Thread: Left Right Sums

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|||10|||9||||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.

I got 5.75 and 6.25.

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

I got 13.25 and 16.5

(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?

The answer should be in minutes. I am not exactly sure how to do this, something with f(b)-f(a) delta t, I seemed to have lost the notes taken in class.

a and b are correct

for left and right hand sums the error bound is

En =[ K(b-a)^2]/2n

where K is the max of the first derivative and n is the number of sub intervals

K =13

13(1.5)^2/(2n) < .1

solving this inequality we get n = 147 ( to the nearest integer)

which means he would have to take readings about every .6 min or every 36 secs.