Problem 61 - Calculus: Early Transcendental Functions

Question

Involve the just-in-time inventory discussed in the chapter introduction. For a business using just-in-time inventory, a delivery of \(Q\) items arrives just as the last item is shipped out. Suppose that items are shipped out at the constant rate of \(r\) items per day. If a delivery arrives at time \(0,\) show that \(f(t)=Q-r t\) gives the number of items in inventory for \(0 \leq t \leq \frac{Q}{r} .\) Find the average value of \(f\) on the interval \(\left[0, \frac{Q}{r}\right]\).

Step-by-Step Solution

Answer

The average value of \(f(t)\) on the interval [0, \(Q/r\)] is \(\frac{Q}{2}\).

Step 1: Understand the Function \(f(t)\)

In the just-in-time inventory model specified, \(f(t) = Q - rt\) represents the number of items in inventory at any time \(t\) in the interval \([0, \frac{Q}{r}]\). Here, \(Q\) is the initial number of items, and \(r\) is the rate at which items are shipped out.

Step 2: Calculate the Integral of \(f(t)\)

First, calculate the integral of \(f(t)\) over the interval \([0, \frac{Q}{r}]\): \[\int_{0}^{\frac{Q}{r}} (Q - rt) dt = [Qt - \frac{1}{2}r t^2]_ {0}^{\frac{Q}{r}}\]. After substitution of the upper and lower limit of the interval, the result is \[Q \cdot \frac{Q}{r} - \frac{1}{2} \cdot r \left(\frac{Q}{r}\right)^2 = \frac{Q^2}{r} - \frac{1}{2} \cdot Q^2 = Q^2 \left(\frac{1}{r} - \frac{1}{2}\right)\].

Step 3: Calculate the Average of \(f(t)\)

Finally, apply the formula for the average value of a function over an interval: \[f_{avg} = \frac{1}{b-a}\int_{a}^{b} f(t) dt\]. Substitute the previously calculated integral and \(a = 0\), \(b = \frac{Q}{r}\): \[f_{avg} = \frac{r}{Q} \cdot Q^2 \left(\frac{1}{r} - \frac{1}{2}\right) = Q\left(\frac{1}{r} - \frac{1}{2}\right) = \frac{Q}{2}\]\(r\) is the constant shipment rate.

Key Concepts

Average Value of Functions

When discussing the average value of a function, we essentially want to find what a "typical" value of the function is over a specific interval. Suppose we have a function, like our inventory function \(f(t) = Q - rt\), which describes how many items remain over time.

• The average value of this function will tell us the typical number of items in the inventory during the entire time period.
• In mathematical terms, the average value \(f_{avg}\) of a function \(f(t)\) over an interval \([a, b]\) is given by the formula:

\[f_{avg} = \frac{1}{b-a} \int_{a}^{b} f(t) \, dt\]In our case, \(a = 0\) and \(b = \frac{Q}{r}\), which means the interval covers the time from when the delivery arrives until the last item is shipped out.Finding the average value is crucial in managing inventory efficiently, as it gives insight into how stocked the inventory typically is over time. This way, businesses can adjust supply or mode of delivery to better serve their needs.

Integration

Integration is the process of finding the integral of a function, and it is a huge part of calculus, often described as the inverse of differentiation. In simpler words, while differentiation breaks things down into rates of change, integration builds them up to find areas under curves or accumulated totals.

• In our problem, we need to integrate the function \(f(t) = Q - rt\) over the interval \([0, \frac{Q}{r}]\).
• The integral of a linear function like this one involves calculating the area under the line across the interval of interest.

We compute the integral as follows:\[\int_{0}^{\frac{Q}{r}} (Q - rt) \, dt = \left[Qt - \frac{1}{2}rt^2\right]_ {0}^{\frac{Q}{r}}\]This integration finds the total stock measured over the time until all items are shipped. It tells us how many items on average were present, providing a cumulative insight into inventory at any given time.Integration is the backbone of understanding real-world dynamic changes, such as managing and keeping track of inventory effectively.

Calculus

Calculus is a branch of mathematics that deals with change and motion, through the study of derivatives and integrals. In the realm of inventory management, as seen in our problem with the just-in-time strategy, calculus plays a pivotal role.

Derivatives, representing rates of change, help us understand how fast inventory is depleted, with the rate \(r\) in the function \(f(t) = Q - rt\) marking how quickly items leave the inventory per day.

On the flip side, integration pools information together to look at the cumulative impact over time, like the calculation of the total or average inventory present during an interval.

This dual power of calculus to dissect and synthesize information is why it remains indispensable in solving complex tasks like inventory management.Whether you're estimating changes using derivatives or accumulating data using integrals, calculus serves as a fundamental tool to systematize operations and improve efficiencies in many business areas.