Problem 54 - Calculus: Early Transcendentals

Question

The number of bacteria after \( t \) hours in a controlled laboratory experiment is \( n = f(t) \). (a) What is the meaning of the derivative \( f'(5) \)? What are its units? (b) Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, \( f'(5) \) or \( f'(10) \)? If the supply of the nutrients is limited, would that affect your conclusion? Explain.

Step-by-Step Solution

Answer

(a) \( f'(5) \) is the growth rate at 5 hours; units are bacteria/hour. (b) With unlimited resources, \( f'(10) > f'(5) \); limited resources may stabilize growth, \( f'(5) \) could be larger or equal.

Step 1: Understanding f'(5)

In this context, the function \( n = f(t) \) represents the number of bacteria present after \( t \) hours. The derivative \( f'(t) \) indicates the instantaneous rate of change of the number of bacteria over time. Therefore, \( f'(5) \) signifies the rate at which the bacteria population is growing at exactly \( 5 \) hours. The units would be 'bacteria per hour' since it measures the change in bacteria count with respect to time, in hours.

Step 2: Compare f'(5) and f'(10) with Unlimited Resources

When the resources (space and nutrients) are unlimited, the bacteria can grow exponentially without constraints. In a typical exponential growth scenario, the rate of growth, indicated by \( f'(t) \), increases over time as the population size becomes larger. Therefore, \( f'(10) \) is expected to be larger than \( f'(5) \), as the population would be larger at \( t = 10 \) hours leading to a higher rate of growth.

Step 3: Effects of Limited Nutrients on f'(5) and f'(10)

If the supply of nutrients is limited, it directly impacts the bacteria's ability to continue growing at an exponential rate. Initially, \( f'(5) \) might reflect rapid growth, but as resources become scarce over time, the growth rate \( f'(10) \) could slow down or even stabilize. Therefore, with limited resources, \( f'(5) \) might be equal to or greater than \( f'(10) \) as the population begins to reach carrying capacity, limiting further growth.

Key Concepts

Instantaneous Rate of Change

When we talk about the instantaneous rate of change in the context of bacteria growth, we're focusing on how fast the number of bacteria is increasing at a specific moment in time. This is represented mathematically by the derivative of the function describing the number of bacteria, denoted as \( f'(t) \). Suppose we have a bacteria population modeled over time \( t \) by a function \( n = f(t) \). The derivative \( f'(t) \) is then the rate at which the bacteria population is changing at any given hour \( t \).
For example, \( f'(5) \) means the instantaneous growth rate at exactly 5 hours. It tells us how many bacteria are being added every hour at that point in time. These units are typically expressed as 'bacteria per hour'. This concept is crucial in predicting future populations and understanding growth patterns in biology or any situation involving dynamic change.

• Instantaneous rate gives a snapshot of growth speed at a particular moment.
• It's crucial for identifying trends and predicting future states of the system.

Exponential Growth

Exponential growth is a fascinating concept often seen in bacterial populations when conditions are ideal. This mathematical form of growth happens when the rate of increase of the population is proportional to the size of the population itself at any given time. In other words, the bigger the population, the faster it grows.
Consider a scenario with unlimited space and nutrients. Here, bacteria reproduce rapidly, and the rate of growth, \( f'(t) \), actually increases as time progresses, because each bacterium can give rise to offspring, multiplying the population size quickly. Because of this, \( f'(10) \), the rate at 10 hours, is usually larger than \( f'(5) \) in these conditions, since the larger population at 10 hours grows faster.

• Exponential growth is characterized by rapid increases over time.
• This growth is driven by the continuous reproduction of individuals within the population.

Carrying Capacity

Carrying capacity plays a critical role in limiting exponential growth. It refers to the maximum population size that an environment can sustain indefinitely. When resources such as nutrients or space become limited, they can restrain the population growth of bacteria.
In a setting where resources are finite, even if growth starts out exponentially, it may slow down as the population approaches the carrying capacity. Consequently, while \( f'(5) \) might show a high rate of growth when resources are still abundant, by the time we reach \( f'(10) \), the rate might have slowed down as resources are used up, slowing or stalling further growth.
This concept emphasizes the realistic constraints faced by populations in nature:

• Carrying capacity imposes limits on growth based on resource availability.
• It can convert exponential growth into a logistic growth pattern, where growth stabilizes near the carrying capacity.