https://file.aiquickdraw.com/mathbot/file/17265892106520xitlbrf.jpg

In the image, we are dealing with a cost function for the production of a commodity given by:

\[
C(x) = 301 + 24x - 0.09x^2 + 0.0001x^3
\]

### Part (a)
The derivative of the cost function $ C(x) $, denoted $ C'(x) $, gives the marginal cost. This means how much the cost changes when an additional unit is produced.

- $ C'(100) = 9 $ was correctly computed.

The interpretation for $ C'(100) $, which was selected correctly, is:

- **"This is the rate at which costs are increasing with respect to the production level when $ x = 100 $."**

### Part (b)
In part (b), you're asked to find the actual cost of producing the **101st** item. This is given by:

\[
C(101) - C(100)
\]

However, the provided answer is incorrect. We will need to calculate the costs for $ C(101) $ and $ C(100) $ to find the difference.

Let me calculate this for you.The actual cost of producing the 101st item is approximately **$8.94**, not $9 as entered.

Would you like more details or explanations?

Here are five related questions to consider:
1. How do you compute the marginal cost at any production level?
2. What does the second derivative $ C''(x) $ of the cost function represent?
3. How do you calculate the exact cost for any given number of items?
4. Why is it important to know the cost of producing one more item (marginal cost)?
5. Can you interpret the difference between the total cost and the marginal cost?

**Tip:** The marginal cost function can be used to estimate the cost of producing small changes in quantity, but actual cost differences may vary slightly.

The cost function for production of a commodity is C(x) = 301 + 24x - 0.09x^2 + 0.0001x^3. (a) Find C'(100). (b) Find the actual cost of producing the 101st item.

This problem explores the cost function C(x) = 301 + 24x - 0.09x^2 + 0.0001x^3, including finding the derivative at x = 100 to get the marginal cost and calculating the actual cost of producing the 101st item. Learn how derivatives are used to analyze cost functions and marginal cost for advanced high school or college students.

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