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​(a) The average rate of change of employees with respect to revenue for the changes from ​$4 billion to ​$34 billion is about 
  
767,000 ​employees/billion dollars.
​(Round to the nearest thousands as​ needed.)
Part 2
​(b) The average rate of change of employees with respect to revenue for the changes from ​$34 billion to ​$108 billion is about 
  
351,000 ​employees/billion dollars.
​(Round to the nearest thousands as​ needed.)
Part 3
​(c) The average rate of change of employees with respect to revenue for the changes from ​$4 billion to ​$108 billion is about 
  
106000 ​employees/billion dollars.
​(Round to the nearest thousands as​ needed.)

Let's analyze the problem step by step based on the graph and information given. The problem involves finding the average rate of change of the number of employees with respect to the revenue for different intervals of revenue.

The formula for the average rate of change of a function $ f(x) $ between two points $ x_1 $ and $ x_2 $ is:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

Where:
- $ f(x_1) $ is the value of the function (number of employees) at $ x_1 $ (the first revenue),
- $ f(x_2) $ is the value of the function at $ x_2 $ (the second revenue).

### (a) Average rate of change from $4 billion to $34 billion:
- For $ x_1 = 4 $ billion, the number of employees is $ f(4) = 11 $ million (from the graph).
- For $ x_2 = 34 $ billion, the number of employees is $ f(34) = 34 $ million.

Using the formula:

\[
\text{Average Rate of Change} = \frac{34 - 11}{34 - 4} = \frac{23}{30} = 0.767 \text{ million employees per billion dollars}.
\]

This is equivalent to:

\[
767,000 \text{ employees per billion dollars}.
\]

### (b) Average rate of change from $34 billion to $108 billion:
- For $ x_1 = 34 $ billion, the number of employees is $ f(34) = 34 $ million.
- For $ x_2 = 108 $ billion, the number of employees is $ f(108) = 60 $ million.

Using the formula:

\[
\text{Average Rate of Change} = \frac{60 - 34}{108 - 34} = \frac{26}{74} \approx 0.351 \text{ million employees per billion dollars}.
\]

This is equivalent to:

\[
351,000 \text{ employees per billion dollars}.
\]

### (c) Average rate of change from $4 billion to $108 billion:
- For $ x_1 = 4 $ billion, the number of employees is $ f(4) = 11 $ million.
- For $ x_2 = 108 $ billion, the number of employees is $ f(108) = 60 $ million.

Using the formula:

\[
\text{Average Rate of Change} = \frac{60 - 11}{108 - 4} = \frac{49}{104} \approx 0.471 \text{ million employees per billion dollars}.
\]

This is equivalent to:

\[
471,000 \text{ employees per billion dollars}.
\]

However, you mentioned a result of 106,000 employees per billion dollars, so we might need to verify if there's an error in interpretation of the graph. Let me know if you'd like further clarification or recalculation!

The figure to the right shows the number of full-time employees of a company (in millions) as a function of the revenue generated by the company (in billions). Find and interpret the average rate of change of employees with respect to revenue for the following changes in revenue: (a) $4 billion to $34 billion, (b) $34 billion to $108 billion, (c) $4 billion to $108 billion.

This problem involves calculating the average rate of change of full-time employees with respect to the company's revenue. Using the formula for the average rate of change, the solution explores changes in employees for revenue increases from $4 billion to $34 billion, $34 billion to $108 billion, and $4 billion to $108 billion. Key concepts include functions and linear approximations.

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