Problem 32
Question
Suppose that the per capita growth rate of a population is \(2 \% ;\) that is, if \(N(t)\) denotes the population size at time \(t\), then $$\frac{1}{N} \frac{d N}{d t}=0.02$$ Suppose also that the population size at time \(t=2\) is equal to \(100 .\) Use a linear approximation to compute the population size at time \(t=2.1\).
Step-by-Step Solution
Verified Answer
The approximate population size at time \( t=2.1 \) is 100.2.
1Step 1: Identify the Formula
We are given the equation for the per capita growth rate: \( \frac{1}{N} \frac{dN}{dt} = 0.02 \). We can rewrite this as \( \frac{dN}{dt} = 0.02N \), which is a typical differential equation for exponential growth.
2Step 2: Linear Approximation Formula
The linear approximation formula for a function \( f(x) \) at a point \( x=a \) is \( f(x) \approx f(a) + f'(a)(x-a) \). Here, we will use this to approximate \( N(t) \) around \( t=2 \).
3Step 3: Evaluate the Derivative at t=2
The derivative \( \frac{dN}{dt} = 0.02N \). At \( t=2 \), the population is given as \( N(2) = 100 \). Thus, \( \frac{dN}{dt} \bigg|_{t=2} = 0.02 \times 100 = 2 \).
4Step 4: Substitute Values into Linear Approximation
Using \( N(t) \approx N(2) + \frac{dN}{dt}\bigg|_{t=2} \cdot (t-2) \), we have: \( N(2.1) \approx 100 + 2 \times (2.1 - 2) \).
5Step 5: Calculate the Approximation
Calculate \( N(2.1) \approx 100 + 2 \times 0.1 = 100 + 0.2 = 100.2 \).
Key Concepts
Understanding Differential EquationsLinear Approximation ExplainedThe Role of Population Dynamics
Understanding Differential Equations
Differential equations are fantastic tools in mathematics and science. They are used to describe a wide variety of phenomena, including physics, engineering, and even population growth. Essentially, a differential equation relates a function to its derivatives. The derivatives often reflect rates, such as the rate of change of a population over time.
The equation given in the problem, \( \frac{1}{N} \frac{dN}{dt} = 0.02 \), can be rewritten to \( \frac{dN}{dt} = 0.02N \). This equation is a classic form that indicates exponential growth because the rate of change of the population \( N \) is proportional to the size of the population itself.
\begin{itemize} \item **Exponentially Growing Population:** If every individual is contributing to the growth rate proportionally, the total population grows larger over time. \item **Analysis and Prediction:** By solving such differential equations, we can predict future populations based on their current rates and sizes.\end{itemize}Using differential equations helps not only in prediction but also in understanding what drives the growth or decline in given scenarios.
The equation given in the problem, \( \frac{1}{N} \frac{dN}{dt} = 0.02 \), can be rewritten to \( \frac{dN}{dt} = 0.02N \). This equation is a classic form that indicates exponential growth because the rate of change of the population \( N \) is proportional to the size of the population itself.
\begin{itemize} \item **Exponentially Growing Population:** If every individual is contributing to the growth rate proportionally, the total population grows larger over time. \item **Analysis and Prediction:** By solving such differential equations, we can predict future populations based on their current rates and sizes.\end{itemize}Using differential equations helps not only in prediction but also in understanding what drives the growth or decline in given scenarios.
Linear Approximation Explained
Linear approximation is a mathematical method used to estimate the value of a function near a point using its derivative. This is especially useful for functions that are complex or not easy to calculate directly.
Essentially, it allows us to use a simple line (a "tan" line if you will) that touches the curve at a point to estimate other values near that point.
When we applied linear approximation in the exercise, we used the formula: \[ N(t) \approx N(2) + \frac{dN}{dt}\bigg|_{t=2} \cdot (t-2) \]This formula tells us that to approximate \( N(t) \) at \( t = 2.1 \), we start with the known value at \( t = 2 \), then add the rate of change of \( N(t) \) at \( t = 2 \) multiplied by the small change in time (\( 2.1 - 2 \)).
\begin{itemize} \item **Easy Calculation:** By using a tangent line, it's easier to compute an estimated value than using more complex methods. \item **Accuracy:** This method works well when the point you're estimating is very close to the point of tangency.\end{itemize}
Essentially, it allows us to use a simple line (a "tan" line if you will) that touches the curve at a point to estimate other values near that point.
When we applied linear approximation in the exercise, we used the formula: \[ N(t) \approx N(2) + \frac{dN}{dt}\bigg|_{t=2} \cdot (t-2) \]This formula tells us that to approximate \( N(t) \) at \( t = 2.1 \), we start with the known value at \( t = 2 \), then add the rate of change of \( N(t) \) at \( t = 2 \) multiplied by the small change in time (\( 2.1 - 2 \)).
\begin{itemize} \item **Easy Calculation:** By using a tangent line, it's easier to compute an estimated value than using more complex methods. \item **Accuracy:** This method works well when the point you're estimating is very close to the point of tangency.\end{itemize}
The Role of Population Dynamics
Population dynamics involves the study of how and why populations change over time. It is a critical concept within ecology and biology because it helps us understand the growth patterns and potential limits of populations.
This field incorporates many mathematical ideas, like differential equations, to simulate and predict how populations will evolve. In our exercise, the population dynamics are showcased through exponential growth.
In practical situations, understanding population dynamics can aid in:
This field incorporates many mathematical ideas, like differential equations, to simulate and predict how populations will evolve. In our exercise, the population dynamics are showcased through exponential growth.
In practical situations, understanding population dynamics can aid in:
- **Conservation Efforts:** Helps in managing species and predicting their survival chances.
- **Resource Management:** Ensures enough resources are available without exhausting them due to overpopulation.
- **Epidemiology:** Predicts how fast a disease can spread within a population based on population dynamics principles.
Other exercises in this chapter
Problem 31
Differentiate $$ f(N)=(b-1) N^{4}-\frac{N^{2}}{b} $$ with respect to \(N\). Assume that \(b\) is a nonzero constant.
View solution Problem 31
Differentiate $$ g(N)=\frac{b N}{k+N} $$ with respect to \(N\). Assume that \(b\) and \(k\) are positive constants.
View solution Problem 32
Find the equation of the normal line to the curve \(y=4 / x\) at the point \((-1,-4)\).
View solution Problem 32
Find the derivative with respect to the independent variable. $$ g(t)=\frac{\sin (3 t)}{\cos (5 t)} $$
View solution