This assignment features an exponential function that is closely related to Moore's Law, which states that the numbers of transistors per square inch in Central Processing Unit (CPU) chips will double every 2 years. This law was named after Dr. Gordon Moore.
Table 1 below shows selected CPUs from this leading processor company introduced between the years 1974 and 2008 in relation to their corresponding processor speeds of Million Instructions per Second (MIPS).
Table 1: Selected CPUs with corresponding speed ratings in MIPS.
|
Processor
|
Year
|
t Years After 1974 When Introduced
|
Million Instructions per Second (MIPS)
|
|
1
|
1974
|
0
|
0.29
|
|
2
|
1978
|
4
|
0.33
|
|
3
|
1979
|
5
|
0.75
|
|
4
|
1982
|
8
|
1.28
|
|
5
|
1985
|
11
|
2.15
|
|
6
|
1989
|
15
|
8.7
|
|
7
|
1992
|
18
|
25.6
|
|
8
|
1994
|
20
|
188
|
|
9
|
1996
|
22
|
541
|
|
10
|
1999
|
25
|
2,064
|
|
11
|
2003
|
29
|
9,726
|
|
12
|
2006
|
32
|
27,079
|
|
13
|
2008
|
34
|
59,455
|
(Instructions per second, n.d.)
This information can be mathematically modeled by the exponential function:
MIPS(t)=(0.112)(?1.405?^1.1395t)
Be sure to show your work details for all calculations and explain in detail how the answers were determined for critical thinking questions. Round all value answers to three decimals.
1. Generate a graph of this function, MIPS(t)=(0.112)(?1.405?^1.1395t), t years after 1974, using Excel or another graphing utility. (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".)
2. Find the derivative of MIPS(t) with respect to t.
3. Choose a t-value between 20 and 34. Calculate the value of MIPS'(t).
4. Interpret the meaning of the derivative value that you just calculated from part 3 in terms of the MIPS(t) function.
5. If the MIPS(t) function is reasonably accurate, for what value of t will the rate of increase in MIPS per year reach 1,000,000 MIPS? Approximately which year does that correspond to?
6. For the t-value you chose in part 3 above, find the equation of the tangent line to the graph of MIPS(t) at that value of t. What information about the MIPS(t) function can be obtained from the tangent line?
7. Using Web or Library resources research to find the years of introduction and the processor speeds for both the CPU A and the CPU B. Be sure to cite your creditable resources for these answers. Convert the years introduced to correct values of t and determine how well the MIPS(t) function predicts when these CPUs' processor speeds occurred.
8. What explanation can you give for the differences observed in part 7?
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/
Instructions per second. (n.d.). Wikipedia. Retrieved from http://en.wikipedia.org/wiki/Instructions_per_second
Intel. (2008). Mircoprocessor quick reference guide. Retrieved from http://www.intel.com/pressroom/kits/quickrefyr.htm
Laird, J. (2011, January 3). Intel Core i5-2500K review. Techradar. Retrieved from http://www.techradar.com/us/reviews/pc-mac/pc-components/processors/intel-core-i5-2500k-917570/review
Laird, J. (2013, June 3). Intel Core i7-4770K review. Techradar. Retrieved from http://www.techradar.com/us/reviews/pc-mac/pc-components/processors/intel-core-i7-4770k-1156062/review
Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: http://microsoft-mathematics.en.uptodown.com/