The population of the United States (in millions) by decade is...

Question

The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure.

Explain why the average rate of growth from 1950 to 1960 is a good approximation to the (instantaneous) rate of growth in 1955.

Accepted Answer

Welcome back, everyone. In this problem, a scientist is studying the growth of a bacterial population in a lab culture. The experiment started at 8 a.m. and the table shows the population T hours since the experiment started. The population size B of T as a function of time T is modeled by the curve below. Here we have a table with our values and an accompanying curve. Justify why the average growth rate from 2 p.m. to 3 p.m. is a reasonable estimate to the instantaneous growth rate at 2:30 p.m. Now, if we're going to justify why this average growth rate is a reasonable estimate to the instantaneous growth rate, we need to figure out where our times are on our graph. Recall, we were told that the experiment started at 8 a.m. and the table shows the population T hours since the experiment started. In other words, here, when T equals 1, it means 1 hour after 8 a.m. or 9 a.m. Likewise, T equals 2 is 10 a.m. T equals 3 is 11 a.m., 4 is 120 p.m., 5 is 10 p.m., and 6 is 2 p.m., OK? So this is when, uh, let's let's write it here. At 2 p.m. T equals 6. Likewise, it follows then that at 3 p.m. OK. At 3 p.m. oops, let's write that properly here. At 3 p.m., T equals 7. And finally, at 2:30 p.m. OK. T would be equal to 6.5, halfway through 6 and 7. So now that we know where all of these points are, OK, the question is how can we use that to show that the average growth rate is a reasonable estimate of the instantaneous growth growth rate? Well, let's think about both of them. Recall that the tangent line at a point represents the instantaneous growth rate growth rate at that point. And the slope of the C line represents the average growth rate between two points. So if we were to go to our graph here, if we draw, OK. And let me try to draw that properly here. If we draw our average. Or the sorry, the slope of the second line to represent the average growth rate. What do you notice? Well, essentially it's the same as the tangent line or it's pretty close to the tangent line at the point where T equals 6.5. So in that case, this tells us then that the tangent line at T equals 6.5 would have a slope approximately equal to the slope of the C line between T equals 6 and T equals 7, which means that the curve is approximately linear from T equals 6 to T equals 7. So that is why the average growth rate would be a reasonable estimate to the instantaneous growth rate. Thanks a lot for watching everyone. I hope this video helped.

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