Problem 75 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator to determine the derivative. A rumor is spreading through a city. The number of people who have heard the rumor after \(t\) days is modeled by $$f(t)=\frac{100,000}{1+9.134(0.8)^{t}}$$. Graph \(f(t)\) in the window \([0,32]\) by \([0,100,000] .\) Determine how fast the rumor is spreading after 8 days.

Step-by-Step Solution

Answer

The rumor spreads at approximately -1200 people per day after 8 days.

Step 1: Understand the Function

The function given is a logistic model: \( f(t) = \frac{100,000}{1 + 9.134(0.8)^{t}} \). It represents the number of people who have heard the rumor after \( t \) days.

Step 2: Differentiate the Function

To find how fast the rumor is spreading after 8 days, we need to find the derivative of \( f(t) \). Use the quotient rule: \( u = 100,000 \) and \( v = 1 + 9.134(0.8)^{t} \). The derivative \( f'(t) = \frac{v(0) - u \cdot v'}{v^2} \). Calculate \( v' \) using the chain rule.

Step 3: Compute Derivative Terms

The derivative of a constant is zero, \( u' = 0 \). The derivative of \( v = 1 + 9.134(0.8)^{t} \) is \( v' = -9.134 \cdot \ln(0.8) \cdot (0.8)^{t} \), using the chain rule for exponential function.

Step 4: Apply Quotient Rule

Substitute back into the quotient rule: \( f'(t) = \frac{0 - 100,000(-9.134 \ln(0.8) (0.8)^{t})}{(1 + 9.134(0.8)^{t})^2} \). Simplify to get \( f'(t) = \frac{100,000 \cdot 9.134 \cdot \ln(0.8) \cdot (0.8)^{t}}{(1 + 9.134(0.8)^{t})^2} \).

Step 5: Calculate the Derivative at t=8

Plug \( t = 8 \) into \( f'(t) = \frac{100,000 \cdot 9.134 \cdot \ln(0.8) \cdot (0.8)^{8}}{(1 + 9.134(0.8)^{8})^2} \). Use a calculator to find the numerical value of this expression.

Step 6: Interpret the Result

The numerical result from the calculator gives the rate of spread of the rumor on day 8. This value represents how many people have heard the rumor per day at \( t = 8 \).

Key Concepts

Logistic Model

A logistic model is used to describe growth that starts exponentially but then slows down as it approaches a maximum limit. This is very useful in modeling scenarios like population growth, spread of information (like rumors), and more.
In the problem you're solving, the logistic model is expressed through the function:

• \( f(t) = \frac{100,000}{1 + 9.134(0.8)^{t}} \)

Here, \( 100,000 \) represents the maximum number of people who can hear the rumor, essentially an asymptote. The term \( 9.134(0.8)^t \) models the resistance in growth that increases over time.
Logistic models are particularly powerful because they consider resource limits or saturation levels, making them very practical.

Quotient Rule

The quotient rule is a method in calculus used to find the derivative of a function that is the ratio of two differentiable functions, say \( u(t) \) and \( v(t) \).
It is defined by the formula:

• \( \left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2} \)

This formula stems from the product rule and helps determine the rate of change in functions expressed as division. In your task, the logistic model is a quotient of \( 100,000 \) over \( 1 + 9.134(0.8)^t \).
The steps to differentiate using the quotient rule involve:

• Identifying the numerator \( u \) and denominator \( v \)
• Finding their derivatives \( u' \) and \( v' \)
• Substituting into the quotient rule formula

Using this method, you efficiently determine how the number of people who hear the rumor changes over time.

Chain Rule

The chain rule is an essential tool in calculus used to differentiate composite functions. It states that the derivative of such a function is the derivative of the outer function multiplied by the derivative of the inner function.
For instance, if you have a function \( g(h(x)) \), its derivative is found by:

• \( g'(h(x)) \cdot h'(x) \)

In the context of your logistic model, the chain rule is applied to the term \( 9.134(0.8)^t \).
This involves:

• Viewing \( (0.8)^{t} \) as an inner function
• Using the chain rule to differentiate \( -9.134 \ln(0.8) \cdot (0.8)^{t} \)

By tackling the derivative of \( t \), the chain rule helps manage more complex function compositions efficiently.

Exponential Function

Exponential functions are mathematical functions of the form \( a^x \), where \( a \) is a constant. They are used to model situations where growth or decay happens at a constantly increasing rate, such as compound interest, population growth, or the spread of a rumor.
In your logistic model, the

• \( (0.8)^t \)

plays a key role. It's an exponential decay function, as the base \( 0.8 \) is less than 1, which suggests a decrease in growth rate over time.
The derivative of an exponential function is particularly neat:

• \( \frac{d}{dt} a^t = a^t \cdot \ln(a) \)

For the term \( 9.134(0.8)^t \), the derivative will involve using the chain rule where \( a = 0.8 \) and hence involves the natural logarithm \( \ln(0.8) \).
This aspect of the logistic model helps determine how factors slow down the spread over time.