Problem 48 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

INVENTORY ANALYSIS A businessman maintains inventory over a particular 30 -day month as follows: \(\begin{array}{ll}\text { days } 1-9 & 30 \text { units } \\ \text { days } 10-15 & 17 \text { units } \\ \text { days } 16-23 & 12 \text { units } \\\ \text { days } 24-30 & \text { steadily decreasing from } 12 \text { units } \\\ & \text { to } 0 \text { units }\end{array}\) Sketch the graph of the inventory as a function of time \(t\) (days). At what times is the graph discontinuous?

Step-by-Step Solution

Answer

The graph is discontinuous at Days 10, 16, and 24.

Step 1 - Understand the Inventory Levels

Observe the inventory levels for the different time periods given in the problem: Days 1-9: 30 units, Days 10-15: 17 units, Days 16-23: 12 units, Days 24-30: steadily decreasing from 12 units to 0 units.

Step 2 - Plot the Inventory Levels

Start by plotting horizontal lines for the constant inventory levels for Days 1-9, Days 10-15, and Days 16-23. The inventory level for each interval remains constant.

Step 3 - Include the Decreasing Inventory

From Days 24-30, the inventory level decreases steadily from 12 units to 0 units. Draw a line with a negative slope to represent this decrease.

Step 4 - Identify Discontinuities

The graph will have discontinuities where there are sudden jumps in inventory levels: at Days 10, 16, and 24.

Step 5 - Sketch the Complete Graph

Combine all parts to produce the final sketch: horizontal lines for each time period with inventory jumps at Days 10, 16, and 24, and a sloping line for the final period.

Key Concepts

Discontinuous Functions

A function is discontinuous when there are sudden jumps or breaks in its graph. In mathematical terms, these are points where the function is not continuous. In the context of inventory analysis, we see discontinuities where inventory levels change abruptly. For instance, in our example, the graph is discontinuous at Days 10, 16, and 24. This is because the inventory jumps from one level to another instantly.

• Days 10: Inventory drops from 30 units to 17 units.
• Days 16: Inventory drops further from 17 units to 12 units.
• Days 24: Inventory starts to decrease steadily to 0 units by Day 30.
These points of discontinuity are crucial in understanding how inventory behaves over time and can help in better managing inventory levels by highlighting times that require attention.

Graphing Step Functions

A step function is a type of piecewise function that remains constant within intervals but jumps to different values at specific points. In inventory analysis exercises, step functions are commonly used to graph inventory levels that remain constant over certain periods and then change suddenly.

To graph a step function:

• Plot horizontal lines for time intervals where the inventory level stays the same.
• Identify and mark the points of discontinuity where the inventory level changes.
• Connect these points using lines that 'step' up or down at the changes, without slopes between intervals.
In our example:

• From Days 1 to 9, draw a horizontal line at 30 units.
• From Days 10 to 15, draw a horizontal line at 17 units.
• From Days 16 to 23, draw a horizontal line at 12 units.
By following these steps, you can accurately represent the inventory levels as a step function.

Calculus Applications in Business

Calculus is essential in business, especially for understanding and managing various dynamic processes. In inventory analysis, calculus helps in defining and analyzing rates of change over time. For example, the rate at which inventory levels decrease or increase can be crucial for forecasting and planning.

There are a few key concepts from calculus that are applicable:

• **Derivative**: Represents the rate of change. It helps in understanding how quickly inventory levels change at a point in time.
• **Integral**: Helps in determining the total quantity, such as total demand or supply over a period. In the inventory context, it can help find total inventory usage.
In our example, during Days 24 to 30, the inventory steadily decreases from 12 to 0. Here, the derivative shows us the rate of decrease. If inventory decreases linearly, the derivative would be a constant negative value. Using these calculus tools aids businesses in optimizing inventory levels, predicting shortages, and planning future inventory needs based on past trends.