Estimating Rate of Change for a Cost Function

A company's total production cost for a certain good is a function of the quantity produced, as detailed in the table below.

Using the principle that a rate of change can be approximated by the change resulting from a one-unit increase, what is the approximate rate of change of the cost when production is increased from 11 to 12 units?

The Airline's Pricing Decision

For a company's profit function, $P(q)$, where $q$ is the quantity of goods sold, the calculation $P(101) - P(100)$ provides the exact rate of change of profit at the point where 100 units are sold.

A firm's total revenue from selling a product is described by the function $R(q) = 150q - q^2$, where $q$ is the number of units sold. Using the change from a one-unit increase to estimate the instantaneous rate of change, the approximate rate of change in revenue when moving from 30 to 31 units sold is ____.

A company analyzes its cost, revenue, and profit functions, denoted by C(q), R(q), and P(q) respectively, where q represents the quantity of units produced and sold. Match each economic concept on the left with its correct mathematical approximation on the right, based on the principle of using a one-unit change to estimate an instantaneous rate of change.

A farming community works on a fixed plot of land. In the first year, 50 workers produce a total of 10,000 units of food. In the second year, the workforce grows to 60 workers, and they produce a total of 11,400 units of food. Assuming income is directly tied to the food produced, what change occurred in the average output per worker and what was the most likely effect on the community's average living standard from the first year to the second?

An analyst wants to estimate the instantaneous rate of change for a function at a specific point. The method available is to calculate the change in the function's value resulting from a one-unit increase in the input variable. Consider two different contexts:

1. Context A: Estimating the change in a nation's Gross Domestic Product (GDP) when the labor force increases by one person.
2. Context B: Estimating the change in a small bakery's daily revenue when they sell one additional loaf of bread.

In which context is this one-unit approximation method likely to be a more accurate estimate of the true instantaneous rate of change, and why?

Evaluating a Manager's Production Decision

A manager at a manufacturing plant observes that the total cost to produce 100 widgets is $5,000, and the total cost to produce 101 widgets is $5,045. Based on this, they conclude that the cost to produce one additional widget is always $45, regardless of the current production level. Which of the following statements best evaluates the manager's conclusion?

Programmatic Numerical Approximation of a Derivative