Advertising Costs The cost, in thousands of dollars, of...

Advertising Costs The cost, in thousands of dollars, of airing $x$ television commercials during a Super Bowl game is given by $^{14}$ $$ C(x)=150+2,250 x-0.02 x^{2} $$ a. Find the marginal cost function and use it to estimate how fast the cost is increasing when $x=4$. Compare this with the exact cost of airing the fifth commercial. HINT [See Example 1.] b. Find the average cost function $\bar{C}$, and evaluate $\bar{C}(4)$. What does the answer tell you? HINT [See Example 4.]

Advertising Costs The cost, in thousands of dollars, of airing $x$ television commercials during a Super Bowl game is given by $^{14}$
$$
C(x)=150+2,250 x-0.02 x^{2}
$$
a. Find the marginal cost function and use it to estimate how fast the cost is increasing when $x=4$. Compare this with the exact cost of airing the fifth commercial. HINT [See Example 1.]
b. Find the average cost function $\bar{C}$, and evaluate $\bar{C}(4)$. What does the answer tell you? HINT [See Example 4.]

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Cost Function

A cost function represents the total cost incurred as a function of the number of units produced or services rendered. It typically comprises a fixed cost component, which does not depend on the number of units, and a variable cost component, which increases with each additional unit.

Marginal Cost

Marginal cost is the derivative of the cost function with respect to the number of units, representing the instantaneous rate of change of the cost. It approximates the additional cost incurred by producing one more unit at a specific production level.

Derivative as Instantaneous Rate of Change

The derivative in this context is used to determine the marginal cost at a particular number of units. Evaluating the derivative at a certain point provides an estimate of how fast the total cost is increasing, serving as an important tool when changes are small and continuous.

Average Cost Function

The average cost function is obtained by dividing the total cost by the number of units produced. It gives the cost per unit, offering insight into the efficiency of production and how costs are spread over each unit, which is crucial for assessing economies of scale.

Transcript

Please give Ace some feedback