The cost of producing a quantity, q, of a product is given...

The cost of producing a quantity, $q$, of a product is given by
$$C(q)=1000+30 e^{0.05 q} \quad \text { dollars. }$$
Find the cost and the marginal cost when $q=50 .$ Interpret these answers in economic terms.

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Key Concepts

Cost Function

A cost function represents the total costs incurred by a firm in producing a given quantity of goods. It typically includes fixed costs that do not change with output and variable costs that vary with the level of production. Understanding the cost function helps in analyzing how total expenses evolve as the production quantity changes.

Marginal Cost

Marginal cost is the rate of change of the total cost with respect to the quantity produced; it is found by taking the derivative of the cost function. This concept is essential in economics as it indicates the additional cost incurred by producing one more unit, guiding decisions about output levels and efficient resource allocation.

Differentiation in Economic Analysis

Differentiation is the mathematical process used to find the marginal cost from the cost function. In economic analysis, it provides insights into how small changes in production output affect total costs, thereby allowing businesses to optimize pricing, production, and profit strategies.

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