MATH233 Unit 3 Individual Project 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 0.292 1978 4 0.333 1979 5 0.754 1982 8 1.28 5 1985 11 2.156 1989 15 8.77 1992 18 25.68 1994 20 1889 1996 22 54110 1999 25 2,06411 2003 29 9,72612 2006 32 27,07913 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?
TO BE RE-WRITTEN FROM THE SCRATCH
MATH 233 Unit 3 Individual Project
Generating a graph for the function MIPS(t)=(0.112)(〖1.405〗^1.1395t)
Finding the derivative of MIPS(t) with respect to t
MIPS(t)=(0.112)(〖1.405〗^1.1395t)
Taking the constant out (af)’ = a.f’
Applying the exponent rule (ab=ebln(a))
Applying the chain rule()
Let 1.1395t ln(1.405) = u
Simplifying
Calculating the value of MIPS\'(t) for t= 25
Interpreting for meaning of above derivative
The first derivative of a function can be considered as the rate of change. For instance, f’ (x) is the rate of change of the function at x. It follows therefore that the is representative of the rate of increase of MIPS at 25 years from 1974. The slope of the tangent is another major interpretation of the derivative.
Calculating for the value of t that the rate of increase in MIPS per year will be 1,000,000 MIPS
Finding the equation of the tangent line to the graph of MIPS (t) at that value t =25.
We know the slope to f(x) at x=a is f (a). The tangent line equation is given by
y= f(a) + f’(a) (x-a)
We first find the value of PIMS at t= 25 years and then find the slope of the tangent line, which is the derivative of PIMS evaluated at t =25.
We compute for the slope of PIMS (t)
Computing the slope
Finding the line with slope m =698.877 and passing thro…………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….. Exponential function that is closely related to Moore’s Law …………………………….
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