Problem 83
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
Verified Answer
(a) The semilog plot linearizes the exponential function; (b) The slope of the line is 3.
1Step 1: Explore the Semilog Plot
A semilog plot is a way to plot data on a graph using a logarithmic scale for the y-axis and a linear scale for the x-axis. When the equation is of the form \(N(t) = ae^{bt}\), taking the natural log on both sides gives \(\log(N(t)) = \log(a) + bt\). This is the equation of a straight line: \(y = mx + c\), where \(x = t\), \(m = b\), and \(c = \log(a)\). Hence, graphing \(N(t)\) on a semilog plot results in a straight line.
2Step 2: Determine the Straight Line Equation
Given \(N(t) = 2e^{3t}\), take the natural log of both sides to linearize it in terms of \(t\). This gives us \(\log(N(t)) = \log(2) + 3t\). This equation shows that when \(\log(N(t))\) is plotted against \(t\), it is a straight line with the slope 3 and y-intercept \(\log(2)\).
3Step 3: Determine the Slope of the Line
From the linearized equation \(\log(N(t)) = 3t + \log(2)\), it is clear that the slope \(m\) of the line on the semilog plot is the coefficient of \(t\), which is 3.
Key Concepts
semilog plotpopulation dynamicslinearization
semilog plot
A semilog plot is a special type of graph that is incredibly useful for understanding exponential relationships. It uses a logarithmic scale on the y-axis, while the x-axis remains on a linear scale. This special plotting technique helps to transform exponential curves into straight lines, making it easier to analyze growth patterns.
In the context of our exercise, the function given is \(N(t) = 2e^{3t}\). By converting the exponential equation into its logarithmic form, \(\log(N(t)) = \log(2) + 3t\), we can easily see that it resembles the linear equation format \(y = mx + c\).
With this transformation, the semilog plot displays what looks like a straight line, where the slope is determined by the constant multiplying \(t\), and the y-intercept is determined by the logarithm of the initial term. This linear relationship on a semilog plot provides a simplified view that helps us understand complex exponential growth phenomena.
In the context of our exercise, the function given is \(N(t) = 2e^{3t}\). By converting the exponential equation into its logarithmic form, \(\log(N(t)) = \log(2) + 3t\), we can easily see that it resembles the linear equation format \(y = mx + c\).
With this transformation, the semilog plot displays what looks like a straight line, where the slope is determined by the constant multiplying \(t\), and the y-intercept is determined by the logarithm of the initial term. This linear relationship on a semilog plot provides a simplified view that helps us understand complex exponential growth phenomena.
population dynamics
Population dynamics is a fundamental concept in understanding how populations change over time. It helps us model the behavior of a population, whether it grows exponentially or exhibits other growth patterns. One powerful model for rapid growth is the exponential growth model, highlighted by the equation \(N(t) = ae^{bt}\). This is particularly applicable in ecological and biological studies.
In the given exercise, the equation \(N(t) = 2e^{3t}\) signifies exponential growth, where the population multiplies by a constant factor in equal intervals of time. Understanding this model aids in explaining how populations can grow from a few individuals to a large number quickly, given sufficient resources and without constraints like predation or resources shortage. It forms a foundational concept for various applications including predicting human population growth, studying invasive species spread, or managing biological conservation efforts.
In the given exercise, the equation \(N(t) = 2e^{3t}\) signifies exponential growth, where the population multiplies by a constant factor in equal intervals of time. Understanding this model aids in explaining how populations can grow from a few individuals to a large number quickly, given sufficient resources and without constraints like predation or resources shortage. It forms a foundational concept for various applications including predicting human population growth, studying invasive species spread, or managing biological conservation efforts.
linearization
Linearization is a mathematical technique used to simplify complex equations, particularly those that involve exponential elements. By converting an exponential equation into its logarithmic form, we can transform it into a linear equation, making it easier to graph and analyze using straightforward methods.
This is an exceptionally useful tool in mathematics and engineering, especially when working with exponential growth models such as our equation \(N(t) = 2e^{3t}\). By taking the natural logarithm of both sides, \(\log(N(t)) = \log(2) + 3t\), we linearize the function, revealing a simple linear equation in terms of \(t\) with a slope and y-intercept.
Linearization reveals important characteristics of the exponential function, such as the growth rate and initial size of the population, which are otherwise harder to discern when examining the original exponential form. This provides a much clearer understanding of the population's growth pattern over time, offering insights into predicting future dynamics.
This is an exceptionally useful tool in mathematics and engineering, especially when working with exponential growth models such as our equation \(N(t) = 2e^{3t}\). By taking the natural logarithm of both sides, \(\log(N(t)) = \log(2) + 3t\), we linearize the function, revealing a simple linear equation in terms of \(t\) with a slope and y-intercept.
Linearization reveals important characteristics of the exponential function, such as the growth rate and initial size of the population, which are otherwise harder to discern when examining the original exponential form. This provides a much clearer understanding of the population's growth pattern over time, offering insights into predicting future dynamics.
Other exercises in this chapter
Problem 82
Simplify the following expressions: (a) \(e^{3 \ln x}\) (b) \(e^{-\ln \left(x^{2}+1\right)}\) (c) \(e^{-2 \ln (1 / x)}\) (d) \(e^{-2 \ln x}\)
View solution Problem 82
Solve for \(x\). (a) \(5^{x}=625\) (b) \(4^{4 x}=256\) (c) \(10^{2 x}=0.0001\)
View solution Problem 83
Write the following expressions in terms of base \(e\), and simplify: (a) \(3^{x}\) (b) \(4^{x^{2}-1}\) (c) \(2^{-x-1}\) (d) \(3^{-4 x+1}\)
View solution Problem 83
Solve for \(x\). (a) \(\ln (x-3)=5\) (b) \(\ln (x+2)+\ln (x-2)=1\) (c) \(\log _{3} x^{2}-\log _{3} 2 x=2\)
View solution