Question

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 1982 0 1.28

5 1985 3 2.15

6 1989 7 8.7

7 1992 10 25.6

8 1994 12 188

9 1996 14 541

10 1999 17 2,064

11 2003 21 9,726

12 2006 24 27,079

13 2008 26 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.

Generate a graph of this function, MIPS(t) = (0.112)(1.405^(1.14t+9.12)), t 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 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”.)

Find the derivative of MIPS(t) with respect to t. Show your work details.

Choose a t-value between 10 and 26. Calculate the value of MIPS'(t). Show your work details.

Interpret the meaning of the derivative value that you just calculated from part 3 in terms of the MIPS(t) function and this scenario.

If the MIPS(t) function is reasonably accurate, for what value of t will the rate of increase in MIPS per year reach 6,000,000 MIPS? Approximately which year does that correspond to? Show your work details.

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? Show your work details.

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 by subtracting 1982 from each year. Then, determine how well the MIPS(t) 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.

Answer 1

The graph of MIPS (t) is as shown below:

The derivative of MIPS (t) is given as

Let t = 25

Then MIPS'(t) =

The first derivative shows the slope or the rate of change of the curve at that particular value of t. So 47,118 is the slope, or the rate of increase of the MIPS (t) equation at t=25.

We have MIPS'(t) = 6,000,000

So,

Equation of the tangent line to the graph of MIPS(t) is the same as MIPS'(t)