The daily cost (in dollars) of producing x units of a...

The daily cost (in dollars) of producing $x$ units of a certain product is given by the function $$\alpha(x)=225+36.5 x-9 x^{2}+.01 x^{3}.$$ (a) Graph $\alpha(x)$ in the window $[0,70]$ by $[-400,2000].$ (b) What is the cost of producing 50 units of goods? (c) Consider the situation as in part (b). What is the additional cost of producing one more unit of goods? (d) At what production level will the daily cost be $\$ 510 ?$

The daily cost (in dollars) of producing $x$ units of a certain product is given by the function
$$\alpha(x)=225+36.5 x-9 x^{2}+.01 x^{3}.$$
(a) Graph $\alpha(x)$ in the window $[0,70]$ by $[-400,2000].$
(b) What is the cost of producing 50 units of goods?
(c) Consider the situation as in part (b). What is the additional cost of producing one more unit of goods?
(d) At what production level will the daily cost be $\$ 510 ?$

Key Concepts

Marginal Cost

Marginal cost is the additional cost incurred from producing one more unit of a product. It is often determined by finding the derivative of the cost function, which provides the instantaneous rate of change. This concept is essential in decision making as it helps determine the efficiency and profitability of increasing production levels.

Function Evaluation

Function evaluation is the process of substituting a specific value into a function to determine the corresponding output. It is a fundamental concept that helps in understanding how the function behaves at particular points, such as finding the cost of producing a specific number of units.

Solving Polynomial Equations

Solving polynomial equations involves finding the values of the variable that satisfy the equation. This skill is important when determining specific production levels that correspond to a given cost, or when analyzing the behavior of cost functions, as it allows one to solve for unknowns in the context of total cost and production quantity.

Graphing Functions

Graphing functions involves plotting the relationship between input and output values over specified domains and ranges. This provides a visual understanding of the behavior of the function, including trends, turning points, and overall shape. It is particularly useful for analyzing how cost or any other quantity changes with varying production levels or other inputs.

Cost Function

A cost function represents the total cost associated with producing a certain number of units. It typically includes fixed costs (which do not change with production volume) and variable costs (which increase with each additional unit produced). This function is crucial in economics and business as it helps in planning, budgeting, and determining profitability.

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