The problem provides the total cost function for manufacturing $q$ units of a commodity:

$C(q) = 3q^2 + 5q + 10$

You are asked to estimate how the total cost will change if 40.5 units are produced, given that the current level of production is 40 units.

To estimate the change, we can use differentials (i.e., the derivative of $C(q)$) to approximate the change in cost:

Step 1: Find the derivative of the cost function

$C'(q) = \frac{d}{dq} \left( 3q^2 + 5q + 10 \right)$ Using the power rule: $C'(q) = 6q + 5$

Step 2: Evaluate the derivative at $q = 40$

$C'(40) = 6(40) + 5 = 240 + 5 = 245$

Step 3: Use the derivative to estimate the change

The change in $q$ is $\Delta q = 40.5 - 40 = 0.5$. We estimate the change in cost $\Delta C$ using: $\Delta C \approx C'(40) \cdot \Delta q = 245 \cdot 0.5 = 122.5$

Conclusion:

The total cost will increase by approximately $122.50 if the production is increased from 40 units to 40.5 units.

Would you like any further clarification or additional details?

Here are 5 related questions:

1. What does the second derivative of the cost function represent in this context?
2. How does the rate of cost increase change as production levels increase?
3. What would be the exact cost of producing 40.5 units using the original cost function?
4. How can you interpret the meaning of the constant 10 in the cost function?
5. How would the total cost change if production increased from 40 to 41 units instead?

Tip: Derivatives provide an efficient way to estimate small changes in functions, especially when calculating exact values may be more complex.