Problem 33 - Calculus

Question

The length of a plant, \(L,\) is a function of its mass, \(M,\) so \(L=f(M) .\) A unit increase in a plant's mass stretches the plant's length more when the plant is small, and less when the plant is large. Assuming \(M>0,\) decide if \(f^{\prime}\) agrees with this description. \(f^{\prime}\) is increasing

Step-by-Step Solution

Answer

No, \(f'(M)\) being increasing disagrees with the description.

Step 1: Understand the Problem

We need to determine if the function's behavior aligns with the description provided. Specifically, we need to determine how a unit increase in mass affects the plant's growth when it's smaller versus when it's larger.

Step 2: Analyze the Description

The description suggests a diminishing rate of growth as the plant's mass increases. This means the rate of change of the plant's length with respect to its mass (\(f'(M)\)) should be decreasing as the mass increases.

Step 3: Interpret \(f'\) Increasing Condition

If \(f'(M)\) is increasing, it would mean that the rate of change of the plant's length with respect to its mass increases as the mass increases. This contradicts the description given in the problem.

Step 4: Conclusion

Since \(f'(M)\) is stated to be increasing, it does not agree with the scenario where a unit increase in mass stretches the length more when the plant is small and less when the plant is large. Therefore, \(f'(M)\) does not match the description.

Key Concepts

Rate of Change

A fundamental concept in calculus is the rate of change, which helps us understand how one quantity varies relative to another. In our context, we're examining how the plant's length changes as its mass changes. This relational change can be quantified using the derivative, represented here as \( f'(M) \).

• When \( f'(M) \) is positive, it suggests that the plant's length is increasing as its mass increases.
• If \( f'(M) \) is negative, it would mean the length is decreasing as the plant's mass increases.

Understanding whether the rate of change increases or decreases helps predict the overall behavior of the function, providing insights into how much more or less a plant grows as it becomes heavier. This is crucial for making informed conclusions about the growth patterns of plants based on their mass.

Function Behavior

Function behavior refers to how a function acts as its input changes. In this scenario, we look at how the function \( L = f(M) \) behaves as the mass \( M \) of the plant increases. Primarily, we're interested in how the derivative \( f'(M) \) of the function changes, as it tells us about the rate at which the plant's length grows with changing mass.

• If \( f'(M) \) is increasing, it implies that the rate of growth of the plant's length is also increasing.
• This would mean larger gains in length for equal gains in mass.
• Conversely, if \( f'(M) \) is decreasing, it indicates diminishing returns in length with each unit of increased mass, as described in the problem.

By understanding these behaviors, we can better comprehend scenarios like diminishing growth, where a function does not continue increasing at a steady pace but rather slows down as the input (in this case, mass) becomes larger.

Increasing and Decreasing Functions

A key part of understanding functions in calculus is determining when a function is increasing or decreasing. This is particularly relevant when considered alongside the derivative.

• An increasing function is one where the output (the plant's length in our case) grows as the input (mass) grows.
• A decreasing function would imply that the output shrinks or grows at a slower pace as the input grows.

The derivative \( f'(M) \) tells us about the increasing or decreasing nature of the function \( f(M) \). If \( f'(M) \) itself is increasing, then the function is experiencing increasing rate of growth. But in the context of the original problem, a scenario where \( f'(M) \) increases doesn't align with the expected behavior of slower growth as mass increases.Hence, recognizing whether the function is increasing or decreasing helps students predict and understand how the function works and whether it fits real-world descriptions, like the diminishing growth in plant length as mass increases.