite!

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) 13 12 11 11 9 8 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.
miles (lower estimate)
miles (upper estimate)

(b) Give upper and lower estimates for the distance Roger ran in total during the entier hour and a half.
miles (lower estimate)
miles (upper estimate)

(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?
every minutes

2. Originally Posted by mathlete
(a) Assuming that Roger's speed is never increasing, give upper and lower estimates for the distance Roger ran during the first half hour.
Because his speed is never increasing, the fastest he could ever run between 0 and 15 minutes is 13mph. So put down 15 minutes at 13mph. During the interval from 15-30 minutes, the most he could ever run is 12mph. So the fastest he could have run was 13mph*.25hrs+12mph*.25hrs = 6.25 miles.

The lowest he could have run was if he immediately dropped to 12 mph, and ran the entire distance from 0-15minutes at 12mph. And then at 15 min, dropped to 11 mph and ran 15 min at 11mph. So 12*.25+11*.25=5.45 miles

Those are your upper and lower estimates, use the same method on the next question.

I expect that this will be preparing you for Riemann Sums, which will lead into integration, so if you want to do a little extra work to get a good feel for where you are going, you might check out Riemann sums. On this page (Riemann sum - Wikipedia, the free encyclopedia), you will see some graphs, with area calculated using rectangles. Notice how on one of the graphs, the rectangles go above the line for the graph, and on the other, they stay below that line. That is exactly what we just did on this question, we got our overestimate by calculating from the left, and our lower estimate by calculating from the right. You will probably also take some where you calculate it in the middle, and after you get comfortable with that, you will be introduced to the concept of continuous sums, or taking an infinite number of rectangles to get as close as possible to the correct value. This process is integration (you'll probably first hear it called "anti-derivative" because it reverses the process of differentiation).

Anyway, you could read over that page in 20 min, and get a good feel for what you're going to learn next, and it would probably help you out a bit. And at the very least, should make it more clear why we took those values the way we did.

As for question C, I don't know the answer.