The cost (in dollars) of producing q items is given by...

The cost (in dollars) of producing $q$ items is given by $C(q)=0.08 q^{3}+75 q+1000$
(a) Find the marginal cost function.
(b) Find $C(50)$ and $C^{\prime}(50) .$ Give units with your answers and explain what each is telling you about costs of production.

Key Concepts

Cost Function

A cost function represents the total cost associated with producing a given number of items. It is a mathematical model that includes all production costs, often written as a function of the quantity produced. This concept is widely used in economics and business to analyze and predict how total production costs change with production levels.

Marginal Cost

The marginal cost is the additional cost incurred to produce one extra unit of a product. It is found by taking the derivative of the cost function with respect to quantity and gives insight into cost behavior as production scales. Marginal cost helps businesses make decisions about increasing or decreasing production based on the incremental cost.

Differentiation in Context of Economics

Differentiation is a fundamental calculus tool used to analyze how a function changes. In economics, differentiating the cost function yields the marginal cost function, which indicates the rate at which total costs change with respect to the number of items produced. This technique is essential for optimizing production and understanding cost dynamics.

Evaluation of Functions at Specific Points

Evaluating functions at specific points involves substituting a particular production quantity into the cost and marginal cost functions. This practice provides concrete numerical values that inform decision-making processes by showing the total cost at a given production level and the cost of producing an additional unit at that level.

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