Problem 52 - Brief Applied Calculus

Question

BIOMEDICAL: Flu Epidemic The number of people newly infected on day \(t\) of a flu epidemic is \(f(t)=13 t^{2}-t^{3} \quad\) (for \(\left.0 \leq t \leq 13\right)\). Find the instantaneous rate of change of this number on: a. Day 5 and interpret your answer. b. Day 10 and interpret your answer.

Step-by-Step Solution

Answer

Day 5: 55 people/day (increasing). Day 10: -40 people/day (decreasing).

Step 1: Identify the Function

The given function represents the number of people newly infected on day \( t \) during a flu epidemic: \[ f(t) = 13t^2 - t^3 \] We need to find the instantaneous rate of change at specific days, which involves finding the derivative of this function.

Step 2: Differentiate the Function

To find the instantaneous rate of change, differentiate the function \( f(t) = 13t^2 - t^3 \) with respect to \( t \). Using the power rule, the derivative \( f'(t) \) is calculated as follows: \[ f'(t) = \frac{d}{dt}(13t^2) - \frac{d}{dt}(t^3) = 26t - 3t^2 \] Hence, the derivative of the function, \( f'(t) = 26t - 3t^2 \), represents the rate of change at any day \( t \).

Step 3: Calculate Instantaneous Rate of Change on Day 5

Substitute \( t = 5 \) into the derivative to find the rate of change on day 5:\[ f'(5) = 26(5) - 3(5)^2 = 130 - 75 = 55 \] Thus, on day 5, the instantaneous rate of change is 55 people per day. This means that the number of newly infected people is increasing at a rate of 55 people per day on day 5.

Step 4: Calculate Instantaneous Rate of Change on Day 10

Substitute \( t = 10 \) into the derivative to find the rate of change on day 10:\[ f'(10) = 26(10) - 3(10)^2 = 260 - 300 = -40 \] Thus, on day 10, the instantaneous rate of change is -40 people per day. This means that the number of newly infected people is decreasing at a rate of 40 people per day on day 10.

Key Concepts

Derivative

A derivative is a fundamental concept in calculus that measures how a function changes as its input changes. It provides the rate of change or the slope of the function at any given point. To find the derivative of a function, we essentially determine the function's behavior near a specific point by examining its tangent.

In the context of the problem, we are given a function that models the number of newly infected individuals during a flu epidemic:

• Function: \( f(t) = 13t^2 - t^3 \)

The derivative of this function, noted as \( f'(t) \), helps us understand how quickly or slowly the infection is spreading or decreasing on any specific day. By differentiating, we apply rules such as the power rule, leading to:

• Derivative: \( f'(t) = 26t - 3t^2 \)

This derivative function can then be used to calculate the instantaneous rate of change for any given day \( t \). Understanding the concept of derivatives is crucial as it allows insights into the dynamics of situations that change over time, like the spread of a disease.

Flu Epidemic Modeling

Flu epidemic modeling involves using mathematical functions to predict and analyze trends in the spread of influenza. In this exercise, the function \( f(t) = 13t^2 - t^3 \) models how many new infections occur as days pass.

Modeling epidemics is vital for public health planning and control strategies. By learning how the number of infections changes over time, officials can make informed decisions on interventions or resource allocation.

On days 5 and 10, substituting these values for \( t \) in the derivative \( f'(t) = 26t - 3t^2 \) reveals:

• Day 5: The rate of increase in new infections is 55 people per day.
• Day 10: The rate of change is negative, \(-40\) people per day, indicating a decrease in new infections.

Flu epidemic modeling provides critical data on how quickly an epidemic can expand or contract, potentially signaling when preventative actions are working or if further measures are needed. This approach uses calculus tools effectively to quantify real-world problems.

Calculus Applications

Calculus is not just about solving equations; it is about applying mathematical principles to solve real-world problems. One vital application of calculus is in the field of epidemic research, as seen in this flu epidemic problem.

By using derivatives, we can model scenarios and understand processes occurring in various time frames, not just momentary snapshots. This can inform how we respond to infectious disease outbreaks.

• Immediate applications include:
  • Predicting peak infection times.
  • Monitoring decreases in infection as interventions are applied.

The flu calculation exercise illustrates these principles in practice, showing not only when infections increase but also the point where they begin to decrease, as identified on days 5 and 10. Calculus is a versatile tool that translates complex theoretical concepts into practical insights, crucial for timely decision-making in dynamic situations.