Problem 72 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Epidemiologists estimate that \(t\) days after the flu begins to spread in a small town, the percent of the population infected by the flu is approximated by $$p(t)=t^{2}+2 t$$ for \(0 \leq t \leq 5 .\) Find the instantaneous rate of change of the percent of the population infected at time \(t=3\).

Step-by-Step Solution

Answer

The instantaneous rate of change at \(t=3\) is 8% per day.

Step 1: Understand the Problem

The problem asks us to find the instantaneous rate of change of the given function \(p(t) = t^2 + 2t\) at \(t=3\). This is essentially asking for the derivative \(p'(t)\) evaluated at \(t=3\).

Step 2: Differentiate the Function

To find the instantaneous rate of change, we need to compute the derivative of the function \(p(t) = t^2 + 2t\). Using basic differentiation rules, \(\frac{d}{dt} (t^2) = 2t\) and \(\frac{d}{dt} (2t) = 2\), therefore the derivative is \(p'(t) = 2t + 2\).

Step 3: Evaluate the Derivative at t=3

Substitute \(t=3\) into the derivative function: \(p'(3) = 2(3) + 2\). Simplifying this gives \(p'(3) = 6 + 2 = 8\).

Step 4: Interpret the Result

The result \(p'(3) = 8\) represents the instantaneous rate of change of the percent of the population infected at \(t=3\). This means that at \(t=3\) days, the percentage of the population infected is increasing at a rate of 8% per day.

Key Concepts

Derivative

In calculus, the derivative of a function measures how the function's output value changes as the input changes. It's like taking the slope of a curve at a specific point, telling us how steep the curve is at that point. The formula for a simple derivative, using a basic power rule, involves decreasing the power by one and multiplying by the original power.
This is what we did in the original exercise to find the instantaneous rate of change in the flu infection model. By finding the derivative of the function representing the percentage of infected people, we are determining how quickly the infection grows at a specific time.

• For example, if we have a function described by a term like \( t^2 \), its derivative is \( 2t \), which shows the rate of change for that particular term.
• Likewise, for a simple linear term like \( 2t \), the derivative is \( 2 \), indicating the constant rate of change.

The derivative therefore plays a crucial role in understanding how populations affected by the flu will change over time.

Differentiation

Differentiation is the process of finding the derivative of a function. It breaks down complex functions into simpler parts, making it easier to analyze rates of change. When applied to a polynomial function, each term is differentiated individually using basic calculus rules.
In the flu infection model, let's consider the function \( p(t) = t^2 + 2t \). Differentiation helps us find how each component of this function contributes to the overall rate of infection change.

• Start by applying the power rule for differentiation to the term \( t^2 \), giving you \( 2t \).
• Then differentiate the linear term \( 2t \), resulting in \( 2 \).

Hence, the derivative, denoted as \( p'(t) = 2t + 2 \), represents the instantaneous rate of change of the percentage of infected people. Differentiation is thus essential for calculating such rates in epidemiology and various other fields in science and engineering.

Flu Infection Model

The flu infection model in this exercise is represented by a mathematical function that estimates how flu spreads over time within a population. The function provided, \( p(t) = t^2 + 2t \), shows how increasing time impacts the percentage of the population infected.
This model is designed for a small town and provides insights based on how the flu progresses from the start, where \( t \) is the number of days after the initial infection outbreak.

• It's essential to evaluate the function at various points to understand the infection trajectory.
• Using the derivative, we can pinpoint critical points in time, such as \( t=3 \), where the infection is at a crucial growth phase with an instantaneous rate of change of 8% per day.

This model simplifies a real-life phenomenon into a concise mathematical form, allowing for better forecasting and understanding of infection patterns. Understanding such models can significantly impact public health strategies and resource allocation during an outbreak.