Problem 50 - 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 population is growing increasingly fast.

Step-by-Step Solution

Answer

Both the first and second derivatives are positive.

Step 1: Identify the Situation

The problem describes a population that is growing increasingly fast. This indicates that not only is the quantity increasing over time, but the rate at which it increases is also accelerating.

Step 2: Determine the First Derivative

The first derivative, denoted as \( f'(t) \), represents the rate of change of the population over time. Since the population is growing, \( f'(t) \) is positive.

Step 3: Determine the Second Derivative

The second derivative, denoted as \( f''(t) \), represents the rate of change of the first derivative, essentially indicating whether the rate itself is increasing or decreasing. Because the growth rate is described as increasingly fast, \( f''(t) \) is positive.

Key Concepts

First Derivative

When dealing with population growth, the first derivative of a function, represented as \( f'(t) \), provides significant insights. This derivative shows the rate at which the population is changing over time. Consider it as a snapshot of the population's growth pace at any given moment.

In our scenario, where the population grows increasingly fast, the first derivative is positive:

• A positive \( f'(t) \) means that the population is actually growing over time.
• It implies that each day or unit of time that passes sees more people being added to the population.

Understanding the sign of \( f'(t) \) helps determine whether populations are increasing, decreasing, or remaining constant over time.

Second Derivative

The second derivative \( f''(t) \) delves deeper into the nature of change in a population. It indicates how the rate of population growth is evolving, providing insights into the acceleration or deceleration of growth.

For instance, in our problem, the population is described as growing increasingly fast. This means not only is the population growing, but it is growing at an ever-faster pace. Therefore, the second derivative in this case is positive:

• A positive \( f''(t) \) shows that the rate of growth itself is increasing.
• It's like saying not just that more people are joining daily, but even more people are joining each subsequent day than the previous.

This aspect of acceleration in growth is crucial for understanding future population trends.

Rate of Change

The concept of the rate of change underpins much of calculus and is essential in understanding functions describing population growth. The rate of change essentially captures how one quantity changes in relation to another—typically time. In our example, it clarifies how quickly the population size alters over a period.

By evaluating the first derivative (\( f'(t) \)), we observe the instant rate of population increase. The second derivative (\( f''(t) \)), in turn, reflects how this rate itself modifies over time:

• If \( f'(t) \) is positive, there's an increase in population size.
• If \( f''(t) \) is positive, the rate of this increase is accelerating.

Understanding both derivatives provides a comprehensive picture of population dynamics, offering vital data for planning and resource allocation in real-life applications.