55–58. Marginal cost Consider the following marginal cost functions.


b. Find the additional cost incurred in dollars when production is increased from 500 units to 550 units.


C′(x)=200−0.05x

Verified step by step guidance

1. Identify the marginal cost function given: \(C\' (x) = 200 - 0.05x\), which represents the rate of change of the cost with respect to the number of units produced.
2. Understand that the additional cost incurred when production increases from 500 to 550 units can be found by integrating the marginal cost function over the interval from \(x = 500\) to \(x = 550\).
3. Set up the definite integral to find the additional cost: \(\int_{500}^{550} (200 - 0.05x) \, dx\).
4. Integrate the function: find the antiderivative of \$200 - 0.05x\(, which is \)200x - 0.025x^2$.
5. Evaluate the antiderivative at the upper and lower limits and subtract: calculate \(\left[200x - 0.025x^2\right]_{500}^{550}\) to find the additional cost.

Key Concepts

Marginal Cost

Marginal cost represents the rate of change of the total cost with respect to the quantity produced. It is given by the derivative of the cost function, C'(x), and indicates the additional cost of producing one more unit at a certain production level.

Definite Integral for Accumulated Change

To find the total additional cost over an interval, integrate the marginal cost function over that range. The definite integral of C'(x) from x = a to x = b gives the total increase in cost when production increases from a to b units.

Interpreting and Applying the Marginal Cost Function

Understanding how to apply the marginal cost function involves evaluating or integrating it over a specific interval to find actual cost changes. This requires recognizing that marginal cost varies with production level and using calculus tools to compute total cost increments.