Problem 15 - Applied Calculus

Question

Table 2.13 shows the cost. \(C(q),\) and revenue, \(R(q),\) in terms of quantity \(q .\) Estimate the marginal cost. \(C^{\prime}(q)\) and marginal revenue, \(R^{\prime}(q),\) for \(q\) between 0 and 7. $$\begin{array}{r|r|r|r|r|r|r|r|r}\hline q & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\\hline C(q) & 9 & 10 & 12 & 15 & 19 & 24 & 30 & 37 \\\\\hline R(q) & 0 & 5 & 10 & 15 & 20 & 25 & 30 & 35 \\\\\hline \end{array}$$

Step-by-Step Solution

Answer

Marginal cost increases as quantity increases, while marginal revenue remains constant.

Step 1: Understand Marginal Concepts

Marginal cost, \(C'(q)\), is the change in total cost over the change in quantity. Similarly, marginal revenue, \(R'(q)\), is the change in total revenue over the change in quantity. We calculate marginal values using approximate numerical derivatives between each consecutive quantity value.

Step 2: Calculate Marginal Cost

The marginal cost \(C'(q)\) for each interval between \(q\) values is calculated as:\[ C'(q) = \frac{C(q+1) - C(q)}{(q+1) - q} \]For example, \(C'(0)\) is \( \frac{10 - 9}{1 - 0} = 1 \).1. \(C'(0) = \frac{10-9}{1} = 1\)2. \(C'(1) = \frac{12-10}{1} = 2\)3. \(C'(2) = \frac{15-12}{1} = 3\)4. \(C'(3) = \frac{19-15}{1} = 4\)5. \(C'(4) = \frac{24-19}{1} = 5\)6. \(C'(5) = \frac{30-24}{1} = 6\)7. \(C'(6) = \frac{37-30}{1} = 7\)

Step 3: Calculate Marginal Revenue

Similarly, for the marginal revenue \(R'(q)\), the calculation is:\[ R'(q) = \frac{R(q+1) - R(q)}{(q+1) - q} \]For example, \(R'(0)\) is \( \frac{5 - 0}{1 - 0} = 5 \).1. \(R'(0) = \frac{5-0}{1} = 5\)2. \(R'(1) = \frac{10-5}{1} = 5\)3. \(R'(2) = \frac{15-10}{1} = 5\)4. \(R'(3) = \frac{20-15}{1} = 5\)5. \(R'(4) = \frac{25-20}{1} = 5\)6. \(R'(5) = \frac{30-25}{1} = 5\)7. \(R'(6) = \frac{35-30}{1} = 5\)

Step 4: Interpret the Results

The marginal cost \(C'(q)\) increases with each step, indicating increasing costs per unit as more units are produced. The marginal revenue \(R'(q)\), however, remains constant, suggesting that the revenue gained per additional unit sold is steady.

Key Concepts

Marginal Revenue

When discussing marginal revenue, think about the extra money generated from selling one more item. It's a helpful concept because it shows us how each additional item sold contributes to overall earnings. Marginal revenue is calculated as the change in revenue when one more unit is sold, using the formula:

• Marginal Revenue formula: \[ R'(q) = \frac{R(q+1) - R(q)}{(q+1) - q} \]

In the given table, you see this extra earning is consistent at 5 units of revenue per additional item sold across different selling levels.
This means for each unit increase from 0 to 7, the revenue climbs by a flat rate of 5, which is crucial for knowing consistent earnings against growing quantities.

Numerical Derivatives

Numerical derivatives help us find the rate of change between discrete data points when it's hard to apply standard calculus derivatives. Instead of an exact derivative, we use numerical methods to approximate the change rates. This method works well with tabular data where measurements are taken at different intervals.
To calculate the numerical derivative, we use the formula:

• Numerical Derivative formula: \[ f'(q) = \frac{f(q+1) - f(q)}{(q+1) - q} \]

For instance, finding the numerical derivative for cost tells us how much each additional unit costs in terms of how costs change between these intervals.
It's useful in economics and production, allowing us to estimate cost functions and revenue functions when given in charts or tables.

Total Cost

Total cost refers to all expenses incurred in production. It includes both fixed and variable costs. Fixed costs don't change with production levels, while variable costs do. To understand production costs effectively, analyze how total cost increases with added units, since each new unit represents both fixed and fluctuating costs.
This concept is particularly crucial in estimating economies of scale—determining if costs per unit decrease when volume increases.
In our example, by examining total costs as quantity increases from 0 to 7, you can see how each new item produced affects overall cost.

• Total Cost formula break down:
  • Total Cost = Fixed Costs + (Variable Cost per Unit * Units Produced)

Understanding total cost helps businesses price their products effectively, manage their budget, and maximize their profit by understanding how costs change with production levels.