Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter P - P.2 - Linear Models and Rates of Change - Exercises - Page 16: 24


a) The graph is on the figure. b) The least rapid increase coincides with the least slope and that happened in the year of 2008. c) The average change rate is the increase of $2.8$ million people per year. d) The prediction is that there would be $337.8$ million people living in US in $2020$.

Work Step by Step

a) The plot is given on the figure and for technical reason is done on computer. To draw it by hand just draw points $(t,y)$ from the table putting $t$ horizontally and $y$ vertically as in the plot given down. Then connect the points (red points on the figure) drawing a straight segment between two neighboring points. b) To find the slope of the line between two points divide the difference in their $y$ coordinate with the difference in their $t$ coordinate. We will do that for each pair of neighboring points and find which slope is the lowest and that means that during that year the change was the least rapid: $$m_1 = \frac{295.8-293}{5-4} = 2.8;$$ $$m_2 = \frac{298.6-295.8}{6-5} = 2.8;$$ $$m_3 = \frac{301.6-298.6}{7-6} = 3;$$ $$m_4 = \frac{304.4-301.6}{8-7} =2.8;$$ $$m_5=\frac{307.0-304.4}{9-8} = 2.6.$$ Here we see that $m_5$ which corresponds to the segment from $t=8$ to $t=9$ is the lowest which means that during the year of $2008$ the increase was the least rapid. c) The average rate of change is just the total difference in $y$ from $t=9$ to $t=4$ divided by the difference in $t$ which is: $$m_{avg} = \frac{307.0-293.0}{5} = 2.8.$$ This tells us that the average rate of change in the US population during the period of $2004-2009$ is $2.8$ million people per year. d) If the population keeps changing in this rate of $2.8$ million people per year then it will increase on average for $2.8$ million people every year from $2009$ to $2020$ which is the difference of $11$ years so the population in $2020$ would be that in the $2009$ plus the total increase: $$y=307.0+2.8*11 = 337.8$$ in millions.
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