Problem 44 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(x\) and \(f(x)\) represent the given quantities. Fix \(x=a\) and let \(h\) be a small positive number. Give an interpretation of the quantities $$ \frac{f(a+h)-f(a)}{h} \text { and } \lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} $$ \(x\) denotes the level of production of a certain commodity, and \(f(x)\) denotes the total cost incurred in producing \(x\) units of the commodity.

Step-by-Step Solution

Answer

The expression \(\frac{f(a+h)-f(a)}{h}\) represents the average rate of change in the total cost when the production level increases by a small amount \(h\), from \(x=a\) to \(x=a+h\). On the other hand, the limit \(\lim_{h \rightarrow 0} \frac{f(a+h)-f(a)}{h}\) represents the instantaneous rate of change in the cost at the production level \(x=a\), which indicates the marginal cost of producing an additional unit at that level of production.

Step 1: Interpret the difference quotient

\(\frac{f(a+h)-f(a)}{h}\) represents the \(\textbf{average rate of change}\) of total cost when production increases from \(a\) units to \(a+h\) units. This is the average marginal cost over that interval.

Step 2: Interpret the limit

\(\lim_{h \to 0} \frac{f(a+h)-f(a)}{h} = f'(a)\) represents the \(\textbf{instantaneous rate of change}\) of total cost at production level \(a\). This is the marginal cost — the cost of producing one additional unit when already producing \(a\) units.

Key Concepts

Average Rate of Change

Understanding the average rate of change is fundamental when dealing with varying quantities and their relationships. Imagine you are monitoring the total cost of production within a factory over time, and you'd like to measure how this cost increases as the quantity of products made ramps up.

Mathematically, the average rate of change is the difference in total costs divided by the difference in production levels. In simpler terms, it tells you, on average, how much more it will cost to produce one more unit over a given interval. It's like looking at the slope of the line that connects two points on a cost graph. If production increases from 100 to 150 units, you would calculate the average rate of change by seeing how much the total cost increased in this jump and divide it by the increase in units, namely 50 in this case.

Interpreting The Average Rate of Change

Think of it as driving a car over a mile. You start at one point and after 20 minutes you're at another point one mile away. The average rate of change here is your speed, which would be 3 miles per hour. Similarly, in production, if you know the average rate of change in cost, you can estimate the increase in expense for additional units over the interval measured.

Derivative of a Function

While the average rate of change gives us a snapshot over an interval, the derivative tells us the rate at a specific, exact point. It's like a microscope zooming in on the graph until that average slope becomes the slope of a tangent touching the curve at just one point.

A derivative is a concept from calculus that examines how a function's output value changes as its input changes by an infinitesimally small amount. In our production cost example, it would mean examining how the total cost reacts when we nudge production up by just a tiny bit, essentially producing fractions of a unit. The resulting value tells us precisely how sensitive the total cost is to even the smallest changes in production at any given level of output.

Calculating The Derivative

Mathematically, the derivative is the limit of the average rate of change as the interval approaches zero. It's the engine behind calculating things like velocity, where you're not interested in average speed over a few seconds but the exact speed at a split-second moment.

Marginal Cost

Marginal cost is a term often heard in the realms of economics and business, and it plays a crucial role in decision-making processes. It refers to the cost of producing one additional unit of a good. In essence, marginal cost is the derivative of the total cost function with respect to the quantity of goods produced.

For a company, understanding marginal cost is essential because it helps determine at what point producing more goods becomes less profitable. For instance, initially, as production increases, the cost per unit often decreases due to the economies of scale. However, after a certain point, each additional unit starts to cost more to make due to factors like overuse of machinery or labor overtime.

Why Marginal Cost Matters

This is the point where the curve of total cost starts to steepen significantly, indicating that each additional unit is adding more to the cost than previous units did. Businesses aim to produce up to the level where marginal cost equals marginal revenue, which is the most cost-effective point of operation.

Difference Quotient

The difference quotient is a formula that gives an average rate of change between two points on a function. It's like a mathematical recipe for exploring the behavior of a curve between two distinct spots. If you were to measure the steepness of a hill at various sections, the difference quotient would be your method of choice.

The difference quotient is expressed as \(\frac{f(a+h)-f(a)}{h}\) where \(f(a)\) is the function value at the starting point, \(f(a+h)\) is the function value after a certain increase, \(h\), and the whole expression is divided by that same \(h\). In our production cost example, think of it as a way to average out the rise or fall in costs over the discrete interval defined by \(h\).

Applying The Difference Quotient

It allows for a firm understanding of how a function behaves locally but with a broader perspective, keeping in view the entire interval, rather than zooming in on an infinitely small point. However, as \(h\) gets smaller and closer to zero, the difference quotient starts to morph into the derivative, offering a window into the instantaneous rate of change.