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The concept of marginal cost is vital in understanding how businesses make production decisions. Marginal cost, represented by \( C'(q) \), indicates the cost of producing one additional unit of a good. It’s a derivative, so it tells us the rate of change of the total cost with respect to the number of goods produced.
For example, in part (a) of the exercise, when it’s stated that \( C'(50) = 75 \), this means that producing the 51st item will cost an additional $75. For businesses, understanding this helps in determining whether it’s profitable to increase production.
Producing additional units is only beneficial if the revenue from those units exceeds their marginal costs. This leads us directly to understanding marginal revenue.

Marginal revenue, symbolized by \( R'(q) \), measures the additional revenue a firm earns from selling one more unit of a product. Much like marginal cost, it’s a derivative indicating the rate of change of total revenue with respect to quantity sold.
In part (a) of the exercise, \( R'(50) = 84 \) means selling the 51st item yields an additional \(84 in revenue. Comparing marginal revenue to marginal cost allows businesses to ascertain whether producing more items is profitable.

• If marginal revenue is greater than marginal cost, producing more is profitable.
• If marginal revenue is less than marginal cost, producing more incurs losses.

In essence, a positive difference between marginal revenue and marginal cost increases profit, as seen with the 51st item, yielding \)9 in profit.

Profit maximization occurs where the profit function \( \pi(q) \) is at its peak. This happens when the derivative \( \pi'(q) \) equals zero, since no additional profit can be made by altering production levels.
In scenario (c), the profit function reaches its maximum at \( q = 78 \). At this point, marginal cost \( C'(78) \) equals marginal revenue \( R'(78) \). This equilibrium indicates optimal production.

• If a company produces more, costs increase over revenues.
• If producing less, potential revenue is lost.

Therefore, the key to profit maximization is finding that precise balance where producing one more unit results in no extra profit—indicative of optimal efficiency.

Derivatives play an essential role in economic analysis, especially in understanding changes in cost and revenue structures. By using derivatives like \( \pi'(q) = R'(q) - C'(q) \), firms calculate the profit from producing an additional unit.
The power of derivatives arises from their ability to model real-world changes succinctly and efficiently. They provide insights into optimal production levels, pricing strategies, and resource allocation.

• Economic derivatives help in forecasting short-term and long-term impacts of production changes.
• They highlight critical points like maximum profit and break-even levels.

In summary, derivatives are indispensable tools in economics for decision-making, helping bridge theoretical models and practical business dynamics.