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The Internet is defined as a worldwide interconnection of individual networks operated by governments, industries, academia, and private parties (Internet World Stats, 2015). As the Internet grows and becomes more accessible to many people in the world, there is an increasing interest in forecasting its number of users as it helps improve the technology and its security.
In a recent study, researchers found that the rate of change of Internet users in the world (in millions) is modeled by the function
P^' (x)=0.24x^2+4.12x+25.49
where x is the number of years from 1994.
For the questions that require calculations, be sure to show your work details. If applicable, round your final answers to the nearest hundredth (2 digits after the decimal point) to receive full credit. For critical thinking questions, explain in detail how the answers were determined.
Using integration on P'(x), find the function P(x) that predicts the number of Internet users in the world x years after 1994. (Note: Since this is an indefinite integral, do not forget to add a constant of integration, C, in your final answer.) Show your work details.
Based on the first letter of your last name, calculate the value of C from #1 by choosing an ordered pair (x, P(x)) from the table below. Show your work details.
First letter of your last name x (years from 1994) P(x) (in millions)
A–B 6 361
C–D 7 513
E–F 8 587
G–H 9 719
I–J 10 817
K–L 11 1,018
M–N 12 1,093
O–P 13 1,319
Q–R 14 1,574
S–T 15 1,802
U–V 16 2,267
W–X 17 2,497
Y–Z 18 2,802
From your answers in #1 and #2, state the mathematical model P(x) (in millions) that will predict the number of Internet users in the world.
Using P(x), complete the following table. Show the work details for full credit.
Year x Predicted Number of Internet Users P(x) (in millions)
2016
2018
2020
Below is a graph of P'(x) over the interval [6,22]. Set up the definite integral that will calculate the area of the shaded region on the graph.
Calculate the area of the shaded region. Show your work details.
What information does the area give in terms of number of Internet users in this scenario? Would you estimate that this is a reasonable growth of Internet users from 2000 to 2016? Explain and justify your answer with critical thinking.

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