## 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

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