math 1720 final.pdf

UNIVERSITY OF WINDSOR DEPARTMENT OF MATHEMATICS AND STATISTICS Di ff erential Calculus MATH 1720/1760 Final Exam Wednesday, December 15, 2021, 12:00 pm - 2:30 pm Last name : First name : Student No. : Instructions : Download this exam to avoid any problem with internet while you are solving it. You can print this exam and write your solutions on it, or you can work on your own paper and put just the question number next to your solutions. In both cases make sure you have written your name and ID on your papers. Try to keep the questions in the same order when you scan your solutions. This exam has 10 questions. You have 150 minutes (2 hours and 30 minutes). Read carefully and answer all questions. Show all your work to receive full credit . Give exact answers not the decimal approximations unless a question asks to give answer in decimal approximations. Only non-programmable and non-graphic calculators are allowed. You must stop writing at 2:30 pm and must upload a single pdf file (not a Zip file) with your solved exam by 3:00 pm on Blackboard . Make sure the pdf is clear and your solutions can be seen clearly to avoid any confusion during marking. If your exam cannot be read, it will not be marked. Better to use pen or dark pencil if possible. You can submit your exam solutions once (you can save draft and check that it is fine before you submit). Make sure to check your submission is submitted and not in progress (you should see a yellow icon in My Grades if it is submitted). You are responsible for your submission, this means it is your responsibility to make sure your submission is on time and accurate (the file is correct and includes all pages, pages on email will not be accepted). Late submissions will not be accepted . Code of Conduct: This open book examination must be written without collaboration. While writing the exam you must not communicate with any other person and you must not consult the internet or any material other than the course textbook, lectures, your course notes, assignments, and any material posted on the course Blackboard pages. If there is any evidence of academic misconduct, we will file an allegation of misconduct according to Bylaw 31. All rights reserved. This assessment is the intellectual property of the author and may not be repro- duced in any manner whatsoever without the express written permission of the author. Downloaded by ze md (mmhm7548@gmail.com)

2 1. (17) Evaluate the following limits: (a) lim x → 0 x + tan x 3 x + 4 x 2 (b) lim x → 7 - x 2 - 5 x - 14 x 2 - 49 (Do not use L’Hospital rule in part(b)) (c) lim x →∞ e - 3 sin x x 2 (d) lim x → 1 + sin( π x ) ln ( x - 1) Downloaded by ze md (mmhm7548@gmail.com)

3 2. (5) For what value(s) of the constant c is the function f continuous on ( -∞ , ∞ ) f ( x ) = c 2 - ( x + 1) c, if x > 1 (2 x + 1) c - x 2 - 5 , if x ≤ 1 3. (6) Use the definition of derivative to find the derivative of f ( x ) = √ 3 x - 7 Downloaded by ze md (mmhm7548@gmail.com)

4 4. (9) Consider the equation e x + 2 x = 0 (a) Use the Intermediate Value Theorem to show that the equation has at least one real root. (b) Use Rolle’s theorem to show that the equation has at most one real root. Downloaded by ze md (mmhm7548@gmail.com)

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