Chapter 3.2, Problem 57E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# In this exercise we estimate the rate at which the total personal income is rising in the Richmond-Petersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,400, and the population was increasing at roughly 9200 people per year. The average annual income was $30,593 per capita, and this average was increasing at about$1400 per year (a little above the national average of about $1225 yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in the Richmond-Petersburg area in 1999. Explain the meaning of each term in the Product Rule. To determine To estimate: The total personal income was rising in the year 1999. Explanation Derivative rule: Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)] Calculation: Let x be the number of year, f(x) be the function of growth of population and g(x) be the function of growth of average income. f(x)=(Population increased per year)×(Number of years)+(Population in 1999)=(9200)×(x)+(961,400)=9200x+961,400 g(x)=(National average per year)×(Number of years)+(Average annual income per capita)=(1225)×(x)+(30,593)=1225x+30,593 Since the population was increasing at roughly 9200 per year and the average annual income was increasing at about$1400 per year, f(x)=9200 and g(x)=1400.

In 1999, f(0)=961,400, g(0)=30,593, f(0)=9200 and g(0)=1400.

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

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

That is, F(x)=f(x)g(x)

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