Math 233 Unit 2 Individual Project A computer virus is a malicious program that can cause high levels of destruction to your computer or your mobile device. Even with an antivirus program installed, infections can still spread very rapidly as it is restricted to detecting infections, based on information on previous virus code. A team of experts wants to study a recently discovered computer virus by building a mathematical model that predicts the spread of infection, even with an antivirus installed, as a function of time. 1. Set up a function for the spread of this new computer virus by choosing a value for A, based on the first letter of your last name, from the table below. (A does not necessarily have to be a whole number). In your equation, S(t) is the number of machines projected to be infected and t is the time, in days. S ( t )=9 t 3+ A t 2 First letter of your last name Mathematical Model 3 Possible values for A 2 A-F S ( t )=7 t + A t , 0 t 12 101-120 G-L S ( t )=9 t 3+ A t 2 , 121-140 M-R S ( t )=11 t 3+ A t 2 , 0 t 12 141-160 S-Z S ( t )=13 t 3+ A t 2 ,0 t 12 161-180 S ( t )=9 t 3+ A t 2 , 0 t 12 3 9 t +121 t 0 t 12 2 [A=121] Therefore, the number of machines projected to be infected in t number of days is expressed as S ( t )=9 t 3+ 121t 2 2. Using the function that you have created from #1, find the value of S(9). 3 S ( 9 )=9 ( 9 ) +121 ( 9 ) 2 = -9(729) + 121(81) Page 1 of 4 = -6561 + 9801 = 3240 Therefore the value of S(9) = 3240 3. Find S ' (t ) , the derivative of S(t). Simplify your answer. Using the power rule d n ( x ) =n x n 1 as dt S ' t= d (9 t 3 +121 t2 ) dt 9 d 3 ( t ) +121 d ( t 2 ) dt dt 9 ( 3t 2 ) +121 ( 2t ) 2 27 t +242 t ' 2 Therefore the derivative of S(t) is S ( t )=27 t +242 t ' 2 4. Find the value of S '(9) . Substitute t with 9 in the function S ( t )=27 t +242 t ' 2 S ( 9 )=27 ( 9 ) +242 ( 9 ) = -27(81) + 2178 = -2187 + 2178 = -9 Therefore the value of S'(9) = -9 5. Interpret the meaning of S(9) and S ' (9) . Page 2 of 4 S(9) represents that the number of infected systems to reach 3240 machines in 9 days. While the derivative of S'(9) represents the rate of change in the number of machines infected in -9? That would make sense if a remediation was mentioned but I can't make sense of this without some kind of abatement factored in. 6. Using the tangent line approximation method, find the projected number of machines that will be infected after 10 days. S (10 )=S ( 9 ) + S' ( 9 ) (109) 3240+9 (1 ) = 3231 (Therefore the total number of machines infected after 10 days will be 3231 7. Generate a graph for S(t) and S ' (t) . (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0; or, there are also online utilities such as this site and many others.) Insert the graph into your Word document that contains all of your work details and answers. Be sure to label and number the axes appropriately. (Note: Some graphing utilities require that the independent variable must be \"x\" instead of \"t\".) 8. After how many days is the virus projected to infect the maximum number of machines? What maximum number of machines is projected to be infected? 9. At what intervals is the function increasing and decreasing? What do you think is the reason why the number of infected machines will start to decrease at some point in the domain? References Desmos. (n.d.). Retrieved from https://www.desmos.com/ Graph 4.4.2. (n.d.). Retrieved from the Graph Web site: http://www.padowan.dk/ Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: http://microsoftmathematics.en.uptodown.com/ Page 3 of 4 Page 4 of 4