Problem 83 - Calculus for Biology and Medicine

Question

Suppose that \(N(t)\) denotes a population size at time \(t\) and satisfies the equation $$ N(t)=2 e^{3 t} \quad \text { for } t \geq 0 $$ (a) If you graph \(N(t)\) as a function of \(t\) on a semilog plot, a straight line results. Explain why. (b) Graph \(N(t)\) as a function of \(t\) on a semilog plot, and determine the slope of the resulting straight line.

Step-by-Step Solution

Answer

A semilog plot of \( N(t) = 2e^{3t} \) results in a straight line because \( \log(N(t)) = \log(2) + 3t \), with slope 3.

Step 1: Understanding the equation

The given function is \( N(t) = 2 e^{3t} \), where \( e \) is the base of natural logarithms and \( t \geq 0 \). This is an exponential function where the population size \( N(t) \) grows exponentially with time \( t \).

Step 2: Understanding a Semilog Plot

A semilog plot is a graph where one of the axes (usually the y-axis) has a logarithmic scale. In this case, plotting \( N(t) \) on a semilog plot implies that we plot \( \log(N(t)) \) against \( t \).

Step 3: Converting to Logarithmic Form

To analyze \( N(t) \) on a semilog plot, take the natural logarithm of both sides: \( \log(N(t)) = \log(2 e^{3t}) = \log(2) + \log(e^{3t}) \). Using the rule \( \log(a^b) = b \log(a) \), this becomes \( \log(e^{3t}) = 3t \log(e) = 3t \) since \( \log(e) = 1 \). So, \( \log(N(t)) = \log(2) + 3t \).

Step 4: Identifying Linear Relationship

In the equation \( \log(N(t)) = \log(2) + 3t \), if we plot \( \log(N(t)) \) versus \( t \), it follows the linear form \( y = mx + c \), where \( m = 3 \) is the slope and \( c = \log(2) \) is the y-intercept. Hence, a straight line appears on the semilog plot.

Step 5: Determining the Slope

From the form \( \log(N(t)) = \log(2) + 3t \), the slope \( m = 3 \) is evident. Thus, on the semilog plot, which graphically represents this linear relationship, the slope of the line is \( 3 \), implying the exponential growth rate is \( 3 \).

Step 6: Drawing the Graph

To graph \( N(t) \) on a semilog plot, use \( t \) for the x-axis and \( \log(N(t)) \) for the y-axis. Begin plotting from \( t = 0 \), ensuring the line starts at \( y = \log(2) \) with a slope that ascends at a rate of 3 units per 1 unit increase in \( t \). This will confirm the line's slope is consistent across the plot.

Key Concepts

Semilog Plot

When graphing exponential functions like \(N(t) = 2e^{3t}\), it's often helpful to use a semilog plot. A semilog plot is a type of graph where one of the axes is on a logarithmic scale, which transforms exponential growth into a straight line.
For example, if you plot \(N(t)\) on a semilog plot, you will be plotting \(\log(N(t))\) against \(t\). This transformation uses the properties of logarithms to help reveal underlying trends.
In this case, taking the natural logarithm of both sides gives us \(\log(N(t)) = \log(2) + 3t\). This equation is in the form of a linear equation \(y = mx + c\), where \(m = 3\) is the slope and \(c = \log(2)\) is the y-intercept.
The result is a straight line on the semilog plot, simplifying the visualization of the exponential growth.

Exponential Function

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. A classic example is \(N(t) = 2e^{3t}\), where \(e\) is Euler's number, a natural logarithm base approximately equal to 2.718.
These types of functions are characterized by their rapid growth, where the rate of increase is proportional to their current value. Simply put, the larger the quantity becomes, the faster it grows.
In mathematical terms, for an exponential function \(N(t) = ae^{bt}\):

• \(a\) is the initial amount at \(t = 0\).
• \(b\) is the growth rate determining how quickly the function rises.

This exponential behavior is prevalent in many natural phenomena such as population growth, radioactive decay, and financial interest calculations, to name a few.

Population Dynamics

Population dynamics is the study of how populations change over time. Models using exponential growth, such as \(N(t) = 2e^{3t}\), are commonly employed in this field.
In these models, the population size at any time \(t\) is a result of initial size and growth rate. For instance:

• The initial population size is represented by the constant factor, here 2.
• The growth rate, controlled by \(3t\), shows how quickly the population expands.

Exponential growth models like these assume unlimited resources, resulting in continuous and unbounded growth, a scenario that might be idealistic in real-world applications.
Nonetheless, these models provide significant insights into theoretical population trends, helping scientists understand potential future outcomes based on current growth rates.