Problem 29 - Applied Calculus

Question

Consumer demand for a product is changing over time, and the rate of change of demand, \(f^{\prime}(t),\) in units/week, is given, in week \(t,\) for \(0 \leq t \leq 10,\) 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

Demand is increasing on \(0 \leq t < 3\) and \(6 < t \leq 10\), decreasing on \(3 < t < 6\), with a local max at \(t = 2\) and a local min at \(t = 5\).

Step 1: Identify Increasing Demand Intervals

To find when demand is increasing, look for intervals where the rate of change, \(f'(t)\), is positive. From the table, the demand is increasing when \(f'(t) > 0\). These intervals are: \(0 \leq t < 3\) (since \(f'(t)\) is positive at \(t = 0, 1, 2\)) and \(6 < t \leq 10\) (since \(f'(t)\) is positive at \(t = 7, 8, 9, 10\)).

Step 2: Identify Decreasing Demand Intervals

Demand is decreasing when \(f'(t) < 0\). From the table, the demand decreases in the interval \(3 < t < 6\) because \(f'(t)\) is negative at \(t = 4, 5\).

Step 3: Determine Local Maximum

A local maximum occurs where \(f'(t)\) changes from positive to negative. From the table, \(f'(t)\) transitions from positive at \(t = 2\) to negative at \(t = 3\). Therefore, there is a local maximum at approximately \(t = 2\).

Step 4: Determine Local Minimum

A local minimum occurs where \(f'(t)\) changes from negative to positive. From the table, \(f'(t)\) changes from negative at \(t = 5\) to positive at \(t = 6\). Thus, there is a local minimum at approximately \(t = 5\).

Key Concepts

Rate of Change

In calculus, the "rate of change" refers to how a quantity changes over time. For instance, in this exercise, we look at how the rate of consumer demand for a product, represented by \( f'(t) \), varies during different weeks. When \( f'(t) \) is positive, it indicates that the demand is increasing.
Conversely, when \( f'(t) \) is negative, the demand is decreasing. A positive rate indicates that more units are demanded by consumers, while a negative rate shows a decline in demand.
This concept is crucial in understanding market dynamics as businesses need to know when to boost production or cut back based on demand trends.

Increasing and Decreasing Functions

An "increasing function" is one where, as you move to a larger input value (like time), the output or rate of change continues to rise. In this exercise, demand is increasing when \( f'(t) > 0 \). This happens in the intervals when \( t \) is between 0 and 3, and after 6 all the way to 10.
However, for "decreasing functions," the opposite is true. Here, the rate of change falls as time progresses. This occurs when \( f'(t) < 0 \), and it was observed between weeks 3 and 6.

• Increasing Interval: \(0 \leq t < 3\) and \(6 < t \leq 10\)
• Decreasing Interval: \(3 < t < 6\)

Understanding these intervals helps in predicting future trends in demand, assisting businesses in making data-driven decisions.

Local Maximum and Minimum

Local maximum and minimum points are key in determining the highest and lowest points of demand in a given interval. A "local maximum" is a point where the rate of change shifts from positive to negative, indicating a peak in demand. In this example, the local maximum occurs at approximately \( t = 2 \).
This means demand reached its highest at this point before it started decreasing. Similarly, a "local minimum" happens when the rate of change switches from negative to positive. Here, it indicates a lull in demand before it starts picking up again.

• Local Maximum: approximately \( t = 2 \)
• Local Minimum: approximately \( t = 5 \)

Identifying these points helps businesses to optimize their strategies by knowing exactly when to adjust their supply to meet changing demand patterns.