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Problem 90 Hard Difficulty

The table gives the US population from 1790 to 1860.
(a) Use a graphing calculator or computer to fit an exponential function to the data. Graph the data points and the exponential model. How good is the fit?
(b) Estimate the rates of population growth in 1800 and 1850 by averaging slopes of secant lines.
(c) Use (he exponential model in part (a) to estimate the rates of growth in 1800 and 1850. Compare these estimates with the ones in part (b).
(d) Use the exponential model to predict the population in 1870. Compare with the actual population of 38,358,000. Can you explain the discrepancy?

Answer

a. $R^{2}=0.999$
b. $$\begin{array}{l}{\text { For } 1800, m_{s e c}=165,550} \\ {\text { For } 1850, m_{s e c}=719,000}\end{array}$$
c. $$\begin{array}{l}{\text { For } x=1800 : f^{\prime}(x)=169,757} \\ {\text { For } x=1850 : f^{\prime}(x)=742,016}\end{array}$$
d. $45,375,883$

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Video Transcript

in this problem. We have a table of data that gives the U. S. Population from the year 17 92 the year 18 60. And we're going to use a graphing calculator to fit an exponential function to the data. So we start by going to the stat menu and in the edit menu, and we type the years into list one, and we type the populations into list, too. So once we have that, we go out of the list there and then back into the stat menu. And then we go over to calculate, and we go down until we find exponential regression. There it is and we press enter. We're using List one and list, too. And inside the store regression option, we want to go into the virus menu, go over to y bars. That's why variables choose function and choose why one. And that's so that our equation will be pasted into the Y equals menu so we can graph it. And now we calculate and these are the values in the equation. And if we press y equals, we see that that has been pasted into our y equals equation in It's ready to graph. So now what we can do is we can go into the staff plot menu. Second y equals takes us to stop plot. We can press enter to to go into the menu for plot one and then turn it on. So where scatter plot is turned on and now we compress zoom number nine zoom stat so that the calculator will set us a good window. So here we see the scatter plot graft and the exponential model graph. It looks like a really good fit. Okay, so that was part a. Now it says we're supposed to estimate the rates of population growth in the year 1800 the year 18 50 by estimating by averaging slopes of C can't lines. Okay, so we can go back to paper and pencil. And for the year 1800 What we're going to do is we're going to take the year 17 90 the year 1800 find the slope of the sea can't line, and we're going to take the year 1800 the year 18 10 and find the slope of the sea count line. Once we have those two answers, we're going to average them so the slope of the sea can't line between the years 1817 90. That's just why to minus y one over x two minus X one, we get 137,900. That would be people per year, and then we repeat that process for the times 18 10 to 1800. Why? To minus y one over x two minus X one, and we get 193,200 people per year. Let's average those two numbers. Add them together and divide by two. And this is our estimate for the rate of change of the population in the year 1800. We think at that time the rate of change was 165,550 people per year. Now we're going through that process again for the year 18 50. So once again, I've taken three of the data points 18 50 the point right before it. I'm going to find the slope of the sea, can't line and 18 50 the point right after it. And I'm going to find the slope of the sea, can't line and then we're going to average those two slopes so we have the slope of one. C can't line. We have the slope of the other C can't line and then we're going to average those two numbers. So what we're saying is our estimate for the rate of change of the population in the year 18 50 with 719,000 people per year notice the change from what it was in the year 19 or 1800 which was up here 5 165,050 what it is in the year 18. 50. Significantly greater. Okay, so that takes us to part C. We're going to go back to our model and estimate the rates of growth from the graphing calculator. So what we want to dio is we can go into the draw menu, which is second program, and we can choose number five, Draw a tangent and we can draw the tangent for the year 18. What was it? 1800. So type in 18 1800 press enter and the tangent line is drawn and at the bottom of the screen, we can see the equation of that line and look at the slope. 156,848 0.7. So how does that compare with the number we had earlier? This is 156,848 and we had 165,550. Okay, not too far off. About what, Seven or 8000 off from each other. Now let's repeat the process for the year 18 50. So we go back into the draw menu. Second program. We choose tangent again. Press interview. We type in 18 50. Press enter. There's the tangent line. And let's look at the slope at the bottom of the screen. 686,071. Okay. And what did we have before? 719,000. So our values were a little high compared to what we're getting from the graph. Okay, Now, for Part D, let's predict the population in the year 18 70. So we go back to the calculator and we pressed trace and then we use the down arrow to make sure that our cursor is actually on the curve, not the scatter plot. So notice if I move left or right, it moves along the curve, and what we can do is we can type in the number 18 70 and press enter, and we don't have it because we're probably out of range of our screen. So what we need to go do is go to the window and let's change our X max. Let's change that to 1900. Graph it again and now we go to trace and now we type in 18 70. It's try that one more time. Go to graph, go to trace, put the cursor on the curve and now we type in 18 70 and what we have at the bottom of the screen. Let's see, What would that be? 41,946,006. 568. We're comparing that to 38,558,000. Okay, so this value is a little high compared to the actual value. So what would explain the discrepancy? Well, our model is not going to be accurate forever. It's just a model. It's not perfect predictor of the population and different things happen over time. That will change and make the model s accurate.