Help with Upper and Lower Estimates?!

[elb1992]

So I am having a difficult time understanding how to determine the upper and lower estimates. In the example problems in my text book, the numbers seem to come out of nowhere. Here is a problem that I am working on now to review for my upcoming quiz that I am having lots of trouble with...please help?!

Time since brakes applied (sec) 0 2 4 6
Velocity (ft/sec) 88 45 16 0

Give the lower and upper estimates for the distance the car traveled after the brakes were applied.

Thank you SO much to anyone that can help!!!

 

[tkhunny]

The car does not slow in jerks. It is normally a nice, smooth process. Assuming it moves in jerks will create bounds.

All intervals are 2 seconds

2(88) + 2(45) + 2(16) + 2(0)

Here, we assume the car did not actually start to slow until two full seconds after applying the brakes. Obviously, that;s a little silly. We WILL stop better than that. This is an upper bound.

Similarly:

2(45) + 2(16) + 2(0)

This assumes the first decrease happens immedaitely upon applying the brakes. This is obviously quite silly, too. This is a lower bound.

If we create estimates that are not "silly", we create better approximations.

For example, we can apply Merton's Rule, assuming the decleration is uniform.

2(88+45)/2 + 2(45+16)/2 + 2(16+0)/2

In more modern parlance, this might be the midpoint rule. Is it a lower bound or an upper bound? I'm not telling.