The average rate of change of a function $P(x)$ over an interval $[a, b]$ is calculated by the formula:

$\text{Average rate of change} = \frac{P(b) - P(a)}{b - a}$

In this case, the profit function is given as $P(x) = 5x^2 - 5x + 5$, and we are asked to find the average rate of change as $x$ changes from 3 to 5. Here, $a = 3$ and $b = 5$.

Step 1: Find $P(5)$

$P(5) = 5(5)^2 - 5(5) + 5$ $= 5(25) - 25 + 5 = 125 - 25 + 5 = 105$

Step 2: Find $P(3)$

$P(3) = 5(3)^2 - 5(3) + 5$ $= 5(9) - 15 + 5 = 45 - 15 + 5 = 35$

Step 3: Calculate the average rate of change

$\text{Average rate of change} = \frac{P(5) - P(3)}{5 - 3} = \frac{105 - 35}{5 - 3} = \frac{70}{2} = 35$

Thus, the average rate of change of profit as $x$ changes from 3 to 5 is 35 (hundreds of dollars per item).

Would you like more details or have any questions?

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5 Related Questions:

1. What is the interpretation of the average rate of change in this context?
2. How would the average rate of change differ if $x$ changed from 5 to 7?
3. How do you compute the instantaneous rate of change for $P(x)$?
4. What happens to the profit when $x = 0$? Is there a minimum number of items that should be sold for profit?
5. Can we use this average rate of change to predict profit for future values of $x$?

Tip:

The average rate of change can help you understand how quickly profit is growing or shrinking over a range of values. For exact growth at any point, though, you need the derivative.