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NAME (Required): _____________________________
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 1982 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 1982
When Introduced
Million Instructions
per Second (MIPS)
4
5
6
7
8
9
10
11
12
13
1982
1985
1989
1992
1994
1996
1999
2003
2006
2008
0
3
7
10
12
14
17
21
24
26
1.28
2.15
8.7
25.6
188
541
2,064
9,726
27,079
59,455
(Instructions per second, n.d.)
This information can be mathematically modeled by the exponential function:
MIPS(t) = (0.112)(1.405^(1.14t+9.12))
NOTE: This function is created as a “best fit” function for a table of empirical data and,
therefore, does not exactly match many (or any) of the data values in the table above. Rather, the
total cumulative differences from all of the data points is at a minimum for this function.
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^(1.14t+9.12)), years after
1982, 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
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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 ( ) with respect to . Show your work details.
3. Choose a -value between 10 and 26. Calculate the value of ′( ). Show your work
details.
4. Interpret the meaning of the derivative value that you just calculated from part 3 in terms
of the ( ) function and this scenario.
5. If the ( ) function is reasonably accurate, for what value of will the rate of
increase in MIPS per year reach 6,000,000 ? Approximately which year does that
correspond to? Show your work details.
6. For the -value you chose in part 3 above, find the equation of the tangent line to the
graph of ( ) at that value of . What information about the ( ) function can
be obtained from the tangent line? Show your work details.
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 by
subtracting 1982 from each year. Then, determine how well the ( ) function
predicts the forecast CPUs’ processor speeds by comparing the calculated values with the
actual MIPS ratings of these two CPUs. Show your work details.
References
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http://en.wikipedia.org/wiki/Instructions_per_second
http://www.intel.com/pressroom/kits/quickrefyr.htm
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