MAT135H5 S - WINTER 2024 - WRITTEN ASSIGNMENT 2 Submission • You must submit your completed Written Assignment on Crowdmark by 6:00pm (EST) Friday March 1, 2024. You will be emailed a link from Crowdmark with information on how to submit your solutions. • Late penalty: Written assignments may be submitted up to 50 hours late at a penalty of 2% per hour late. • Consider submitting your assignment well before the deadline. • You do not need to print out this assignment; you may submit clear pictures/scans of your work on lined paper, or screenshots of your work. • You do not need to submit the cover page, or the grading scheme. • You must correctly orient/rotate and order your submission. • If you require additional space, please insert extra pages. Additional Instructions You must justify and support your solution to each question. You should use full sentences. Academic Integrity You are encouraged to work with your fellow students while working on questions from the written assignments. However, the writing of your assignment must be done without any assistance whatsoever. Do not post partial or complete solutions to Piazza. I affirm that this assignment represents entirely my own efforts. I confirm that: • I have not copied any portion of this work. • I have not allowed someone else in the course to copy this work. • This is the final version of my assignment and not a draft. • I understand the consequences of violating the University’s academic integrity policies as outlined in the Code of Behaviour on Academic Matters. By submitting solutions for grading I agree that the statements above are true. If I do not agree with the statements above, I will not submit my assignment and will consult the course coordinator (Mike Pawliuk) immediately. This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 Canada License. Original Author: Mike Pawliuk 1 2 MAT135H5 S - WINTER 2024 - WRITTEN ASSIGNMENT 2 Need help? This problem set is designed to make you think, and it contains problems you’ve never seen before. We expect you’ll need to come back to this assignment multiple times and try different approaches; we don’t expect you to solve everything in one sitting. It’s normal to get stuck! Every time you get stuck that means you’re about to learn something when you get unstuck. Look for those moments! If you’re stuck for more than a day or two, you may want to ask for help. Here are some places to do that: • Ask on Piazza. (If you want to post some of your work, please make it a private post.) • Office hours. See Quercus for times and locations. There are about 10 hours a week, and you can attend the office hours of any instructor or TA, not just the one for your LEC section. • Math Learning Center. DH 2027. Good luck, have fun! Grading Scheme This is the grading scheme that TAs will use when grading this assignment. You do not need to submit this page. Question 1. [6 points]. • The warm-ups do not need to be submitted. • Part 3. 1 point for a clear and correct computation. • Part 4. 1 point for a correct answer with a brief explanation. • Part 5. 1 point for a correct line with appropriate computations. • Part 6. 1 point for a correct line with appropriate computations. • Part 7. 1 point for a correct answer, and 1 point for the computation that justifies it. Question 2 [4 points]. • The warm-up does not need to be submitted. • Part 2. 1 point for a correct line (no justification needed). • Part 3. 1 point for finding the tangent line with justification. We prefer a solution with connecting words and sentences, but do not asses that for this part (that will be done in part 4). • Part 4. 1 point for finding the tangent line. 1 point for a complete explanation that uses words and sentences (where appropriate). See tutorial 6 as an example of the expectations. MAT135H5 S - WINTER 2024 - WRITTEN ASSIGNMENT 2 3 In both questions you will explore the function C x where C is a positive parameter. By the end of these questions you will have shown a result known to ancient mathematicians, and you will be better prepared to help a friend move a couch through a tight hallway. H(x) = Throughout this assignment we will use the notation Ra (where a is a positive real number) to represent the rectangle whose bottom left corner is the origin (0, 0) and whose top right corner is the point (a, H(a)). For example, R3 is the rectangle whose bottom left corner is the origin (0, 0) and whose top right corner is the point (3, C/3). You will be using and adding to this Desmos graph: https://www.desmos.com/calculator/0pdnaxk5gx Question 1. (1) Warm-up. Play around with the function at the website above. Choose many different values for the parameters a, b, C. [You do not need to submit anything for this part.] (2) Warm-up. Compute the area of the rectangle R3 . [You do not need to submit anything for this part.] (3) Let a be a positive real number. Compute the area of the rectangle Ra . (4) Let 0 < a < b be any two positive real numbers, and let Ra , Rb be the rectangles described above. How do the areas of Ra and Rb compare to each other? (5) Find the equation of the line that connects the top left corner of Ra with the bottom right corner of Rb . Call this line L(x) and include it in your desmos graph. 4 MAT135H5 S - WINTER 2024 - WRITTEN ASSIGNMENT 2 (6) Find the equation of the line that connects the top right corner of Ra with the top right corner of Rb . Call this line U (x) and include it in your desmos graph. (7) What do you notice about the slopes of L(x) and U (x)? How do these slopes compare to each other? Compute the two slopes to verify your claim. MAT135H5 S - WINTER 2024 - WRITTEN ASSIGNMENT 2 5 In Written Assignment 3 you will use this model again to optimize certain things, so when you’re finished, keep the link to your Desmsos graph. Question 2. This question uses the same notation as Question 1. You should complete that question first before you attempt this one. (1) Warm-up. Play around with the function H(x) at the website above. Find many different values for the parameters a, b, C so that the line L(x) (defined in Q1.5) is tangent to H(x). [You do not need to submit anything for this part.] (2) Let C = 2. By experimenting with different choices of a, b, find the tangent line of H(x) at x = 2. [No justification is needed] (3) Now let C be any positive real number. Find the tangent line of H(x) at x = 2. Call this line T2 (x). (Your slope and y-intercept will depend on the parameter C.) Include T2 (x) in your desmos graph, and check that it is tangent to H(x) for many different choices of C. 6 MAT135H5 S - WINTER 2024 - WRITTEN ASSIGNMENT 2 (4) Now, instead of just x = 2 (as in parts 2.2 and 2.3). Find the tangent line of H(x) for a general x = t. Call this line Tt (x). Include Tt (x) in your desmos graph with t as a parameter, and check that it is tangent to H(x) for many different choices of C. Note. Write up your solution with complete sentences and connecting words (not just naked algebra) as in Tutorial 6. Reflection. Once you have completed Q2.4 you will have constructed the tangent line for the hyperbola H at all possible x values. Way to go!
