Problem 49 - A Graphical Approach to Precalculus with Limits

Question

The number of monthly active Whats App users \(W\) was 200 million in 2012 and grew linearly to 700 million by 2015 . It then continued to grow linearly to 1000 million, or 1 billion, by 2018 . (Source: Statista.) (a) Write a formula for a piece wise-linear function \(W(x)\) that models these data, where \(x\) represents the year. (b) Sketch a graph of \(y=W(x) .\) Is \(W\) a continuous function on the interval \([2012,2018] ?\) (c) Interpret the rates of change in \(W\).

Step-by-Step Solution

Answer

The piecewise function is continuous on [2012, 2018], with rate changes of 166.67M/year (2012-2015) and 100M/year (2015-2018).

Step 1: Understanding the Problem

We have to write a piecewise linear function for the growth of WhatsApp users and analyze its rate of change. The data points provided are: - In 2012, users were 200 million. - By 2015, the users grew to 700 million. - By 2018, users reached 1000 million (1 billion). We are to interpret the linear growth from 2012 to 2015, then from 2015 to 2018.

Step 2: Determine the Linear Equation from 2012 to 2015

For the first part (2012 to 2015), calculate the slope of the line connecting the points (2012, 200) and (2015, 700). The formula for slope is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Where \( y_2 = 700 \), \( y_1 = 200 \), \( x_2 = 2015 \), and \( x_1 = 2012 \). Substitute these in to get the slope:\[ m = \frac{700 - 200}{2015 - 2012} = \frac{500}{3} \approx 166.67 \]

Step 3: Write Equation for 2012 to 2015

Using the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \), substitute:- \( m = 166.67 \),- \( x_1 = 2012 \),- \( y_1 = 200 \) to find y-intercept.Thus, the equation for \( W(x) \) from 2012 to 2015 is: \[ W(x) = 166.67(x - 2012) + 200 \] which simplifies to \[ W(x) = 166.67x - 326733.34 \]

Step 4: Determine the Linear Equation from 2015 to 2018

Calculate the slope for the second part (2015 to 2018) using points (2015, 700) and (2018, 1000).\[ m = \frac{1000 - 700}{2018 - 2015} = \frac{300}{3} = 100 \]Using point-slope form, \[ y - 700 = 100(x - 2015) \]Thus, the equation for \( W(x) \) from 2015 to 2018 is:\[ W(x) = 100x - 2001900 \]

Step 5: Write the Piecewise Function

Combine the linear equations for both segments into a piecewise function:\[ W(x) = \begin{cases} 166.67x - 326733.34 & \text{for } 2012 \leq x < 2015 \ 100x - 2001900 & \text{for } 2015 \leq x \leq 2018 \end{cases} \]

Step 6: Sketch the Graph

Plot the function segments: - From 2012 to 2015, a line passing through (2012, 200) to (2015, 700). - From 2015 to 2018, a line from (2015, 700) to (2018, 1000). The graph is continuous across these intervals with no breaks at any of the data points (2012, 2015, 2018).

Step 7: Describe the Continuity

The function \( W(x) \) is continuous across the interval \([2012, 2018]\) because there are no jumps or gaps between the calculated points from the two linear segments.

Step 8: Interpret the Rates of Change

The rate of change from 2012 to 2015 is about 166.67 million users per year, reflecting a rapid growth phase. From 2015 to 2018, the rate of change is 100 million users per year, indicating a more stable growth rate.

Key Concepts

Rate of Change

In mathematics, the rate of change refers to how a quantity changes with respect to another variable. When analyzing functions, particularly piecewise linear functions like in this exercise, it reflects how quickly or slowly the dependent variable (number of WhatsApp users, in millions) changes relative to the independent variable (time in years).

• Linear Growth (2012-2015): The rate of change here is calculated as 166.67 million more users each year. This rapid increase signifies a significant growth period for WhatsApp during these years.


• Linear Growth (2015-2018): During this period, the growth rate decreased to 100 million users per year. This shows that while still growing, the rate of new user acquisition slowed down compared to earlier years.

Understanding the rate of change is essential as it helps determine the speed and acceleration of growth, which can have strategic implications for a company like WhatsApp.

Slope Calculation

The slope of a line is a measurement of its steepness, often denoted by the letter "m." It is crucial for determining the angle of the line in relation to the x-axis. The slope formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula determines the rate of change between any two points on the line. In the exercise, we calculated the slope to find the linear relationship between the number of users and the years.

• From 2012 to 2015: The points were 2012 (year), 200 (million users) and 2015, 700. The slope was calculated as 166.67, implying a steep rise in users.


• From 2015 to 2018: Using the points 2015, 700 and 2018, 1000, the slope was 100. A less steep line compared to the previous segment, indicating slower growth.

By calculating these slopes, we understand how fast the number of users changes over the years.

Graph Continuity

Graph continuity refers to a graph that is unbroken and smooth over an interval, where there are no gaps or jumps. In piecewise linear functions, checking for continuity involves ensuring that the end point of one segment of the graph connects to the initial point of the subsequent segment.

• In the given context, the function \( W(x) \) is piecewise, described by different linear equations for separate time intervals: 2012 to 2015 and 2015 to 2018.


• To check continuity between segments, we analyze the function at the break point, here 2015. Both the value of the function just before (approaching from the left) and just after (approaching from the right) 2015 are 700 million users.

Hence, the function meets the conditions for continuity over the interval \([2012, 2018]\), signifying a smooth transition without any jumps from one segment to the other.

Function Graphing

Function graphing involves plotting a mathematical relationship onto a coordinate system to visually interpret how the output (y-axis) changes with input (x-axis). With piecewise functions, this often involves plotting separate linear segments defined over different intervals.

• Step 1: Take the linear segments calculated: one from 2012 to 2015 and another from 2015 to 2018.


• Step 2: Plot each segment accordingly on a graph, ensuring that the endpoints meet precisely at the transition years (such as 2015 in this case), maintaining continuity.

Graphing allows us to quickly assess the trends in data over time; in this scenario, illustrating WhatsApp's user growth pattern effectively. Visual aids such as graphs make it easier to understand changes and compare growth over different periods.