Suppose Jen's velocity in mph was measured every ten minutes for one hour, and that her velocity was decreasing over that hour. The recorded values are shown in the table below. Use a right-hand sum to estimate how far Jen travelled during that hour. Is this an over- or under-estimate of the distance she really travelled?
Answer
We're pretending that Jen went 55 mph during the first 10 minutes, 50 mph during the next 10 minutes, and so on.
Let's rewrite the table so that t is measured in hours instead of minutes:
Using the nice formula for taking a right hand sum, we estimate that Jen travelled
Since the velocity function is decreasing, on each sub-interval we're underestimating Jen's velocity. She wasn't going 55 mph for the first ten minutes - she was going faster than 55 mph except at the very end of the sub-interval. This means we under-estimated her distance on the first sub-interval. Similarly, we underestimated her distance on all the other sub-intervals. This means 34.2 miles is an under-estimate of the distance she travelled.
To get a more accurate estimate of how far Jen travelled during that hour, we can average the left- and right- hand sums. We estimate Jen travelled
When you're given information about someone's velocity and asked to estimate the distance they've travelled, if the problem doesn't specifically tell you what sort of sum to use, take both the left-hand and right-hand sum and average them. This average, also known as the trapezoid sum, is usually the best estimate you can get when you only have a few scattered values of the velocity.
Be Careful: Remember to include units in your answers.
