Problem 47 - Applied Calculus

Question

The number of active Facebook users hit 175 million at the end of February 2009 and 200 million \(^{17}\) 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 the end of April 2009, there were ~200 million users, increasing by 12.5 million/month, with a 6.25% growth rate.

Step 1: Determine the Function

Assuming linear growth from February to April 2009, we can use the data points (2, 175) and (4, 200) to determine the function for the number of users. Let's find the linear function \( f(t) = mt + b \) where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Calculate the Slope

The slope \( m \) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{200 - 175}{4 - 2} = \frac{25}{2} = 12.5 \] million users per month.

Step 3: Find the Y-intercept

Using the slope \( m = 12.5 \) and the point (2, 175), solve for \( b \): \[ 175 = 12.5 \times 2 + b \] \[ 175 = 25 + b \] \[ b = 150 \] The function becomes \( f(t) = 12.5t + 150 \).

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

Substitute \( t = 4 \) into the function to find \( f(4) \): \[ f(4) = 12.5 \times 4 + 150 = 50 + 150 = 200 \] million users.

Step 5: Calculate \( f'(t) \)

The derivative \( f'(t) \) represents the rate of change and is equal to the slope of the linear function, which is \( f'(t) = 12.5 \) million users per month.

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

Since \( f(t) \) is linear, \( f'(4) = 12.5 \) million users per month, indicating the rate of increase in active users at \( t = 4 \).

Step 7: Calculate the Relative Rate of Change

The relative rate of change at \( t = 4 \) is given by \( \frac{f'(4)}{f(4)} \). Thus: \[ \frac{f'(4)}{f(4)} = \frac{12.5}{200} = 0.0625 \] or 6.25% per month.

Step 8: Interpretation

At \( t = 4 \), which corresponds to the end of April 2009, there were approximately 200 million active Facebook users. The number of users was increasing at a rate of 12.5 million per month, with a relative rate of change of 6.25%.

Key Concepts

Linear Function

A linear function is a type of function that creates a straight line when graphed. It is described by the equation \( f(t) = mt + b \), where \( m \) is the slope, and \( b \) is the y-intercept.

In the context of the exercise, the number of active Facebook users over time is estimated using a linear function. From the given data points (2, 175) and (4, 200), we find the linear function that represents this trend.

To determine the function, we first calculate the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), which gives us 12.5 million users per month. Using this slope and one of the points, we solve for the y-intercept \( b \). The resulting function is \( f(t) = 12.5t + 150 \), which models the user's growth over months.

Rate of Change

The rate of change is a term that refers to how much a quantity changes with respect to a variable, often time. In linear functions, the rate of change is constant and is equivalent to the slope of the line.

In our Facebook user example, the rate of change, \( f'(t) \), is equal to the slope, which is 12.5. This tells us that the number of users is increasing by 12.5 million users per month throughout the examining period.

Such a consistent rate of change is an important property of linear functions, making it easy to anticipate and predict future values if the slope remains unchanged.

Relative Rate of Change

The relative rate of change is a way to express the rate of change in relation to the value of the function itself. It tells us how fast a function is changing compared to its size and is often expressed as a percentage.

To find the relative rate of change, we use the formula \( \frac{f'(4)}{f(4)} \). For the Facebook users example, this is calculated as \( \frac{12.5}{200} \), which equals 0.0625 or 6.25%.

This means that at \( t = 4 \), the number of Facebook users is increasing by about 6.25% per month.

• It provides a better perspective on growth relative to the total size of the group.
• Helps in understanding the scale of change in practical terms.

Understanding both absolute and relative rates of change is crucial for analyzing functions and their implications in real-world situations.