Problem 13 - Applied Calculus

Question

Figure \(4.65\) shows the average cost, \(a(q)=b+m q\). (a) Show that \(C^{\prime}(q)=b+2 m q\). (b) Graph the marginal cost \(C^{\prime}(q)\).

Step-by-Step Solution

Answer

(a) Confirmed, \(C^{\prime}(q) = b + 2mq\). (b) It graphs as a straight line with slope 2m and y-intercept b.

Step 1: Finding the Total Cost Function

Given that the average cost is represented by \(a(q) = b + mq\), to find the total cost function \(C(q)\), we multiply the average cost by the quantity \(q\):\[ C(q) = q \cdot a(q) = q(b + mq) = bq + mq^2.\] This is the expression for the total cost function.

Step 2: Calculating Marginal Cost

To find the marginal cost, which is the derivative of the total cost function \(C(q)\), we differentiate \(C(q) = bq + mq^2\) with respect to \(q\). Using the power rule, the derivative is:\[ C^{\prime}(q) = \frac{d}{dq}(bq + mq^2) = b + 2mq.\] Thus, we confirm that \(C^{\prime}(q) = b + 2mq\).

Step 3: Graphing the Marginal Cost

To graph the marginal cost function \(C^{\prime}(q) = b + 2mq\), note that it is a linear equation of the form \(y = mx + c\) with slope \(2m\) and y-intercept \(b\). Plot the y-intercept \(b\) on the vertical axis and use the slope \(2m\) to determine the direction and steepness of the line, increasing by \(2m\) units vertically for every unit increase horizontally.

Key Concepts

Average Cost

The average cost, often denoted as \( a(q) \), is the total cost of producing a certain quantity \( q \) of goods, divided by \( q \). This provides an expression that represents the cost incurred, on average, for producing one unit of the good. In our exercise, the average cost function is given as \( a(q) = b + mq \). Here:

• \( b \) represents the fixed cost per unit, which does not change with the level of production.
• \( m \) is the variable cost per unit, meaning it varies depending on how many units are produced. The term \( mq \) indicates how total variable costs increase with the quantity \( q \).

Using this formula, you can calculate the average cost at any desired quantity, which helps businesses understand their cost structure as production varies.

Total Cost Function

The total cost function, \( C(q) \), is an equation that represents the complete cost of producing \( q \) units of a good, encompassing both fixed and variable costs. In the original problem, we determine the total cost function by multiplying the average cost \( a(q) \) by the quantity \( q \): \[ C(q) = q imes a(q) = q(b + mq) = bq + mq^2. \]Key components include:

• \( bq \), which reflects total fixed costs across all units, since \( b \) is constant regardless of \( q \).
• \( mq^2 \), representing the total variable costs, showing how these costs exponentially increase as more units are produced.

Understanding the total cost function is crucial for businesses as it helps in budgeting and forecasting their financial needs.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. When we differentiate a function, we're finding its rate of change or slope at any point. In the context of cost, the derivative gives us the marginal cost, which indicates the additional cost of producing one more unit.For the given total cost function \( C(q) = bq + mq^2 \), the derivative \( C^{\prime}(q) \) is calculated as follows:\[ C^{\prime}(q) = \frac{d}{dq}(bq + mq^2) = b + 2mq. \]This derivative shows that the marginal cost is determined by the constant fixed cost \( b \) and the component \( 2mq \), which links the marginal cost to the quantity produced through the variable cost \( m \). Calculating the derivative is crucial as it guides businesses in marginal decision-making.

Graphing Linear Equations

Graphing linear equations involves plotting equations like \( y = mx + c \) on a graph, where \( m \) is the slope and \( c \) is the y-intercept. In this case, to graph the marginal cost function \( C^{\prime}(q) = b + 2mq \), we follow the steps for graphing a linear equation:

• Start by identifying the y-intercept, \( b \). Plot this point on the vertical axis at \( q = 0 \).
• The slope \( 2m \) determines the line's steepness. For every unit increase horizontally (a change in \( q \)), the line rises by \( 2m \) units.

With these points, you can draw the straight line that represents \( C^{\prime}(q) \). Graphing is an essential skill in visualizing relationships and understanding how changes in one variable correspond to changes in another, such as cost functions in economics.