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The problem gives the formula for the annual inventory cost \( C \) for a manufacturer:

\[
C(Q) = \frac{1,008,000}{Q} + 5.2Q
\]

where \( Q \) is the order size when inventory is replenished. You are asked to:

1. Find the change in annual cost when \( Q \) is increased from 355 to 356.
2. Compare this with the instantaneous rate of change when \( Q = 355 \), i.e., find the derivative \( C'(Q) \) and evaluate it at \( Q = 355 \).

### Step 1: Find the change in \( C \) when \( Q \) changes from 355 to 356
We can calculate the difference \( \Delta C = C(356) - C(355) \).

### Step 2: Find \( C'(Q) \) and evaluate it at \( Q = 355 \)
To compare with the instantaneous rate of change, we'll compute the derivative \( C'(Q) \) and evaluate it at \( Q = 355 \).

Let me start calculating these values for you.The calculations yield the following results:

1. **Change in \( C \)** when \( Q \) increases from 355 to 356:
   \[
   \Delta C = C(356) - C(355) = -2.78
   \]
   
2. **Instantaneous rate of change at \( Q = 355 \)** (i.e., \( C'(355) \)):
   \[
   C'(355) = -2.80
   \]

Both the change in cost and the instantaneous rate of change are negative, indicating that the cost decreases slightly as the order size increases. The actual change \( \Delta C = -2.78 \) is very close to the instantaneous rate of change \( C'(355) = -2.80 \), meaning that the rate of change gives a good approximation of the actual cost change over this small interval.

Do you want more details or have any questions?

Here are 5 follow-up questions:
1. What does the derivative \( C'(Q) \) represent in the context of inventory cost?
2. Why is the cost decreasing as the order size increases?
3. How would the change in \( C \) behave if we increased \( Q \) over a larger interval, say from 355 to 365?
4. Can we predict how the total cost would behave for very large or very small values of \( Q \)?
5. What happens to \( C'(Q) \) as \( Q \) becomes extremely large?

**Tip:** The derivative of a function provides the best linear approximation of the function's change over small intervals.

The annual inventory cost C for a manufacturer is given by the equation C = 1,008,000/Q + 5.2Q, where Q is the order size when the inventory is replenished. Find the change in annual cost when Q is increased from 355 to 356, and compare this with the instantaneous rate of change when Q = 355.

This problem involves calculating the change in annual inventory cost when the order size increases from 355 to 356, and comparing it with the instantaneous rate of change at Q = 355 using calculus. Suitable for high school or early college students.

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