91Ó°ÊÓ

Population growth: The following table \({ }^{6}\) shows the population of reindeer on an island as of the given year. $$ \begin{array}{|l|c|c|c|c|} \hline \text { Date } & 1945 & 1950 & 1955 & 1960 \\ \hline \text { Population } & 40 & 165 & 678 & 2793 \\ \hline \end{array} $$ We let \(t\) be the number of years since 1945 , so that \(t=0\) corresponds to 1945 , and we let \(N=N(t)\) denote the population size. a. Approximate \(\frac{d N}{d t}\) for 1955 using the average rate of change from 1955 to 1960 , and explain what this number means in practical terms. b. Use your work from part a to estimate the population in \(1957 .\) c. The number you calculated in part a is an approximation to the actual rate of change. As you will be asked to show in the next exercise, the reindeer population growth can be closely modeled by an exponential function. With this in mind, do you think your answer in part a is too large or too small? Explain your reasoning.

Step by step solution

01

Identify Years and Populations

First, understand that the years relevant to the exercise are 1955 and 1960. The corresponding populations are 678 (in 1955) and 2793 (in 1960).

02

Calculate Average Rate of Change

To approximate \( \frac{dN}{dt} \) for 1955, calculate the average rate of change of the population from 1955 to 1960. Use the formula for the average rate of change: \[ \frac{N(t_2) - N(t_1)}{t_2 - t_1} \] where \( N(t_2) = 2793 \), \( N(t_1) = 678 \), \( t_2 = 1960 \) (or \( t = 15 \)), and \( t_1 = 1955 \) (or \( t = 10 \)). This gives \[ \frac{2793 - 678}{15 - 10} = \frac{2115}{5} = 423 \].

03

Interpret the Average Rate of Change

The number 423 means that the population increased on average by 423 reindeer per year from 1955 to 1960.

04

Estimate Population for 1957

Using the average rate of change found (423 reindeer per year), estimate the population in 1957 by adding two years' worth of growth to the 1955 population: \[ N(1957) = 678 + 2 \times 423 = 678 + 846 = 1524 \].

05

Evaluate Approximation Validity

The rate calculated from part a is large because it is based on linear growth, not exponential. Given an exponential growth model, the actual instantaneous rate of change (derivative) would increase over time, meaning our linear approximation might be an underestimate for the instantaneous rate of change in 1955 since the growth generally accelerates.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Rate of Change

When considering population growth, particularly in the context of reindeer on an island, one useful metric is the "average rate of change." This reflects how much a quantity changes over a certain interval of time.
To calculate the average rate of change of the population between 1955 and 1960, we use the formula: \[ \frac{N(t_2) - N(t_1)}{t_2 - t_1} \] where \( N(t_2) \) is the population in 1960 and \( N(t_1) \) is the population in 1955. Substituting the given values, we find: \[ \frac{2793 - 678}{15 - 10} = 423 \] This result, 423, means that on average, the reindeer population increased by 423 per year during this period. It provides a simplified view of how the population was changing over time.
However, this average does not account for any fluctuations that might have occurred within the time interval; it's a smooth approximation of growth over those five years.

Exponential Growth

Exponential growth means that the population increases at a rate proportional to its current size. This is often encountered in natural populations and certain financial contexts, where growth builds upon itself.
In our exercise, understanding whether the population grew exponentially can help explain why the reindeer numbers soared from 1955 onwards. Exponential growth occurs when the growth rate is constant in proportion to the current population, meaning any increase compounds over time.
During exponential growth, the population of reindeer would not just increase linearly (by the same number each year), but rather it would accelerate—each year's growth increment is larger than the last, compounding on previous growth. This concept becomes essential in predicting future populations accurately, as it suggests continuous and accelerating change.

Derivative

In the context of population growth, a "derivative" helps determine the rate at which the population is changing at a specific moment. This is known as the "instantaneous rate of change."
The derivative can be seen as the limit of the average rate of change as the time interval approaches zero. It is crucial for capturing how populations grow or shrink instantaneously without the smooth approximation provided by an average rate.
Since the exercise suggests the population grows exponentially, the derivative provides a more precise measurement than the linear approximation. Generally, in scenarios like these, the actual rate of change would increase over time due to growing population numbers, making the instantaneous rate larger than the average rate observed over several years.

Estimation Techniques

In many real-world scenarios, precise calculations on population or growth may not always be possible or practical due to varying data quality or availability. Hence, estimation techniques become invaluable.

• Average rate of change provides a simplified, quick estimate of how a population was changing over a specific period.
• Linear interpolation, like estimating populations in 1957 using growth rates, offers a straightforward method by attaching average growth to known values.
• Understanding models like exponential growth allows for educated guesses about future rates, even when precise data is scarce or uneven.

These techniques help create useful insights into likely future outcomes or historical analyses without the need for computing every potential parameter. Using estimation instead of exact calculations is often a practical solution, especially when dealing with complex systems like population dynamics.