Chapter 3.2, Problem 49E

To determine

To estimate: The total personal income was rising in the year 1999.

Answer to Problem 49E

The total personal income was rising by about $1,627,415,600 per year in 1999.

Explanation of Solution

Derivative rule:

Product Rule: $\frac{d}{dx}\left[f_1\left(x\right)f_2\left(x\right)\right]=f_1\left(x\right)\frac{d}{dx}\left[f_2\left(x\right)\right]+f_2\left(x\right)\frac{d}{dx}\left[f_1\left(x\right)\right]$

Calculation:

Let x be the number of year, $f\left(x\right)$ be the function of growth of population and $g\left(x\right)$ be the function of growth of average income.

$\begin{array}{c}f\left(x\right)=\left(\text{Population increased per year}\right)\times \left(\text{Number of years}\right)+\left(\text{Population in 1999}\right)\\ =\left(9200\right)\times \left(x\right)+\left(961,400\right)\\ =9200x+961,400\end{array}$

$\begin{array}{c}g\left(x\right)=\left(\text{National average per year}\right)\times \left(\text{Number of years}\right)+\left(\begin{array}{l}\text{Average annual }\\ \text{income per capita}\end{array}\right)\\ =\left(1225\right)\times \left(x\right)+\left(30,593\right)\\ =1225x+30,593\end{array}$

Since the population was increasing at roughly 9200 per year and the average annual income was increasing at about $1400 per year, $f^{\prime }\left(x\right)=9200$ and $g^{\prime }\left(x\right)=1400$.

In 1999, $f\left(0\right)=961,400$, $g\left(0\right)=30,593$, $f^{\prime }\left(0\right)=9200$ and $g^{\prime }\left(0\right)=1400$.

Obtain the total personal income was rising in the year 1999.

Let $F\left(x\right)$ be total personal income which is product of $f\left(x\right)$ and $g\left(x\right)$.

That is, $F\left(x\right)=f\left(x\right)g\left(x\right)$.

Apply the product rule and simplify the terms,

$\begin{array}{c}{F}^{\prime }\left(x\right)=\frac{d}{dx}\left(f\left(x\right)g\left(x\right)\right)\\ =f\left(x\right)\frac{d}{dx}\left(g\left(x\right)\right)+g\left(x\right)\frac{d}{dx}\left(f\left(x\right)\right)\\ =f\left(x\right)g^{\prime }\left(x\right)+g\left(x\right)f^{\prime }\left(x\right)\end{array}$

Substitute 0 for x,

$F^{\prime }\left(0\right)=f\left(0\right)g^{\prime }\left(0\right)+g\left(0\right)f^{\prime }\left(0\right)$

Substitute the values $f\left(0\right)=961,400$, $g\left(0\right)=30,593$, $f^{\prime }\left(0\right)=9200$ and $g^{\prime }\left(0\right)=1400$,

$\begin{array}{c}{F}^{\prime }\left(0\right)=\left(9200\left(0\right)+961,400\right)1400+\left(1225\left(0\right)+30,593\right)9200\\ =961,400\times 1400+30,593\times 9200\\ =1,346,960,000+281,455,600\\ =1,627,415,600\end{array}$

Therefore, the total personal income was rising by about $1,627,415,600 per year in 1999.

To Explain: The meaning of each term in the product.

In 1999, the term $f\left(0\right)g^{\prime }\left(0\right)=\$1,346,960,000$ represents the portion of the rate of change of total income due to existing population’s increasing income.

In 1999, the term $f^{\prime }\left(0\right)g\left(0\right)=\$281,455,600$ represents the portion of the rate of change of total income due to increasing population.