Question

Marginal Analysis, use differentials to approximate the change in cost, revenue, or profit corresponding to an increase in sales of one unit. For instance, in Exercise 29, approximate the change in cost as \(x\) increases from 12 units to 13 units. Then use a graphing utility to graph the function, and use the trace feature to verify your result. \(C=0.025 x^{2}+8 x+5 \quad x=10\)

Short answer

The approximate increase in cost when \(x\) increases from 12 to 13 units is $8.6.

Step-by-step solution

Compute the derivative of the cost function

We first need to find the derivative of the function (\(C=0.025 x^{2}+8 x+5\)). The derivative gives us the rate of change of the cost function with respect to \(x\), which we'll call \(dC/dx\). Using the power rule for differentiation, we get \(dC/dx= 0.05x + 8\).

Calculate the marginal cost at \(x = 12\)

Substitute \(x = 12\) into the first derivative equation found in step 1 to find the rate of change of the cost when 12 units are being produced. This will yield \(dC/dx = 0.05(12) + 8 = 8.6\). This means for each unit increase in \(x\) from 12, the cost increases by approximately $8.6.

Use the graphing utility

Plot the original cost function \(C=0.025 x^{2}+8 x+5\) using a graphing utility. Select the trace feature and enter \(x = 12\) and \(x = 13\). The differences in cost at these two points should roughly equal to the result obtained in step 2.

Key concepts

Differential Approximation

Differential approximation is a valuable tool in calculus, especially when we wish to estimate small changes in a function's value due to slight shifts in its variable. In the context of economics and business, it's used to predict how a minor increment or decrement in the quantity of goods sold can influence the associated costs, revenue, or profit.

In the example provided, when the sales of an item increase from 12 to 13 units, it’s not efficient to recalculate the whole cost function for 13 units from scratch. Instead, we apply the concept of differential approximation. Here's how it works: we take the derivative of the cost function, which represents the rate of change of cost with respect to the number of items. By evaluating the derivative at 12 units, we obtain the approximate rate of change of cost per unit around that point.

The formula for differential approximation in this case is given by \(dC = (dC/dx) dx\), where \(dC/dx\) is the derivative of the cost function, and \(dx\) is the change in units sold. For an increase of one unit (from 12 to 13), \(dx = 1\), and thus, the approximate increase in cost is \(dC\). This approximation becomes increasingly accurate for smaller changes in \(x\), highlighting the practicality of differential approximation in marginal analysis.

Cost Function Derivation

Understanding how the cost function derivation works is fundamental in marginal analysis. The cost function represents the total cost as a function of the quantity produced. To manage production costs effectively, businesses need to understand how the cost changes as they adjust output levels.

The step-by-step solution for our exercise begins with finding the derivative of the cost function \(C=0.025x^2+8x+5\). Applying calculus principles, such as the power rule, we derive the marginal cost as the function \(dC/dx=0.05x+8\). This derived function symbolizes the increase in total cost for every additional unit produced after a certain production level (here, at 12 units).

For practical purposes, the marginal cost is paramount for decision-making in business operations. Knowing the derivative of the cost function allows us to calculate the marginal cost at any level of production, thus informing how much the next unit will add to the total cost. This is essential for pricing strategies, production planning, and profit maximization efforts.

Graphing Utility

Graphing utilities are digital tools that aid in visualizing mathematical functions, which is especially handy when studying complex relationships like those within a cost function. By plotting the function, we can visually assess its behavior and validate calculations made through analytical methods, such as derivatives for marginal analysis.

In our example, by using a graphing utility, we input the cost function to generate a curve representing total cost versus the number of units produced. The 'trace' feature can be utilized to pinpoint the precise cost values at given quantities, such as 12 and 13 units, offering a graphical verification of the differential approximation previously calculated.

A graphing utility does not only validate our previous findings but also enhances comprehension by offering a visual context. This can be essential for students who often benefit from seeing abstract concepts displayed in a more tangible form. Graphically inspecting the slope of the tangents to the curve at particular points (such as at 12 units) provides an intuitive grasp of marginal cost, making the connection between calculus and real-world applications clearer.