Problem 20 - Applied Calculus

Question

Consumer demand for a certain product is changing over time, and the rate of change of this demand, \(f^{\prime}(t)\), in units/week, is given, in week \(t\), in the following table. $$\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c} \hline t & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline f^{\prime}(t) & 12 & 10 & 4 & -2 & -3 & -1 & 3 & 7 & 11 & 15 & 10 \\ \hline \end{array} $$ (a) When is the demand for this product increasing? When is it decreasing? (b) Approximately when is demand at a local maximum? A local minimum?

Step-by-Step Solution

Answer

(a) Demand is increasing during weeks 0-2 and 6-10, decreasing during weeks 3-5. (b) Maximum at week 2, minimum at week 5.

Step 1: Understanding the Rate of Change

The table provides us with the derivative of the demand function, \( f'(t) \), at different weeks \( t \). This derivative represents the rate of change in the consumer demand.

Step 2: Identifying Increasing and Decreasing Intervals

To determine when the demand is increasing or decreasing, look at the sign of \( f'(t) \). Demand is increasing when \( f'(t) > 0 \) and decreasing when \( f'(t) < 0 \). From the table, \( f'(t) > 0 \) for intervals \([0, 2] \), \([6, 10] \) and \( f'(t) < 0 \) for intervals \([3, 5] \). This shows demand is increasing during these intervals and decreasing during \( t = 3, 4, 5 \).

Step 3: Finding the Local Maximum and Minimum

A local maximum occurs when \( f'(t) \) changes from positive to negative, and a local minimum occurs when it changes from negative to positive. From the table, \( f'(t) \) changes from positive to negative at \( t = 2 \) (since \( f'(2) = 4 \) and \( f'(3) = -2 \)), indicating a local maximum around \( t = 2 \). Similarly, \( f'(t) \) changes from negative to positive at \( t = 5 \) (since \( f'(5) = -1 \) and \( f'(6) = 3 \)), indicating a local minimum around \( t = 5 \).

Key Concepts

Rate of Change

In consumer demand analysis, understanding the rate of change is crucial. This concept refers to how quickly or slowly consumer demand is increasing or decreasing over a certain period. The rate of change is represented by the derivative of the demand function, noted as \( f'(t) \). In our table, \( f'(t) \) gives us a snapshot of how consumer demand fluctuates from week to week. When thinking about the rate of change, imagine checking how steep a hill is at various points; this is similar to how economists evaluate how swiftly demand is shifting during different time intervals. A positive \( f'(t) \) tells us demand is climbing, while a negative \( f'(t) \) signals a drop in demand. Each piece of data helps businesses strategize whether to ramp up production or hold back as demand ebbs and flows.

Increasing and Decreasing Intervals

Determining when consumer demand is on the rise or decline involves examining the signs of the function's derivative. For a demand curve, it is important to identify when it is increasing or decreasing, which can be done by analyzing \( f'(t) \).

Here's a simple way to remember it:

• If \( f'(t) > 0 \), demand is increasing. The slope is like riding uphill; sales are expected to go up.
• If \( f'(t) < 0 \), demand is decreasing. Think of sliding downhill, indicating a dip in sales.

In the given data, demand for the product increases during the intervals \([0, 2] \) and \([6, 10] \) because that's when \( f'(t) \) holds positive values. Conversely, demand decreases in the range \([3, 5] \) where \( f'(t) \) becomes negative.

Local Maximum and Minimum

Local maximum and minimum points are valuable indicators of consumer demand peaks and troughs within limited time frames. A local maximum is the highest point of demand over a particular interval. This occurs when \( f'(t) \) switches from positive to negative, signaling that demand stops rising and begins to fall.

On the other hand, a local minimum is where demand bottoms out and starts to climb again, observed when \( f'(t) \) changes from negative to positive. In our example, a local maximum is around week \( t = 2 \) because \( f'(t) \) moves from \( 4 \) at \( t = 2 \) to \( -2 \) at \( t = 3 \), indicating a crest in demand.

Similarly, a local minimum appears near week \( t = 5 \) since \( f'(5) = -1 \) switches to a positive \( 3 \) at \( t = 6 \). Recognizing these local extremes helps businesses better predict inventory needs, optimize pricing strategies, and plan for future market fluctuations.