MAT135 Assignment 8 Written Component Due: Friday 1 April 2022, at 6pm (Toronto time) Submitting this assignment Due date and time: Friday 1 April 2022, at 6pm (Toronto time) Where to submit: On Crowdmark. You should receive an email with your personal linkto the assignment on Crowdmark. What to do: Write solutions tothe problems on blank paper, or write them digitally (on an iPad for example, but using your own handwriting). For each question, take a photo, scan, or save a le of your solutions, starting each question on a new page. Click on the link in the email from Crowdmark. and submit your solutions one question at a time. Notes 1. Deadline: Deadlines to submit assignments online are strict. Missing the deadline by even a few seconds will result in a grade of zero. Please leave plenty of time to upload your assignment! 2. Time zones: Times are given in Toronto time (EST or EDT depending on the time of year). If you are in a different time zone, Crowdmark will automatically work out the time that the assignment is due. If you are unsure, log in to Crowdmark and check the assignment deadline. 3. Upload: If you handwrite your assignment on paper, you can scan or photograph it to submit it. It is important that the scans or photos you use are clear and easy to read (in focus, not too dark, not too bright, etc.). The les you upload are what will be marked; if the grader cannot read them, you will not get full marks. It is also acceptable to handwrite your solutions using a tablet (like an iPad). You should not type your solutions using IATEX. If you use a tablet, export PDFs of your solutions rather than taking screenshots. 4. Format: Write the solution for each question on a separate page. Explain all your work, showing your steps as well as your reasoning. You should write in words what you are doing and why. Someone reading your solution should easily be able to understand what you have done. The grading schemes given below reect this standard. 5. Grading and feedback: The assignment has two questions and is out of 10 marks. Grading schemes are given after each problem. Your grade will depend on a combination of your answers, and your explanations. The graders may leave feed back for you on your assignment work. Please log in to Crowdmark to view the feedback. (We will announce when it is ready.) 6. Purpose: The purpose of the written assignments is to give you some practice in writing solutions to mathe matical problems, without any time pressure. You will receive feedback on your writing and on your solutions. You are encouraged to take this opportunity to write your solutions carefully and think about how to best present your reasoning. 7. Academic Integrity: The solutions that you submit to this assignment must be all your own work. By submit- ting this assignment you declare that: (a) Your solutions are all you own work, explained in your own words. (b) You have not copied any part of the assignment solutions from anyone or anywhere. (c) You have not let anyone else copy any part of your solutions to the assignment. Problems 1. You are in charge of modernizing a busy 20 kilometrelong stretch of highway near a large city. The speed limit on there is 90 km/h and residents have been complaining of excessive speeding on the highway, and the police haven't been able to keep up with the problem. Yourjob is to design some speed traps (described below) that can catch speeding drivers and automatically send them speeding tickets. There are existing metal structures like this one placed every 2 kilometres along this stretch of highway (11 of them in total). The city has a large number of smart cameras that can read the license plates of cars and note the exact times (to the second) the cars pass by the cameras. Your plan is to attach one of these cameras to each of the structures mentioned above. The data from the cameras is then transmitted to a computer system that records the exact time of day at which each car passes each camera in a central database. Your goal is to write a very simple program that will use this data to detect cars that exceed the speed limit, so that their owners can be sent speeding tickets. To avoid errors, you want to design the system to only assign speeding tickets to cars that go at least 10% faster than the speed limit. How should your program work? Describe how a computer can analyze the data collected by the cameras and be sure that a car was speeding at some point. (Your explanation should reference a theorem you've studied recently. You're not being asked to actually write computer code, but describe, mostly in words, how the computer would analyze the data above to nd speed- ers, and we can be sure of its conclusions.) Here is some sample data you can think about. This is the data collected about two cars (the column headers are their license plates) as they travel down this stretch of highway and pass all the cameras. One if these cars should get a speeding ticket and the other should not. You are not required to discuss this sample data in your solution. It's just to give you something to work with. ABCD 123 DCBA 777 Cam 1 20220325 12:45:01 20220323 21:03:26 Cam 2 2022-03-2512:46:28 2022-03-23 21:04:46 Cam 3 2022-03-2512:47:48 2022-03-23 21:06:01 Cam 4 2022-03-2512:49:06 2022-03-23 21:07:12 Cam 5 20220325125036 202203723 21:08:27 Cam 6 2022-03-2512:52:08 20220323 21:09:43 Cam 7 2022-03-2512:53:43 2022-03-23 21:11:03 Cam 8 2022-03-2512:55:12 2022-03-23 21:12:22 Cam 9 2022-03-2512:56:37 2022-03-23 21:13:43 Cam 1O 2022-032512:58:00 202203-23 21:15:09 Cam 11 2022-03-2512:59:21 2022-03-23 21:16:39 Grading scheme for Q1 5 points: A mathematically correct solution that is written well enough to be published. A good standard to aim for is: another student in the class who hasn't thought about this problem could read your solution and completely understand all of your reasoning, understand exactly how the concepts you've recently learned this course apply to this problem, and not have any questions about why what you're saying is true. 4 points: A solution explained as well as a 5-point solution, but with at most one or two minor errors (like minor arithmetic and algebra errors). 3 points: A solution with at most one or two minor errors (as described above), and that is explained but not to the standard of a 5-point solution. 2 points: A good effort has been made toward a solution, but there are one or two major errors in the reasoning (e.g., applying a theorem incorrectly, or an answer that doesn't pass a "sanity check"). 1 point: Any relevant work not meeting the standards above. 0 points: No relevant work submitted. 2. Let f be the function dened by f(a:) : w"(a: + 3), where n is an integerthat is greater than or equal to 2. (a) Find all critical points of f. (b) Find the the intervals on which f is increasing or decreasing, (c) Use the information from the previous two parts to nd determine whether the critical points are local maxima, local minima, or neither. (Do not use any information about the second derivative; just the sign of the rst derivative.) Note: You may not assume that n is any particular integer. Your answers to some of the parts above will be different for different n's. For example, you may nd different behaviours when n is even or when n is odd. Please make sure to be as clear as possible in your explanation. You are encouraged to use your favourite graphing software to get a sense of what this function is doing for different n values, but your answer should justify all of its conclusions using techniques learned in the course. Correct answers without justication will receive no credit. Grading scheme for Q2 (a) 1 point: The correct critical points, with supporting work. 0.5 points: An answer that would be correct other than a minor arithmetic or algebra error, with supporting work. 0 points: Any submission not meeting the standards above. In particular, any answer without supporting work. 3 points: A clear and correct description of the intervals of increase, explained well, with all the necessary supporting work. 2.5 points: A solution that would get 3 points, except for at most one or two minor arithmetic or algebra errors. 2 points: A solution that would get 2.5 or 3 points, with supporting work, but lacking a good explanation. 1 point: A reasonable attempt at the problem, but which contains signicant conceptual errors (e.g., an answer that assumes n equals a specic number but does everything else well). 0 points: Any submission not meeting the standards above. In particular, any answer without supporting work. 1 point: A correct classication of the behaviour at all critical points based on the answers to part (b), with supporting work and explanation. It is possible to get this point even if the answers to part (b) are not correct. 0.5 points: An answerthat would get 1 point, except lacking a good explanation. 0 points: An answer that does not correctly classify the behaviour at all critical points based on the answers to part (b), or any answer without supporting work