The demand for personal computers in the home goes up with...

The demand for personal computers in the home goes up with household income. For a given community, we can approximate the average number of computers in a home as $$q=0.3454 \ln x-3.047 \quad 10,000 \leq x \leq 125,000$$ where $x$ is mean household income. ${ }^{60}$ Your community has a mean income of $\$ 30,000$, increasing at a rate of $\$ 2,000$ per year. How many computers per household are there, and how fast is the number of computers in a home increasing? (Round your answer to four decimal places.)

The demand for personal computers in the home goes up with household income. For a given community, we can approximate the average number of computers in a home as
$$q=0.3454 \ln x-3.047 \quad 10,000 \leq x \leq 125,000$$
where $x$ is mean household income. ${ }^{60}$ Your community has a mean income of $\$ 30,000$, increasing at a rate of $\$ 2,000$ per year. How many computers per household are there, and how fast is the number of computers in a home increasing? (Round your answer to four decimal places.)

Key Concepts

Logarithmic Functions

Logarithmic functions are used to model relationships where the rate of change decreases as the variable increases. In economic or social models, such functions capture diminishing returns, meaning that as income grows, the incremental gain in demand may be lower despite continued growth. They are particularly useful in modeling natural growth processes where exponential characteristics are observed and then transformed using logarithms.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function is changing at any given point. It allows us to analyze how a dependent variable, such as the number of computers per household, responds to changes in an independent variable, like income. This process is essential for creating models that quantify instantaneous change and for understanding trends in applied mathematics.

Related Rates

Related rates problems involve determining the rates at which two or more related quantities change with respect to time. When one variable depends on another variable that is itself changing over time, such as household income increasing per year, related rates techniques and the chain rule are employed to find the derivative with respect to time. This approach is widely used in physics, economics, and other fields where dynamic relationships exist.

Chain Rule

The chain rule is an important technique in calculus used for differentiating composite functions—functions in which one variable depends on another variable that in turn depends on a third variable. In the context of related rates, the chain rule is necessary to differentiate the function with respect to time, linking the rate of change of an intermediate variable (e.g., income) to the ultimate rate of change of the dependent variable (e.g., number of computers).

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