Problem 43 - Applied Calculus

Question

The number of active Facebook users hit 175 million at the end of February 2009 and 200 million \({ }^{14}\) at the end of April \(2009 .\) With \(t\) in months since the start of 2009 , let \(f(t)\) be the number of active users in millions. Estimate \(f(4)\) and \(f^{\prime}(4)\) and the relative rate of change of \(f\) at \(t=4\). Interpret your answers in terms of Facebook users.

Step-by-Step Solution

Answer

At t=4, f(4)=200; f'(4)=12.5; relative rate = 6.25% per month.

Step 1: Understand the Problem

We are given that Facebook had 175 million users at the end of February 2009 and 200 million users at the end of April 2009. We need to find an expression for the number of users, estimate the number of users at the end of April, the rate of change at that time, and the relative rate of change.

Step 2: Identify Initial Data Points

Translate the given information into values. At the end of February, which is 2 months since the start of 2009, the number of users is 175 million, so \( f(2) = 175 \). At the end of April, which is 4 months in, \( f(4) = 200 \).

Step 3: Estimate Linear Function

Assume a linear function for simplicity: \( f(t) = mt + b \). Use the two data points \( f(2) = 175 \) and \( f(4) = 200 \) to find the slope \( m \) and y-intercept \( b \).

Step 4: Calculate Slope

Slope \( m \) is the change in the number of users over time: \[ m = \frac{f(4) - f(2)}{4 - 2} = \frac{200 - 175}{4 - 2} = \frac{25}{2} = 12.5 \]

Step 5: Find Y-intercept

Use one of the data points to solve for \( b \). Using \( f(2) = 175 \): \[ 175 = 2 \cdot 12.5 + b \] \[ 175 = 25 + b \] \[ b = 150 \] So, the equation becomes \( f(t) = 12.5t + 150 \).

Step 6: Estimate \(f(4)\)

Calculate \( f(4) \) using the linear function: \[ f(4) = 12.5 \cdot 4 + 150 = 50 + 150 = 200 \]

Step 7: Estimate \(f'(4)\)

The derivative \( f'(t) \) represents the rate of change, or slope \( m = 12.5 \). This is the rate of new user growth per month at \( t = 4 \).

Step 8: Calculate Relative Rate of Change

Relative rate of change is given by \( \frac{f'(t)}{f(t)} \). At \( t = 4 \), this is \[ \frac{12.5}{200} = 0.0625 \] or 6.25% per month.

Step 9: Interpret the Results

At the end of April 2009, Facebook had 200 million active users. The rate of increase in users was 12.5 million per month, which means the user base was growing by 6.25% per month at that time.

Key Concepts

Rate of Change

The rate of change is a concept that helps us understand how one quantity changes in relation to another over time. In our exercise, we're observing how the number of Facebook users increases as time progresses.

To grasp this better, consider the following:

• The rate of change is calculated as the difference in values divided by the difference in time. This gives us what is known as the 'slope' in the context of linear functions.
• For the exercise's data points, at 2 months there were 175 million users and at 4 months, 200 million users. The rate of change is calculated as \(m = \frac{200 - 175}{4 - 2} = 12.5\).
• This tells us that the number of users is increasing by 12.5 million per month.

Understanding the rate of change can provide insight into growth trends and is particularly useful in predicting future changes.

Linear Functions

Linear functions are mathematical constructs used to model relationships between two variables with a constant rate of change. In practical terms, this means the graph of the function forms a straight line.

For our exercise, the linear function is used to model the growth of Facebook users over time:

• The function is defined as \(f(t) = mt + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
• Using the data points, we determined \(m = 12.5\) and \(b = 150\). This translates the equation to \(f(t) = 12.5t + 150\).
• This equation lets us estimate user growth for any month. For month 4, \(f(4) = 200\), confirming our data point.

This function simplifies complex, real-world data into a manageable form, making predictions and interpretations significantly more straightforward.

Relative Rate of Change

The relative rate of change is a way to compare the rate of change of a function to the function's value at a certain point. It provides a percentage that describes how significant the change is relative to the current amount.

In this context, the relative rate of change is calculated as the ratio of the derivative of the function to the function's value:

• Using the exercise, at \(t = 4\), the derivative, \(f'(4)=12.5\), and \(f(4)=200\).
• Thus, the relative rate of change is given by \(\frac{f'(4)}{f(4)} = \frac{12.5}{200} = 0.0625\).
• This result means that the Facebook user base was growing at approximately 6.25% per month at that time.

The relative rate of change gives a normalized measure of growth, helping to decide if a change is significant in comparison to the current size or amount.