MATH233 Unit 3 Individual Project 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 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)) Page 1 of 3 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)), 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\".) 2. Find the derivative of MIPS (t) with respect to t . Show your work details. 3. Choose a t -value between 10 and 26. Calculate the value of MIPS ' (t). Show your work details. 4. Interpret the meaning of the derivative value that you just calculated from part 3 in terms of the MIPS (t) function and this scenario. 5. If the MIPS (t ) function is reasonably accurate, for what value of t increase in MIPS per year reach 6,000,000 will the rate of MIPS ? Approximately which year does that correspond to? Show your work details. 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? Show your work details. Page 2 of 3 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 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. 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-i52500k-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-i74770k-1156062/review Mathematics 4.0. (n.d.). Retrieved from the Microsoft Web site: http://microsoftmathematics.en.uptodown.com/ Page 3 of 3 here are my comments: RUBRIC GRADING: Purpose of Assignment/Content Development 17/20 *Proficient: Calculations attempted are successful and sufficiently comprehensive to solve the problem. Highly effective and communicative interpretation of mathematical statements. Almost all calculations are successful, with clear and logical documentation of steps. Expertise in applying quantitative data in application to projects is evident. Critical Thinking 17/20 *Proficient: Model ability to transfer learning to application. Organization, Grammar, Presentation 10/10 *Exemplary: English grammatical syntax and spelling are correct. Algebra syntax and notations are all correct. Quantitative Literacy 38/50 *Proficient: Highly effective presentation of analysis, assumptions, mathematical processes, findings and conclusions based on reasoning. 82/100 MATH233-Unit 3 IP Questions' Point-values: Question 1 2 3 4 5 6 7 Total Point-value 20 15 10 10 15 10 20 100 1. MATH233 Unit 3 Individual Project NAME You earned 16 13 8 10 8 7 20 82 This This graph does not show the salient characteristics of this function. Vertical values should go from about -1000000 to 10000000 to see the graph as it applies to this scenario. The table values for the MIPS go up to at least about 60000 so these vertical values should be included in the graph. MIPS ( t )=0.1121.405 2. 1.14 t +9.12 d ( MIPS ) =0.1121.141.4051.14 t +9.12ln ( 1.405 ) dt > d ( MIPS ) =0. 0 4341.405 1.14 t+ 9.12 dt Be careful with reporting decimal fractions. 3. t value chosen =15 1.1415 +9.12 thus, MIPS'(15)= 0. 0 4341.405 (inserting t=15 in equation for derivative of MIPS) MIPS'(15) = 0.04341.405 26.22 = 340.3095 Incorrect; arithmetic errors; MIPS'(15) = 323.268 4. The value calculated in part(3) is overall rate of change of instructions that a processor can process. The relatively high value of rate at t=15 suggests that as years increase the overall rate change of increase in MIPS is increasing at very high rate. And the rate in year t= 15 is 340.3095 Would be okay with correct values for MIPS'(15) 5. As per question rate of increase, d ( MIPS ) =6000000 dt d ( MIPS ) =0.4341.4051.14t +9.12 =6000000 dt => => => ( 1.14 t+ 9.12 )ln ( 1.405 ) = 6000000 0.434 ( 1.14 t+ 9.12 )ln ( 1.405 ) = 0.434 6000000 ( 9.12 )1 1.14 1 ln ln (1.405 ) t= => t = 34.41 6000000 0.434 This corresponds to year = 1982+34.41 = 2016.41 meaning year 2016. Work details and answer are incorrect. The value of t should be 40.355 6. Lets assume the equation of tangent line at t= 15 is, in slope intercept form, y=mt + c where, m is slope and c is intercept, We have from question 3, slope at t= 15 , m = d ( MIPS ) =340.3095 dt 1.1415+ 9.12 =834.24 Thus, y= MIPS= 0.1121.405 Hence, 834.24=340.3095*15+c c = -4270.402 Hence, required equation of tangent is, Y = 340.3095*X-4270.402 From the intercept large negative intercept 'c' of tangent it is clear that the overall slope is very large and is changing very fast. For small value of t=15 MIPS intercept of given function is 4270.402,making slopes of fynction extremely steep. Tangent line equation is incorrect because the slope is incorrect. 7. CPU A: Year of Introduction:2011, Speed:83000 MIPS1 CPU B: Year of Introduction:2013, Speed: 133740 MIPS1 Theoretical calculations: CPU-A-> t = 2011-1982 = 29 CPU-B ->t = 2013-1982 = 31 For CPU-A: MIPS ( 29 )=0.1121.4051.1429+ 9.12=189760 MIPS For CPU-B: MIPS ( 31 t )=0.1121.405 1.1431+9.12 =412009.7 412010 MIPS So MIPS value of CPU-A calculated = 189760/83000 =2.28 times actual speed of processor MIPS value of CPU-B calculated = 412010/133740 =3.08 times actual speed of processor Hence the Moore's law is overstating the processor speeds for a particular year as per the equations. But the deviation is not very large and is multiples of single digit which is quite good. Okay Reference: 1. https://en.wikipedia.org/wiki/Instructions_per_second