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


a. Find the additional cost incurred in dollars when production is increased from 100 units to 150 units.


C′(x) = 300+10x−0.01x²

Verified step by step guidance

1. Understand that the marginal cost function \(C\prime(x)\) represents the rate of change of the total cost with respect to the number of units produced, \(x\). To find the additional cost incurred when production increases from 100 to 150 units, we need to find the total change in cost over this interval.
2. Set up the integral of the marginal cost function \(C\prime(x) = 300 + 10x - 0.01x^{2}\) from \(x = 100\) to \(x = 150\). This integral will give the total additional cost incurred when production increases from 100 to 150 units.
3. Write the integral expression for the additional cost as: \(\int_{100}^{150} (300 + 10x - 0.01x^{2}) \, dx\).
4. Integrate the function term-by-term: - The integral of \$300\( with respect to \)x\( is \)300x\(. - The integral of \)10x\( with respect to \)x\( is \)5x^{2}\(. - The integral of \)-0.01x^{2}\( with respect to \)x\( is \)-0.01 \times \frac{x^{3}}{3} = -\frac{0.01}{3} x^{3}$.
5. Evaluate the definite integral by substituting the upper limit \(x=150\) and the lower limit \(x=100\) into the antiderivative, then subtract the two results to find the total additional cost.

Key Concepts

Marginal Cost Function

The marginal cost function, denoted as C′(x), represents the rate of change of the total cost with respect to the number of units produced. It gives the approximate additional cost of producing one more unit at production level x. Understanding this function is essential to estimate incremental costs over production intervals.

Definite Integral for Total Change

To find the total additional cost when production increases from one quantity to another, integrate the marginal cost function over that interval. The definite integral of C′(x) from x = a to x = b gives the exact total increase in cost between producing a and b units.

Application of Integration in Economics

Integration is used in economics to accumulate marginal values into total quantities. Here, integrating the marginal cost function helps convert a rate of change into a total cost change, linking calculus concepts directly to practical cost analysis in production.