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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 S30,593 per capita, and this average was increasing at about S1400 per year (a little above the national average of about S1225 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.

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00:45

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 2

The Product and Quotient Rules

Derivatives

Differentiation

Missouri State University

Oregon State University

Baylor University

Boston College

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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it's clear. So when you married here. So let's make t b the number of years after 1999. So the population is equal to 9200 t last 96,001 961,400 in the average annual income ISS represented by a of T, we're just equal to 1400 t plus 30,000 593. And if we multiply these together, it gives the total personal income of tea. So the rate at which the total person personal income rises, it's gonna be the derivative of tea, and we use our product rule. When we plug it in, we get 9200 t waas murder 1200 t we simplify and we get plus 9200 t plus 961,400 arms. 1400. We're gonna plug in for 1999. We're gonna plug in zero because there's gonna be zero years and this gives us around 1,000,000,627 million Runge of $15,600. This part right here is the rate of rate of change of total income that comes from the annual income from the additional population and this right here is the rate of change of total income that comes from the increased income of the total population.

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