Accumulated Effect of a Rate Part 3
In the previous activity we came up with a method to use the model of the rate of traffic on Route 15 in Johnson Vermont to estimate the total car count between 6am and 4pm. Now we consider how we can improve the estimate.
In the previous activity we used two samples of the traffic rate, one at 6am and another at 11am. One way to improve our estimate would be to simply sample more often. For instance, we might sample 3 times: the first sample could be at 6am, and the other 2 samples could be equally spaced between 6am and 4pm.
Of course this could then be further improved by sampling 4 times, and 5 times, and so on. This is illustrated in the applet below. Try adjusting
Samples to increase the number of times the model of the traffic rate is sampled, and notice how TotalTraffic changes. This change is the estimate of the car count improving as the number of samples increases. Furthermore, we also see that TotalTraffic increases as Samples increases, but it won't increase to infinity; instead it is converging to a certain number. We'll study what that number is shortly. (Spoiler Alert: it's the integral.)
For now, you're limited to a maximum of 10 Samples (one per hour) in this applet, but we'll shortly lift that restriction. Quick Check: What is the estimate of TotalTraffic with 6 samples?
Clearly there's a lot going on here, but there's two things that are most important to focus on:
- As
Samplesincreases, the estimateTotalTraffic(which can be visualized as the sum of the areas of the green rectangles) becomes more accurate. (Note: You may have noticed thatTotalTrafficis also increasing, but this is a bit of a red herring and isn't really important; more on that later.) - As
Samplesincreases, the green rectangles converge to the 2 dimensional region that's the area under the functiong(x), above the x-axis, and betweenStartTimeandFinishTime.
Samples to 150 in the applet below. I've also shaded the area under the graph of g(x) maroon so you can see how the rectangles "converge" to the shaded region. Be careful as you increase Samples; this applet is known to crash weak computers. But what does this all have to do with integrals? Move forward to find out!