Problem 49 - Brief Applied Calculus

Question

Suppose that the quantity described is represented by a function \(f(t)\) where \(t\) stands for time. Based on the description: a. Is the first derivative positive or negative? b. Is the second derivative positive or negative? The stock market is declining, but less rapidly.

Step-by-Step Solution

Answer

a. First derivative is negative; b. Second derivative is positive.

Step 1: Understanding the Scenario

The problem describes the behavior of the stock market in terms of a function \(f(t)\) over time, \(t\). Specifically, it states that the stock market is declining, which indicates that the function \(f(t)\) representing the stock market's value is decreasing over time.

Step 2: Analyzing the First Derivative

The first derivative of a function, \(f'(t)\), represents the rate of change of the function with respect to time. Since the stock market is declining, \(f(t)\) is decreasing. Therefore, the first derivative \(f'(t)\) must be negative, as a negative derivative indicates a decrease in function value.

Step 3: Analyzing the Second Derivative

The second derivative, \(f''(t)\), represents the rate of change of the first derivative. The problem states that the stock market is declining "less rapidly," indicating that the rate of decline is decreasing. This means \(f'(t)\) is becoming less negative, which implies that \(f''(t)\) is positive, as a positive second derivative indicates that a function is becoming less negative or more positive.

Key Concepts

First Derivative

When trying to understand changes in any function, the first derivative, typically denoted as \(f'(t)\), is a critical concept. Think of it as a measure of how fast something is changing. For a function that describes the stock market over time, \(f'(t)\) will tell us if the market's value is increasing or decreasing at any given moment.

• If \(f'(t) > 0\), the function is increasing; this means that the stock market is on the rise.
• If \(f'(t) = 0\), the function is neither increasing nor decreasing, indicating a potential peak or trough in the market.
• If \(f'(t) < 0\), the function is decreasing, which means the stock market's value is declining over time.

In our scenario, the stock market is explicitly stated to be declining. Thus, we confidently conclude that \(f'(t) < 0\). Understanding this allows us to grasp that a negative first derivative reflects a decrease in the function's value over time, indicating a downward trend in the stock market.

Second Derivative

The second derivative, often represented by \(f''(t)\), shows us the rate at which the first derivative is changing. Essentially, it tells us how the speed of the market's change itself is evolving.

• If \(f''(t) > 0\), the first derivative and thus the function's rate of change is increasing. This means the stock's decline is slowing down.
• If \(f''(t) = 0\), there's a constant rate of change in the function, suggesting steady behavior in market trends.
• If \(f''(t) < 0\), the rate of change is decreasing, meaning the decline in the market is accelerating.

In this context, the stock market is described as declining "less rapidly." This indicates that while the market is still going down, the pace of this decline is decreasing. Thus, \(f''(t) > 0\) because the rate of decrease in \(f'(t)\) is slowing, making the market's downtrend less steep.

Decreasing Function

A decreasing function is one where, as the input (like time \(t\) in \(f(t)\)) increases, the output (in this case, the stock market value) decreases. This can be visualized as a downward-sloping curve on a graph.

• For a function to be decreasing, its first derivative \(f'(t)\) must be negative.

When considering the stock market, if \(f(t)\) represents its value, and it's a decreasing function, then every moment in time sees a lower value than before. However, when the problem mentions that this decline is happening less rapidly, it's crucial to understand the influence of the second derivative as well.
The combination of a negative first derivative and a positive second derivative implies that while values are lowering, the speed of this decrease is reducing, portraying a market that's still falling but with a slowed pace of decline.