Problem 30 - Applied Calculus

Question

Table 2.11 shows the number of Facebook subscribers, \(N\) in millions, worldwide at 3 -month intervals. \(^{27}\) (a) Calculate the average rate of change of \(N\) per month for the time intervals shown between March 2011 and March 2012 (b) What can you say about the sign of \(d^{2} N / d t^{2}\) during the period March 2011 - March \(2012 ?\) $$\begin{array}{c|r|r|r|r|r}\hline \text { Month, } t & \text { Mar } 2011 & \text { Jun } 2011 & \text { Sep} 2011 & \text { Dec } 2011 & \text { Mar } 2012 \\\\\hline N & 664.0 & 710.7 & 756.9 & 799.1 & 835.5 \\ \hline\end{array}$$

Step-by-Step Solution

Answer

Average monthly increase: 14.29 million, \(d^2N/dt^2 < 0\) indicates growth is slowing.

Step 1: Understand the Problem

We need to calculate the average rate of change of the number of Facebook subscribers, denoted by \( N \), per month from March 2011 to March 2012, which involves determining how \( N \) changes over each 3-month interval. Additionally, we need to analyze the behavior of \( d^2N/dt^2 \) based on these intervals.

Step 2: Calculate Total Change in Subscribers

Subtract the initial number of subscribers in March 2011 from the final number in March 2012: \[ \Delta N = N_{\text{Mar 2012}} - N_{\text{Mar 2011}} = 835.5 - 664.0 = 171.5 \] This is the total change in subscribers over the year.

Step 3: Calculate Total Time Duration in Months

There are 12 months between March 2011 and March 2012. The time period is from March 2011 (month 0) to March 2012 (month 12).

Step 4: Calculate Average Rate of Change per Month

Divide the total change in number of subscribers by the total duration in months to find the average monthly rate of change: \[ \text{Average Rate of Change} = \frac{\Delta N}{\Delta t} = \frac{171.5}{12} \approx 14.29 \] Therefore, the average rate of change is approximately 14.29 million subscribers per month.

Step 5: Sign of \(d^2N/dt^2\) - Determining Acceleration

To evaluate \(d^2N/dt^2\), analyze the differences in average rates over each quarter: \1. \( \Delta t_1 = \frac{710.7 - 664.0}{3} = 15.57 \) million per month for Mar-Jun.2. \( \Delta t_2 = \frac{756.9 - 710.7}{3} = 15.4 \) million per month for Jun-Sep.3. \( \Delta t_3 = \frac{799.1 - 756.9}{3} = 14.07 \) million per month for Sep-Dec.4. \( \Delta t_4 = \frac{835.5 - 799.1}{3} = 12.13 \) million per month for Dec-Mar.The rate of change decreases each quarter, indicating \(d^2 N / dt^2 < 0\), suggesting the rate of growth is slowing down.

Key Concepts

Facebook subscribers

Facebook has been a key social media platform, with its subscriber base often being used as a metric to study digital growth trends. From March 2011 to March 2012, a significant increase in global subscribers was observed. This exercise focuses on understanding how the number of subscribers, denoted as \( N \), changes over this period. By analyzing this data, students gain insight into real-world applications of mathematical concepts, such as the average rate of change. In simpler terms, this exercise looks at the growth of Facebook's user base over a year, showing how mathematical calculations can illustrate and analyze this trend. The primary task involved is to determine how Facebook's subscriber count changed monthly and whether this growth is accelerating or decelerating.

differential calculus

Differential calculus is an essential branch of mathematics that involves the study of rates at which quantities change. In our context, we use it to analyze the growth of Facebook subscribers over time. The first derivative, denoted \( \frac{dN}{dt} \), represents the rate of change of subscribers—a concept often reflected as the slope of a tangent to a curve in mathematical terms. Through this exercise, you calculate and analyze the average rate of change over the specified intervals. What differentiates this from simple arithmetic is the broader analysis, allowing you to understand not just static numbers but dynamic trends. By focusing on how the rate changes over each 3-month interval, we can employ differential calculus principles to understand the behavior of real-world data.

growth rate

The concept of a growth rate is pivotal in many fields, including business, economics, and biology. Here, it benefits students to understand how the growth rate of Facebook subscribers can be quantified. The growth rate is essentially the speed at which the subscriber number increases over time. To determine it, one would use the average rate of change. The growth rate formula in this context is: \[\text{Growth Rate} = \frac{\Delta N}{\Delta t} \]Where \( \Delta N \) is the change in the number of subscribers, and \( \Delta t \) is the respective time interval. With an average rate of 14.29 million subscribers per month, this calculation gives us a concrete measure of Facebook's expansion over the year.

acceleration analysis

In addition to understanding the rate of change, acceleration analysis provides insight into whether this rate is increasing or decreasing over time. Technically, this is the second derivative, \( \frac{d^2N}{dt^2} \). It informs us whether Facebook's subscriber numbers are accelerating or decelerating.From March 2011 to March 2012, the rate of subscriber increase slows, as evidenced by progressively smaller quarterly growth rates. This deceleration can be important for anticipating future trends and making informed decisions. By calculating the rate changes between intervals, students see an example of how acceleration analysis helps assess the sustainability and potential saturation of growth in practical scenarios.