derivatives of exponentials
This assignment features an exponential function that is closely related to Mooreâ€™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.
The following table shows the 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).
Save your time - order a paper!
Get your paper written from scratch within the tight deadline. Our service is a reliable solution to all your troubles. Place an order on any task and we will take care of it. You won’t have to worry about the quality and deadlines
Order Paper Now
Processor |
Year |
t Years After 1982 When Introduced |
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 |
Table: Selected CPUs With Corresponding Speed Ratings in MIPS (Adapted from â€œInstructions per Second,â€ 2018)
This information can be mathematically modeled by the following exponential function:
Note that 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 are at a minimum for this function.
Be sure to show all of your work details for all calculations and explain in detail how the answers for the critical thinking questions were determined. Round all value answers to 3 decimals.
1.Generate a graph of this functionâ€”that is, MIPS(t) = (0.112)(1.405^{1.14t + 9.12})â€” years after 1982 using Excel or another graphing utility. (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0, online utilities like this site, and many others.) Insert both the function and 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 that some graphing utilities require that the independent variable must be x instead of t.)
Find the derivative of with respect to . Show all of your work details.
2.Choose a value between 10 and 26. Calculate the value of Show all of your work details.
3.Interpret the meaning of the derivative value that you just calculated in terms of the function and this scenario.
4.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 all of your work details.
5.For the value that you chose in Question 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 all of your work details.
6.Using Web or library resources, research to find the years of introduction and the processor speeds for both CPU A and 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 CPUsâ€™ processor speeds by comparing the calculated values with the actual MIPS ratings of these two CPUs. Show all of your work details.
7.The interest compound formula is:
A is the amount of money at a certain period of time, P is the principal or the amount originally deposited, n is the number of time the interest is compounded per year, r is the interest rate, and t is the number of years.
A. Does this equation represent an exponential growth? Explain.
B. What does the derivative of this function when t = 5 represent?
C. Knowing that , how can you relate this information to the compound interest amount compounded at a large number of time and to the continuous interest function? Is this conclusion reasonable?
D. What other application of exponential functions can you locate in practice?