Problem 60 - Calculus for Biology and Medicine

Question

Population Growth Suppose that the population size at time is $$ N(t)=N_{0} e^{r t}, \quad t \geq 0 $$ where \(N_{0}\) is a positive constant and \(r\) is a real number. (a) What is the population size at time 0 ? (b) Show that $$ \frac{d N}{d t}=r N $$

Step-by-Step Solution

Answer

(a) The population size at time 0 is \(N_{0}\). (b) \(\frac{dN}{dt} = rN\) holds true.

Step 1: Evaluate Population at Time t=0

To find the population size at time 0, substitute \(t = 0\) into the population formula. We have, \( N(t) = N_{0}e^{rt} \).Therefore, \( N(0) = N_{0}e^{r(0)} = N_{0}e^{0} = N_{0}\). So, the population size at time 0 is \(N_{0}\).

Step 2: Differentiate the Population Function

To demonstrate \(\frac{dN}{dt} = rN\), we need to differentiate \(N(t) = N_{0}e^{rt}\) with respect to time \(t\).Using the chain rule, we have: \( \frac{dN}{dt} = \frac{d}{dt} N_{0}e^{rt} = N_{0} \cdot \frac{d}{dt} e^{rt}\).

Step 3: Apply the Chain Rule

The derivative of \(e^{rt}\) with respect to \(t\) is \(re^{rt}\) because the exponential function \(e^{u}\) differentiates to \(e^{u} \frac{du}{dt}\) where \(u=rt\) and \(\frac{du}{dt} = r\).So, \(\frac{d}{dt}(e^{rt}) = re^{rt}\).Therefore, \( \frac{dN}{dt} = N_{0} \cdot re^{rt} = r N_{0} e^{rt} \).

Step 4: Conclusion

Since \( N(t) = N_{0} e^{rt} \), we can substitute \( N(t) \) back in: \( \frac{dN}{dt} = r(N_{0} e^{rt}) = rN(t) \).Thus, \(\frac{dN}{dt} = rN\), as required by the problem statement.

Key Concepts

Exponential Growth

Exponential growth occurs when the growth rate of a mathematical function is proportional to its current value. In this exercise, the population of a certain group is given by the function \(N(t) = N_0 e^{rt}\). This is a classic example of exponential growth, where:

• \(N_0\) is the initial population size at time \(t = 0\)
• \(r\) is the rate of growth
• \(e^{rt}\) represents the exponential factor, where \(e\) is Euler's number, approximately 2.718

This equation suggests that as time increases, the population grows exponentially, meaning it increases more and more rapidly over time. The growth is constant on a relative scale, indicating that the population is always growing by a consistent percentage rather than a fixed amount.
To understand exponential growth, envision how bacteria multiply: starting with a single bacterium, it doubles over a fixed time interval, causing a rapid increase in population. This model assumes a similar constant rate of growth, fueling an ever-accelerating increase in population.
The formula allows us to determine not only current population sizes but predict future population sizes by simply plugging in different values for \(t\). It's why exponential growth can quickly lead to large numbers over short periods.

Differentiation

Differentiation is a fundamental concept in calculus used to compute the rate at which a quantity changes. In this context, we differentiate the population function to find the rate of change of population over time.
The original function is given by \(N(t) = N_0 e^{rt}\). Differentiation helps us to express how fast the population grows at any time \(t\). To find this, we calculate \(\frac{dN}{dt}\), which represents the derivative of the population with respect to time.
In simpler terms, differentiation answers the question: "How is the population changing at any given moment?" It's essentially about finding the slope or the rate of change at a specific point.
The derivative \(\frac{dN}{dt} = rN\) tells us that the rate of population growth at any time \(t\) is proportional to the current population size \(N\). This result is very telling. It implies that a larger population will grow faster, assuming the growth rate \(r\) remains constant.

Chain Rule

The chain rule is a powerful tool in calculus that allows us to differentiate composite functions. It's particularly useful when a function is nested inside another function, as with the population function here.
In our population model \(N(t) = N_0 e^{rt}\), the term \(e^{rt}\) is a composite function, where \(rt\) serves as the inner function. When we differentiate \(N(t)\) with respect to \(t\), we need to apply the chain rule to tackle the nested function effectively.

• According to the chain rule, if you have \(e^{u}\) where \(u = rt\), the derivative of \(e^{u}\) is \(e^{u} \cdot \frac{du}{dt}\).
• As \(u = rt\), \(\frac{du}{dt} = r\).
• This means when we differentiate \(e^{rt}\), the result is \(re^{rt}\).

By using the chain rule, we neatly manage the complexities of differentiating exponential functions. This systematic approach makes differentiation manageable even for more intricate functions. Ultimately, it allows us to find how our population is changing with precision, guided by the underlying mathematical principles.