Let S(x) represent the total sales (in thousands of dollars)...

Let $S(x)$ represent the total sales (in thousands of dollars) for the month $x$ in the year 2005 at a certain department store. Represent each following statement by an equation involving $S$ or $S_{-}^{\prime}$. (a) The sales at the end of January reached $$\$ 120,560$$ and were rising at the rate of $$\$ 1500$$ per month. (b) At the end of March, the sales for this month dropped to $$\$ 80,000$$ and were falling by about $$\$ 200$$ a day.

Let $S(x)$ represent the total sales (in thousands of dollars) for the month $x$ in the year 2005 at a certain department store. Represent each following statement by an equation involving $S$ or $S_{-}^{\prime}$.
(a) The sales at the end of January reached $$\$ 120,560$$ and were rising at the rate of $$\$ 1500$$ per month.
(b) At the end of March, the sales for this month dropped to $$\$ 80,000$$ and were falling by about $$\$ 200$$ a day.

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Key Concepts

Derivative (Rate of Change)

The derivative of a function measures its instantaneous rate of change with respect to its independent variable. In real-world contexts like sales, the derivative represents how quickly the sales amount changes over time, indicating periods of growth or decline.

Unit Conversion for Rates

When dealing with rates, especially in contexts where different time units are used (like per day versus per month), it is important to convert units appropriately. This ensures that the rate of change is expressed in a form that is consistent with the function's domain.

Function Representation

This concept involves modeling a real-life situation with a function that relates an independent variable to a dependent variable. In many situations, such as in economics or business, a function is used to represent the changing quantity over time, which allows for analysis using algebra and calculus.

Function Evaluation

This concept refers to determining the value of a function at a specific point in its domain. In practical applications, this means substituting a given value (such as a particular month) into the function to obtain the corresponding sales or measurement.

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