Some applications

1.  The price (in dollars) of an item is modeled by the function  \( P(t) = 0.2 t^ 2 - t + 2   (4 \leq  t \leq  10) \),  where \( t \)  is given in years. What is the average rate of change of the price between 5 and 7 years?  solution

2. The gross domestic product (GDP) of a certain country is projected to be \( N(t)=3t^2+2t-8   (0\leq t \leq 6)\) billion dollars \(t\) years from now. What will be the average rate of change of the country's GDP between 3 and 5 years from now?  solution

3. The number of tablets and smartphones in use worldwide (in millions) in year \(t\) from  2010 through 2012 is approximately \( f(t)=128.1 t^{1.94}    (1\leq t \leq 3)
\) where \(t=1\) corresponds to 2010. (a) How many tablets and smartphones were in use in 2011?  (b)  How fast was the number of tablets and smartphones changing in 2011?  solution

4. The share of households in the US that are earning middle-class incomes has been in decline since the 1970s. The percentage of households with annual incomes within  \(50 %\) of the median income is given by  \( P(t)=50.3\,t^{-0.09}  (1\leq t \leq 6) \), where \(t\) is measured in decades, with \(t=1\) corresponding to 1980. (a) What percentage of households had annual incomes within \(50\)  of the median income in 2010? (b) How fast was the percentage of households with annual incomes within \( 50 %\) of the median income decreasing in 2010?  solution

5. The percentage of families that were married couples with children between 1970 and 2010 is approximately \( P(t)=\frac{49.6}{t^{0.27}}  (1\leq t \leq 5) \) where \(t\) is measured in decades, with \(t=1\) corresponds to 1970. (a) What percentage of families were married couples with children in  2000? (b) How fast was the percentage of families that were married couples with children changing in 2000?   solution       

6. Mobile Instant messaging (IM) is a small portion of total IM usage, but it is growing sharply. The function \( P(t)=0.25t^2+0.57t+3.9   (0 \leq t \leq 4) \) gives the mobile IM accounts in year \( t\) as a percentage of total enterprise IM accounts from  \( 2006 (t=0)\) through \(2010 (t=4)\). (a) What percentage of total enterprise IM accounts were the mobile accounts in \(2009\)? (b) How fast was this percentage changing in \(2009\)?   solution